Daily Papers

The rapid scaling of language models (LMs) has resulted in unprecedented computational, memory, and energy requirements, making their training and deployment increasingly unsustainable. Quantization has emerged as an essential compression technique to reduce model size, alleviate memory bottlenecks, and accelerate inference. However, while uniform low-bit quantization (e.g., INT8, INT4) provides significant efficiency gains, it can degrade accuracy in sensitive components of transformer-based LMs. Mixed-precision quantization offers a promising alternative by selectively allocating precision across layers or within tensors to balance efficiency and accuracy. This survey provides a comprehensive overview of Mixed-Precision quantization frameworks for LMs (MXPLMs). We first review quantization fundamentals, including uniform and non-uniform quantizers, quantization granularity, and methods widely used in post-training quantization. We then categorize and compare recent MXPLM frameworks according to their bit allocation strategies and precision configurations across weights, activations, and key-value caches. A comparative analysis highlights differences in perplexity, zero-shot task performance, and deployment trade-offs. Furthermore, we contrast MXPLMs with earlier mixed-precision quantization methods for deep neural networks, identifying strategies that transfer and those that face challenges in the LM setting. Finally, we summarize open issues and future directions, including hardware-aware design, activation quantization, and scalable optimization methods for billion-parameter models. By consolidating recent advances, this work serves as a reference for understanding the current landscape and research prospects of mixed-precision quantization for large-scale language models.

Despite the recent advances in model compression techniques for deep neural networks, deploying such models on ultra-low-power embedded devices still proves challenging. In particular, quantization schemes for Gated Recurrent Units (GRU) are difficult to tune due to their dependence on an internal state, preventing them from fully benefiting from sub-8bit quantization. In this work, we propose a modular integer quantization scheme for GRUs where the bit width of each operator can be selected independently. We then employ Genetic Algorithms (GA) to explore the vast search space of possible bit widths, simultaneously optimising for model size and accuracy. We evaluate our methods on four different sequential tasks and demonstrate that mixed-precision solutions exceed homogeneous-precision ones in terms of Pareto efficiency. In our results, we achieve a model size reduction between 25% and 55% while maintaining an accuracy comparable with the 8-bit homogeneous equivalent.

Mixture-of-Experts (MoE) models face deployment challenges due to their large parameter counts and computational demands. We explore quantization for MoE models and highlight two key insights: 1) linear blocks exhibit varying quantization sensitivity, and 2) divergent expert activation frequencies create heterogeneous computational characteristics. Based on these observations, we introduce MxMoE, a mixed-precision optimization framework for MoE models that considers both algorithmic and system perspectives. MxMoE navigates the design space defined by parameter sensitivity, expert activation dynamics, and hardware resources to derive efficient mixed-precision configurations. Additionally, MxMoE automatically generates optimized mixed-precision GroupGEMM kernels, enabling parallel execution of GEMMs with different precisions. Evaluations show that MxMoE outperforms existing methods, achieving 2.4 lower Wikitext-2 perplexity than GPTQ at 2.25-bit and delivering up to 3.4x speedup over full precision, as well as up to 29.4% speedup over uniform quantization at equivalent accuracy with 5-bit weight-activation quantization. Our code is available at https://github.com/cat538/MxMoE.

Auto-regressive Large Language Models (LLMs) achieve strong performance on coding tasks, but incur high memory and inference costs. Diffusion-based language models (d-LLMs) offer bounded inference cost via iterative denoising, but their behavior under post-training quantization (PTQ) has been sparsely explored. We investigate the application and robustness of PTQ techniques, specifically GPTQ and a modified Hessian-Aware Quantization (HAWQ) algorithm, on a diffusion-based coding LLM (CoDA) and observe that these methods applied to CoDA exhibit greater robustness at low bitwidths compared to Qwen3-1.7B, its auto-regressive counterpart, under a standardized evaluation pipeline. We find that in our setup, CoDA exhibits greater robustness at low bitwidths (2-4 bits), with smaller accuracy degradation across HumanEval and MBPP benchmarks. Additionally, mixed-precision configurations derived from HAWQ provide smooth trade-offs across accuracy, latency, and memory. The results suggest that diffusion LLMs may offer advantages for efficient deployment due to more quantization-resilience.

Serving large-scale machine learning (ML) models efficiently and with low latency has become challenging owing to increasing model size and complexity. Quantizing models can simultaneously reduce memory and compute requirements, facilitating their widespread access. However, for large models not all layers are equally amenable to the same numerical precision and aggressive quantization can lead to unacceptable loss in model accuracy. One approach to prevent this accuracy degradation is mixed-precision quantization, which allows different tensors to be quantized to varying levels of numerical precision, leveraging the capabilities of modern hardware. Such mixed-precision quantiztaion can more effectively allocate numerical precision to different tensors `as needed' to preserve model accuracy while reducing footprint and compute latency. In this paper, we propose a method to efficiently determine quantization configurations of different tensors in ML models using post-training mixed precision quantization. We analyze three sensitivity metrics and evaluate them for guiding configuration search of two algorithms. We evaluate our method for computer vision and natural language processing and demonstrate latency reductions of up to 27.59% and 34.31% compared to the baseline 16-bit floating point model while guaranteeing no more than 1% accuracy degradation.

KV cache quantization can improve Large Language Models (LLMs) inference throughput and latency in long contexts and large batch-size scenarios while preserving LLMs effectiveness. However, current methods have three unsolved issues: overlooking layer-wise sensitivity to KV cache quantization, high overhead of online fine-grained decision-making, and low flexibility to different LLMs and constraints. Therefore, we theoretically analyze the inherent correlation of layer-wise transformer attention patterns to KV cache quantization errors and study why key cache is generally more important than value cache for quantization error reduction. We further propose a simple yet effective framework KVTuner to adaptively search for the optimal hardware-friendly layer-wise KV quantization precision pairs for coarse-grained KV cache with multi-objective optimization and directly utilize the offline searched configurations during online inference. To reduce the computational cost of offline calibration, we utilize the intra-layer KV precision pair pruning and inter-layer clustering to reduce the search space. Experimental results show that we can achieve nearly lossless 3.25-bit mixed precision KV cache quantization for LLMs like Llama-3.1-8B-Instruct and 4.0-bit for sensitive models like Qwen2.5-7B-Instruct on mathematical reasoning tasks. The maximum inference throughput can be improved by 21.25\% compared with KIVI-KV8 quantization over various context lengths. Our code and searched configurations are available at https://github.com/cmd2001/KVTuner.

Mixed-precision inference techniques reduce the memory and computational demands of Large Language Models (LLMs) by applying hybrid precision formats to model weights, activations, and KV caches. This work introduces mixed-precision LLM inference techniques that encompass (i) systematic memory and compute optimization across hierarchical storage and tensor core architectures, and (ii) comprehensive end-to-end mixed-precision optimization across diverse precision formats and hardware configurations. Our approach features two novel mixed-precision pipelines designed for optimal hardware utilization: a General Matrix Multiply (GEMM) pipeline that optimizes matrix operations through offline weight packing and online acceleration, and an attention pipeline that enables efficient attention computation with arbitrary Query, Key, and Value precision combinations. The key implementation of the pipelines includes (i) hardware-aware weight packing for automatic format optimization, (ii) adaptive head alignment for efficient attention computation, (iii) instruction-level parallelism for memory hierarchy exploitation, and (iv) KV memory loading pipeline for enhanced inference efficiency. We conduct comprehensive evaluations across 16 popular LLMs and 4 representative GPU architectures. Results demonstrate that our approach achieves up to 61% lower serving latency (30% on average) and up to 156% higher throughput (58% on average) in mixed-precision workloads compared to existing mixed-precision frameworks, establishing consistent performance improvements across all tested configurations and hardware types. This work is integrated into TurboMind, a high-performance inference engine of the LMDeploy project, which is open-sourced and publicly available at https://github.com/InternLM/lmdeploy.

To enable broader deployment of Large Language Models (LLMs), it is essential to identify the best-performing model under strict memory constraints. We present AMQ, Automated Mixed-Precision Weight-Only Quantization, a framework that assigns layer-wise quantization bit-widths to optimally balance model quality and memory usage. However, the combinatorial search space, with over 10^{100} possible configurations, makes conventional black-box optimization infeasible. AMQ overcomes this challenge through four key innovations:(1) search space pruning using prior knowledge to exclude unpromising configurations, (2) quantization proxy to bypass costly format conversions during search, (3) quality predictor to minimize evaluation overhead, and (4) iterative search-and-update strategy for fast and stable convergence. By integrating these components, AMQ efficiently explores the quality-efficiency landscape, reaching the Pareto frontier and yielding LLMs that are both compact and high-performing. Our code is available at https://github.com/dlwns147/amq.

Mixed-Precision Quantization~(MQ) can achieve a competitive accuracy-complexity trade-off for models. Conventional training-based search methods require time-consuming candidate training to search optimized per-layer bit-width configurations in MQ. Recently, some training-free approaches have presented various MQ proxies and significantly improve search efficiency. However, the correlation between these proxies and quantization accuracy is poorly understood. To address the gap, we first build the MQ-Bench-101, which involves different bit configurations and quantization results. Then, we observe that the existing training-free proxies perform weak correlations on the MQ-Bench-101. To efficiently seek superior proxies, we develop an automatic search of proxies framework for MQ via evolving algorithms. In particular, we devise an elaborate search space involving the existing proxies and perform an evolution search to discover the best correlated MQ proxy. We proposed a diversity-prompting selection strategy and compatibility screening protocol to avoid premature convergence and improve search efficiency. In this way, our Evolving proxies for Mixed-precision Quantization~(EMQ) framework allows the auto-generation of proxies without heavy tuning and expert knowledge. Extensive experiments on ImageNet with various ResNet and MobileNet families demonstrate that our EMQ obtains superior performance than state-of-the-art mixed-precision methods at a significantly reduced cost. The code will be released.

Efficiently serving neural network models with low latency is becoming more challenging due to increasing model complexity and parameter count. Model quantization offers a solution which simultaneously reduces memory footprint and compute requirements. However, aggressive quantization may lead to an unacceptable loss in model accuracy owing to differences in sensitivity to numerical imperfection across different layers in the model. To address this challenge, we propose a mixed-precision post training quantization (PTQ) approach that assigns different numerical precisions to tensors in a network based on their specific needs, for a reduced memory footprint and improved latency while preserving model accuracy. Previous works rely on layer-wise Hessian information to determine numerical precision, but as we demonstrate, Hessian estimation is typically insufficient in determining an effective ordering of layer sensitivities. We address this by augmenting the estimated Hessian with additional information to capture inter-layer dependencies. We demonstrate that this consistently improves PTQ performance along the accuracy-latency Pareto frontier across multiple models. Our method combines second-order information and inter-layer dependencies to guide a bisection search, finding quantization configurations within a user-configurable model accuracy degradation range. We evaluate the effectiveness of our method on the ResNet50, MobileNetV2, and BERT models. Our experiments demonstrate latency reductions compared to a 16-bit baseline of 25.48%, 21.69%, and 33.28% respectively, while maintaining model accuracy to within 99.99% of the baseline model.

Deploying large language models (LLMs) is challenging due to their significant memory and computational requirements. While some methods address this by developing small or tiny language models from scratch, these approaches demand extensive GPU training. Compressing pre-trained LLMs for edge devices offers a compelling alternative. Beyond pruning and quantization, Neural Architecture Search (NAS) enables effective compression, yet prior NAS approaches often limit the search space and decouple architecture from quantization. We introduce a differentiable NAS framework that explores the entire space and jointly optimizes architectural configurations alongside mixed-precision quantization for linear layers of LLMs. Experiments demonstrate superior accuracy-latency trade-offs: our models achieve up to 1.4x faster inference than sequential NAS-then-quantization baselines at comparable accuracy, or up to 6% higher average accuracy across seven reasoning tasks at equivalent latency.

Quantization has become one of the most effective methodologies to compress LLMs into smaller size. However, the existing quantization solutions still show limitations of either non-negligible accuracy drop or system inefficiency. In this paper, we make a comprehensive analysis of the general quantization principles on their effect to the triangle of accuracy, memory consumption and system efficiency. We propose MixLLM that explores the new optimization space of mixed-precision quantization between output features based on the insight that different output features matter differently in the model. MixLLM identifies the output features with high salience in the global view rather than within each single layer, effectively assigning the larger bit-width to output features that need it most to achieve good accuracy with low memory consumption. We present the sweet spot of quantization configuration of algorithm-system co-design that leads to high accuracy and system efficiency. To address the system challenge, we design the two-step dequantization to make use of the int8 Tensor Core easily and fast data type conversion to reduce dequantization overhead significantly, and present the software pipeline to overlap the memory access, dequantization and the MatMul to the best. Extensive experiments show that with only 10% more bits, the PPL increasement can be reduced from about 0.5 in SOTA to within 0.2 for Llama 3.1 70B, while on average MMLU-Pro improves by 0.93 over the SOTA of three popular models. In addition to its superior accuracy, MixLLM also achieves state-of-the-art system efficiency.

This paper presents a novel approach to fine-tuning the Qwen2-1.5B model for Arabic language processing using Quantized Low-Rank Adaptation (QLoRA) on a system with only 4GB VRAM. We detail the process of adapting this large language model to the Arabic domain, using diverse datasets including Bactrian, OpenAssistant, and Wikipedia Arabic corpora. Our methodology involves custom data preprocessing, model configuration, and training optimization techniques such as gradient accumulation and mixed-precision training. We address specific challenges in Arabic NLP, including morphological complexity, dialectal variations, and diacritical mark handling. Experimental results over 10,000 training steps show significant performance improvements, with the final loss converging to 0.1083. We provide comprehensive analysis of GPU memory usage, training dynamics, and model evaluation across various Arabic language tasks, including text classification, question answering, and dialect identification. The fine-tuned model demonstrates robustness to input perturbations and improved handling of Arabic-specific linguistic phenomena. This research contributes to multilingual AI by demonstrating a resource-efficient approach for creating specialized language models, potentially democratizing access to advanced NLP technologies for diverse linguistic communities. Our work paves the way for future research in low-resource language adaptation and efficient fine-tuning of large language models.

Traditional optimization methods rely on the use of single-precision floating point arithmetic, which can be costly in terms of memory size and computing power. However, mixed precision optimization techniques leverage the use of both single and half-precision floating point arithmetic to reduce memory requirements while maintaining model accuracy. We provide here an algorithm to further reduce memory usage during the training of a model by getting rid of the floating point copy of the parameters, virtually keeping only half-precision numbers. We also explore the benefits of getting rid of the gradient's value by executing the optimizer step during the back-propagation. In practice, we achieve up to 25% lower peak memory use and 15% faster training while maintaining the same level of accuracy.

Mixed-precision quantization is a promising approach for compressing large language models under tight memory budgets. However, existing mixed-precision methods typically suffer from one of two limitations: they either rely on expensive discrete optimization to determine precision allocation, or introduce hardware inefficiencies due to irregular memory layouts. We propose SFMP, a search-free and hardware-friendly mixed-precision quantization framework for large language models. The framework is built upon four novel ideas: Fractional bit-width, which extends integer bit-width for weight matrix to fractional value and transforms discrete precision allocation as a continuous problem; 2)Block-wise mixed-precision, enabling fine-grained precision within weight matrices while remaining hardware-friendly; 3)Row-column weight reordering, which aggregates salient weights via row and column reordering, incurring only a small activation reordering overhead during inference; 4)Unified GEMM kernel, which supports mixed-precision GEMM at arbitrary average bit-width. Extensive experiments demonstrate that SFMP outperforms state-of-the-art layer-wise mixed-precision methods under the same memory constraints, while significantly reducing quantization cost and improving inference efficiency. Code is available at https://github.com/Nkniexin/SFMP

Mixed-precision quantization must decide which parts of a model to keep at higher precision. A common premise, shared by sensitivity-based methods such as HAWQ and CoopQ, is that the loss from quantizing a set of layers can be reconstructed from per-layer or pairwise sensitivities measured in isolation. We test this premise at the 4-bit weight-and-activation precisions now being deployed, treating the change in loss f(S) from quantizing a layer set S as a set function on the Boolean cube and analyzing it through two classical changes of basis. This analysis yields two findings. First, across configurations drawn from the deployment distribution, 85--93\% of the variance of f is explained by per-layer effects alone. Second, a monotone transform of a sum of per-layer terms reproduces f's ranking of configurations, misordering at most 2\% of pairs. We propose the coverage model f(S)=cbigl(1-prod_{iin S}(1-a_i)bigr), which reproduces the measured variance profile of f to within a few percent from its L fitted break-rates. This structure supports two predictors of a configuration's loss, each with L+1 parameters. The additive model is the optimal first-order predictor. By Parseval's identity its mean-squared error equals the variance of f left unexplained by per-layer effects, which we measure on full lattices, estimate out of sample at full-network scale, and report with every result as a certificate of how well any additive model can do. The coverage model itself is the second predictor. As allocators at matched memory, they attain the lowest KL divergence among the compared allocators on models from 30B to 355B parameters. Below four bits, the resulting allocations continue to solve code and reasoning tasks at budgets where allocations from gradient sensitivities no longer produce terminating generations.

With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of least squares (LS) problems min_x|b-Ax|_2, where A in R^{mtimes n}, arise in numerous application areas. Overdetermined standard least squares problems can be solved by using mixed precision within the iterative refinement method of Björck, which transforms the least squares problem into an (m+n)times(m+n) ''augmented'' system. It has recently been shown that mixed precision GMRES-based iterative refinement can also be used, in an approach termed GMRES-LSIR. In practice, we often encounter types of least squares problems beyond standard least squares, including weighted least squares (WLS), min_x|D^{1/2}(b-Ax)|_2, where D^{1/2} is a diagonal matrix of weights. In this paper, we discuss a mixed precision FGMRES-WLSIR algorithm for solving WLS problems using two different preconditioners.

As the parameters of Large Language Models (LLMs) have scaled to hundreds of billions, the demand for efficient training methods -- balancing faster computation and reduced memory usage without sacrificing accuracy -- has become more critical than ever. In recent years, various mixed precision strategies, which involve different precision levels for optimization components, have been proposed to increase training speed with minimal accuracy degradation. However, these strategies often require manual adjustments and lack theoretical justification. In this work, we leverage stochastic rounding (SR) to address numerical errors of training with low-precision representation. We provide theoretical analyses of implicit regularization and convergence under the Adam optimizer when SR is utilized. With the insights from these analyses, we extend previous BF16 + SR strategy to be used in distributed settings, enhancing the stability and performance for large scale training. Empirical results from pre-training models with up to 6.7B parameters, for the first time, demonstrate that our BF16 with SR strategy outperforms (BF16, FP32) mixed precision strategies, achieving better validation perplexity, up to 1.54times higher throughput, and 30% less memory usage.

Large models training is plagued by the intense compute cost and limited hardware memory. A practical solution is low-precision representation but is troubled by loss in numerical accuracy and unstable training rendering the model less useful. We argue that low-precision floating points can perform well provided the error is properly compensated at the critical locations in the training process. We propose Collage which utilizes multi-component float representation in low-precision to accurately perform operations with numerical errors accounted. To understand the impact of imprecision to training, we propose a simple and novel metric which tracks the lost information during training as well as differentiates various precision strategies. Our method works with commonly used low-precision such as half-precision (16-bit floating points) and can be naturally extended to work with even lower precision such as 8-bit. Experimental results show that pre-training using Collage removes the requirement of using 32-bit floating-point copies of the model and attains similar/better training performance compared to (16, 32)-bit mixed-precision strategy, with up to 3.7times speedup and sim 15% to 23% less memory usage in practice.

Motivated by recent advances in operational weather forecasting, we study the efficacy of low-precision arithmetic for climate simulations. We develop a framework to measure rounding error in a climate model which provides a stress-test for a low-precision version of the model, and we apply our method to a variety of models including the Lorenz system; a shallow water approximation for flow over a ridge; and a coarse resolution global atmospheric model with simplified parameterisations (SPEEDY). Although double precision (52 significant bits) is standard across operational climate models, in our experiments we find that single precision (23 sbits) is more than enough and that as low as half precision (10 sbits) is often sufficient. For example, SPEEDY can be run with 12 sbits across the entire code with negligible rounding error and this can be lowered to 10 sbits if very minor errors are accepted, amounting to less than 0.1 mm/6hr for the average grid-point precipitation, for example. Our test is based on the Wasserstein metric and this provides stringent non-parametric bounds on rounding error accounting for annual means as well as extreme weather events. In addition, by testing models using both round-to-nearest (RN) and stochastic rounding (SR) we find that SR can mitigate rounding error across a range of applications. Thus our results also provide evidence that SR could be relevant to next-generation climate models. While many studies have shown that low-precision arithmetic can be suitable on short-term weather forecasting timescales, our results give the first evidence that a similar low precision level can be suitable for climate.

This is the 1st part of the dissertation for my master degree and compares the power consumption using the default floating point (32bit) and Nvidia mixed precision (16bit and 32bit) while training a classification ML model. A custom PC with specific hardware was built to perform the experiments, and different ML hyper-parameters, such as batch size, neurons, and epochs, were chosen to build Deep Neural Networks (DNN). Additionally, various software was used during the experiments to collect the power consumption data in Watts from the Graphics Processing Unit (GPU), Central Processing Unit (CPU), Random Access Memory (RAM) and manually from a wattmeter connected to the wall. A benchmarking test with default hyper parameter values for the DNN was used as a reference, while the experiments used a combination of different settings. The results were recorded in Excel, and descriptive statistics were chosen to calculate the mean between the groups and compare them using graphs and tables. The outcome was positive when using mixed precision combined with specific hyper-parameters. Compared to the benchmarking, the optimisation for the classification reduced the power consumption between 7 and 11 Watts. Similarly, the carbon footprint is reduced because the calculation uses the same power consumption data. Still, a consideration is required when configuring hyper-parameters because it can negatively affect hardware performance. However, this research required inferential statistics, specifically ANOVA and T-test, to compare the relationship between the means. Furthermore, tests indicated no statistical significance of the relationship between the benchmarking and experiments. However, a more extensive implementation with a cluster of GPUs can increase the sample size significantly, as it is an essential factor and can change the outcome of the statistical analysis.

In this paper, we explore FP8 low-bit data formats for efficient training of large language models (LLMs). Our key insight is that most variables, such as gradients and optimizer states, in LLM training can employ low-precision data formats without compromising model accuracy and requiring no changes to hyper-parameters. Specifically, we propose a new FP8 automatic mixed-precision framework for training LLMs. This framework offers three levels of FP8 utilization to streamline mixed-precision and distributed parallel training for LLMs. It gradually incorporates 8-bit gradients, optimizer states, and distributed learning in an incremental manner. Experiment results show that, during the training of GPT-175B model on H100 GPU platform, our FP8 mixed-precision training framework not only achieved a remarkable 42% reduction in real memory usage but also ran 64% faster than the widely adopted BF16 framework (i.e., Megatron-LM), surpassing the speed of Nvidia Transformer Engine by 17%. This largely reduces the training costs for large foundation models. Furthermore, our FP8 mixed-precision training methodology is generic. It can be seamlessly applied to other tasks such as LLM instruction tuning and reinforcement learning with human feedback, offering savings in fine-tuning expenses. Our FP8 low-precision training framework is open-sourced at {https://github.com/Azure/MS-AMP}{aka.ms/MS.AMP}.

Quantization has become a mainstream compression technique for reducing model size, computational requirements, and energy consumption for modern deep neural networks (DNNs). With the improved numerical support in recent hardware, including multiple variants of integer and floating point, mixed-precision quantization has become necessary to achieve high-quality results with low model cost. Prior mixed-precision quantization methods have performed a post-training quantization search, which compromises on accuracy, or a differentiable quantization search, which leads to high memory usage from branching. Therefore, we propose the first one-shot mixed-precision quantization search that eliminates the need for retraining in both integer and low-precision floating point models. We evaluate our floating-point and integer quantization search (FLIQS) on multiple convolutional networks and vision transformer models to discover Pareto-optimal models. Our approach discovers models that improve upon uniform precision, manual mixed-precision, and recent integer quantization search methods. With the proposed integer quantization search, we increase the accuracy of ResNet-18 on ImageNet by 1.31% points and ResNet-50 by 0.90% points with equivalent model cost over previous methods. Additionally, for the first time, we explore a novel mixed-precision floating-point search and improve MobileNetV2 by up to 0.98% points compared to prior state-of-the-art FP8 models. Finally, we extend FLIQS to simultaneously search a joint quantization and neural architecture space and improve the ImageNet accuracy by 2.69% points with similar model cost on a MobileNetV2 search space.

Mixture-of-Experts (MoE) architectures scale model capacity through sparse expert activation, but their deployment remains memory-bound because all expert weights must reside in memory. Mixed-precision quantization can substantially reduce this footprint by assigning different bit-widths to different experts. Existing approaches, however, typically rely on calibration data to estimate expert importance and determine bit allocation. For frontier MoE LLMs, the original training data, and hence the true training distribution, is proprietary and inaccessible. As a result, calibration sets are inevitably imperfect surrogates, and this can misestimate expert utilization and lead to suboptimal bit allocation. Motivated by the substantial cross-expert quality variability observed in modern MoE models, and by the success of Heavy-Tailed Self-Regularization (HT-SR) theory at predicting neural network model quality without access to training or testing data, we propose AlphaQ, a calibration-free bit-allocation method for MoE quantization. AlphaQ draws on HT-SR theory and follows a simple principle: experts with more heavy-tailed weight spectra are typically better trained and hence should receive higher bit-widths, while experts with weaker heavy-tailed structure can be quantized more aggressively. AlphaQ operationalizes this principle by measuring expert-wise spectral heavy-tailedness and solving a budget-constrained optimization problem that minimizes total quantization error under a global bit-budget constraint. Across several MoE models, AlphaQ consistently outperforms calibration-based baselines under matched bit budgets. Notably, on Qwen1.5-MoE, AlphaQ achieves near full-precision accuracy with an average expert precision of only 3.5 bits, while delivering more than 4times memory compression. Our code is available at https://github.com/Superone77/AlphaQ.

In astrophysical and cosmological analyses, the increasing quality and volume of astronomical data demand efficient and precise computational tools. This work introduces a novel adaptive algorithm for automatic knots (AutoKnots) allocation in spline interpolation, designed to meet user-defined precision requirements. Unlike traditional methods that rely on manually configured knot distributions with numerous parameters, the proposed technique automatically determines the optimal number and placement of knots based on interpolation error criteria. This simplifies configuration, often requiring only a single parameter. The algorithm progressively improves the interpolation by adaptively sampling the function-to-be-approximated, f(x), in regions where the interpolation error exceeds the desired threshold. All function evaluations contribute directly to the final approximation, ensuring efficiency. While each resampling step involves recomputing the interpolation table, this process is highly optimized and usually computationally negligible compared to the cost of evaluating f(x). We show the algorithm's efficacy through a series of precision tests on different functions. However, the study underscores the necessity for caution when dealing with certain function types, notably those featuring plateaus. To address this challenge, a heuristic enhancement is incorporated, improving accuracy in flat regions. This algorithm has been extensively used and tested over the years. NumCosmo includes a comprehensive set of unit tests that rigorously evaluate the algorithm both directly and indirectly, underscoring its robustness and reliability. As a practical application, we compute the surface mass density Sigma(R) and the average surface mass density Sigma(<R) for Navarro-Frenk-White and Hernquist halo density profiles, which provide analytical benchmarks. (abridged)

The KV cache is a dominant memory bottleneck for LLM inference. While 4-bit KV quantization preserves accuracy, 2-bit often degrades it, especially on long-context reasoning. We close this gap via an algorithm-system co-design for mixed-precision KV caching: Kitty. On the algorithm side, extensive experiments show that Dynamic Channel-wise Precision Boost -- which ranks Key-cache channels by sensitivity and keeps only a small fraction at higher precision -- maintains near-zero loss in accuracy drop while approaching 2-bit memory. The main challenge is handling dynamic 4-bit channel boosts while keeping the page layout coalesced and the dequantization uniform, with no scattered reads or hard-coded masks. Kitty addresses these issues by decompose each mixed-precision Key page into two tensors with unified 2-bit precision. Based on this, Kitty provides a page-centric KV layout, Triton-compatible page dequantization kernels, and a lightweight runtime pipeline that preserves coalescing and avoids divergence. Across seven tasks and two model families (Qwen3, LLaMA3), Kitty cuts KV memory by nearly 8x with negligible accuracy loss, enabling up to 8x larger batches and 2.1x-4.1x higher throughput under the same memory budget. We release the full implementation of Kitty at https://github.com/Summer-Summer/Kitty.

Quantization is an effective method for reducing memory footprint and inference time of Neural Networks, e.g., for efficient inference in the cloud, especially at the edge. However, ultra low precision quantization could lead to significant degradation in model generalization. A promising method to address this is to perform mixed-precision quantization, where more sensitive layers are kept at higher precision. However, the search space for a mixed-precision quantization is exponential in the number of layers. Recent work has proposed HAWQ, a novel Hessian based framework, with the aim of reducing this exponential search space by using second-order information. While promising, this prior work has three major limitations: (i) HAWQV1 only uses the top Hessian eigenvalue as a measure of sensitivity and do not consider the rest of the Hessian spectrum; (ii) HAWQV1 approach only provides relative sensitivity of different layers and therefore requires a manual selection of the mixed-precision setting; and (iii) HAWQV1 does not consider mixed-precision activation quantization. Here, we present HAWQV2 which addresses these shortcomings. For (i), we perform a theoretical analysis showing that a better sensitivity metric is to compute the average of all of the Hessian eigenvalues. For (ii), we develop a Pareto frontier based method for selecting the exact bit precision of different layers without any manual selection. For (iii), we extend the Hessian analysis to mixed-precision activation quantization. We have found this to be very beneficial for object detection. We show that HAWQV2 achieves new state-of-the-art results for a wide range of tasks.

Mixed-precision quantization has been widely applied on deep neural networks (DNNs) as it leads to significantly better efficiency-accuracy tradeoffs compared to uniform quantization. Meanwhile, determining the exact precision of each layer remains challenging. Previous attempts on bit-level regularization and pruning-based dynamic precision adjustment during training suffer from noisy gradients and unstable convergence. In this work, we propose Continuous Sparsification Quantization (CSQ), a bit-level training method to search for mixed-precision quantization schemes with improved stability. CSQ stabilizes the bit-level mixed-precision training process with a bi-level gradual continuous sparsification on both the bit values of the quantized weights and the bit selection in determining the quantization precision of each layer. The continuous sparsification scheme enables fully-differentiable training without gradient approximation while achieving an exact quantized model in the end.A budget-aware regularization of total model size enables the dynamic growth and pruning of each layer's precision towards a mixed-precision quantization scheme of the desired size. Extensive experiments show CSQ achieves better efficiency-accuracy tradeoff than previous methods on multiple models and datasets.

Quantization emerges as one of the most promising approaches for deploying advanced deep models on resource-constrained hardware. Mixed-precision quantization leverages multiple bit-width architectures to unleash the accuracy and efficiency potential of quantized models. However, existing mixed-precision quantization suffers exhaustive search space that causes immense computational overhead. The quantization process thus relies on separate high-performance devices rather than locally, which also leads to a significant gap between the considered hardware metrics and the real deployment.In this paper, we propose an On-chip Hardware-aware Quantization (OHQ) framework that performs hardware-aware mixed-precision quantization without accessing online devices. First, we construct the On-chip Quantization Awareness (OQA) pipeline, enabling perceive the actual efficiency metrics of the quantization operator on the hardware.Second, we propose Mask-guided Quantization Estimation (MQE) technique to efficiently estimate the accuracy metrics of operators under the constraints of on-chip-level computing power.By synthesizing network and hardware insights through linear programming, we obtain optimized bit-width configurations. Notably, the quantization process occurs on-chip entirely without any additional computing devices and data access. We demonstrate accelerated inference after quantization for various architectures and compression ratios, achieving 70% and 73% accuracy for ResNet-18 and MobileNetV3, respectively. OHQ improves latency by 15~30% compared to INT8 on deployment.

The commitment to single-precision floating-point arithmetic is widespread in the deep learning community. To evaluate whether this commitment is justified, the influence of computing precision (single and double precision) on the optimization performance of the Conjugate Gradient (CG) method (a second-order optimization algorithm) and RMSprop (a first-order algorithm) has been investigated. Tests of neural networks with one to five fully connected hidden layers and moderate or strong nonlinearity with up to 4 million network parameters have been optimized for Mean Square Error (MSE). The training tasks have been set up so that their MSE minimum was known to be zero. Computing experiments have disclosed that single-precision can keep up (with superlinear convergence) with double-precision as long as line search finds an improvement. First-order methods such as RMSprop do not benefit from double precision. However, for moderately nonlinear tasks, CG is clearly superior. For strongly nonlinear tasks, both algorithm classes find only solutions fairly poor in terms of mean square error as related to the output variance. CG with double floating-point precision is superior whenever the solutions have the potential to be useful for the application goal.

The design of fusion devices is typically based on computationally expensive simulations. This can be alleviated using high aspect ratio models that employ a reduced number of free parameters, especially in the case of stellarator optimization where non-axisymmetric magnetic fields with a large parameter space are optimized to satisfy certain performance criteria. However, optimization is still required to find configurations with properties such as low elongation, high rotational transform, finite plasma beta, and good fast particle confinement. In this work, we train a machine learning model to construct configurations with favorable confinement properties by finding a solution to the inverse design problem, that is, obtaining a set of model input parameters for given desired properties. Since the solution of the inverse problem is non-unique, a probabilistic approach, based on mixture density networks, is used. It is shown that optimized configurations can be generated reliably using this method.

Large language models deliver strong reasoning and tool-use skills, yet their computational demands make them impractical for edge or cost-sensitive deployments. We present Xmodel-2.5, a 1.3-billion-parameter small language model designed as a drop-in agent core. Training with maximal-update parameterization (μP) allows hyper-parameters tuned on a 20M-parameter proxy to transfer directly to the full model, even under the parameter-tied tie-word-embedding architecture. A 1.4T-token Warmup--Stable--Decay curriculum is used, and we further show that switching from AdamW to Muon during the decay phase improves the 13-task reasoning average by 4.58\,\% while keeping every other hyper-parameter fixed, verifying that early AdamW stability can be paired with late Muon sharpening for better downstream performance. FP8-mixed-precision training balances accuracy and throughput. All checkpoints, recipes, and evaluation code are released under the Apache-2.0 license.https://huggingface.co/XiaoduoAILab/Xmodel-2.5 and https://huggingface.co/XiaoduoAILab/Xmodel-2.5-history (training checkpoints). Training code and evaluation harness: https://github.com/XiaoduoAILab/Xmodel-2.5.

Social scientists are now using large language models to create "silicon samples" - synthetic datasets intended to stand in for human respondents, aimed at revolutionising human subjects research. However, there are many analytic choices which must be made to produce these samples. Though many of these choices are defensible, their impact on sample quality is poorly understood. I map out these analytic choices and demonstrate how a very small number of decisions can dramatically change the correspondence between silicon samples and human data. Configurations (N = 252) varied substantially in their capacity to estimate (i) rank ordering of participants, (ii) response distributions, and (iii) between-scale correlations. Most critically, configurations were not consistent in quality: those that performed well on one dimension often performed poorly on another, implying that there is no "one-size-fits-all" configuration that optimises the accuracy of these samples. I call for greater attention to the threat of analytic flexibility in using silicon samples.

The increasing computational demands of foundation models have spurred research into low-precision training, with 4-bit floating-point (FP4) formats emerging as a frontier for maximizing hardware throughput. While numerous techniques have been proposed to stabilize FP4 training, they often present isolated solutions with varying, and not always clear, computational overheads. This paper aims to provide a unified view of the design space of FP4 training. We introduce a comprehensive, quantisation gradient-based framework for microscaling quantization that allows for a theoretical analysis of the computational costs associated with different stabilization methods on both the forward and backward passes. Using a simulator built on this framework, we conduct an extensive empirical study across a wide range of machine learning tasks, including regression, image classification, diffusion models, and language models. By systematically evaluating thousands of combinations of techniques, such as novel gradient approximations, rounding strategies, and scaling methods, we identify which configurations offer the most favourable performance-to-overhead trade-off. We find that the techniques enabling the best trade-off involve carefully combining Hadamard transformations, tensor scaling and stochastic rounding. We further find that using UE5M3 as a scaling factor potentially offers a good compromise between range and precision with manageable computational overhead.

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were developed over the centuries by mathematicians, who emphasized the attainability of arbitrary precision. Computers, however, operate on few limited precision types, such as the popular float32. In this study, we show that when aiming for limited precision, existing approximation methods can be outperformed by programs automatically discovered from scratch by a simple evolutionary algorithm. In particular, over real numbers, our method can approximate the exponential function reaching orders of magnitude more precision for a given number of operations when compared to previous approaches. More practically, over float32 numbers and constrained to less than 1 ULP of error, the same method attains a speedup over baselines by generating code that triggers better XLA/LLVM compilation paths. In other words, in both cases, evolution searched a vast space of possible programs, without knowledge of mathematics, to discover previously unknown optimized approximations to high precision, for the first time. We also give evidence that these results extend beyond the exponential. The ubiquity of transcendental functions suggests that our method has the potential to reduce the cost of scientific computing applications.

Mixed-precision quantization (MPQ) has become a key technique for deploying large language models under stringent memory and compute constraints. We first identify a phenomenon that we term the Perplexity Illusion: layers ranked as important by perplexity-based sensitivity show little rank correlation with those that are most influential for complex reasoning performance, with Kendall τapprox 0 in our analysis. We further reveal an Alignment-Diversity Tradeoff: using only target-task calibration data can degrade post-quantization performance, whereas incorporating general-domain data stabilizes sensitivity estimation and improves robustness across tasks. Based on these observations, we propose TASA (Task-Aware Sensitivity Analysis), a two-level framework that jointly optimizes calibration-data composition and mixed-precision bit allocation. Specifically, TASA searches for a calibration-data mixture using a training-free gradient-trace alignment criterion, and then aggregates perplexity and reasoning-oriented sensitivity signals to guide both inter-layer and intra-layer bit allocation. Experiments on LLaMA-3-8B and Qwen2.5-7B reveal a precision inversion: appropriately allocated 3.5-bit models can match or surpass less task-aware 4-bit baselines. At an average precision of 3.5 bits, TASA matches or outperforms several competitive 4-bit uniform baselines in aggregate accuracy, and improves over the strongest W3 baseline on GSM8K by more than 20 absolute points on LLaMA-3-8B. These results show that calibration-data composition substantially affects task-sensitive quantization, a factor underexplored in prior work.

We present a reproducible, parameter-driven software workflow for optimizing approximate mutually unbiased basis (AMUB) configurations in arbitrary dimensions d using a Lie-algebra unitary parameterization. The workflow is designed for portable execution across CPU, Apple MPS, CUDA-capable GPU, and HPC backends, using a Taylor-series matrix exponential layer as an accelerator compatibility pathway. As a dimension-six case study, we optimize unanchored configurations across 100 random seeds for basis counts n = 3, 4, 5, 6 in complex128 and complex64 arithmetic. The workflow recovers exact three-basis configurations, identifies a recurrent four-basis partial-exact hub-and-triangle structure, and finds no near-exact pairs for n = 5 or n = 6 in the reported campaigns under the primary tolerance. As a hardware-execution check, we embed the representative d = 6, n = 4 transition unitaries into three-qubit 8x8 unitaries and execute the resulting circuits on the 156-qubit Heron processor ibm-marrakesh using subspace post-selection. The measured QPU pairwise losses are dominated by a hardware and compilation noise floor of approximately 0.02-0.08, associated with compiled circuits averaging 37 native CZ gates, which obscures the distinction between classically near-exact and defective pairs. The results provide a reproducible computational framework for exploring AMUB landscapes, together with an initial assessment of the challenges involved in executing optimized dimension-six unitaries on current quantum hardware.

The state-of-the-art (SOTA) for mixed precision training is dominated by variants of low precision floating point operations, and in particular, FP16 accumulating into FP32 Micikevicius et al. (2017). On the other hand, while a lot of research has also happened in the domain of low and mixed-precision Integer training, these works either present results for non-SOTA networks (for instance only AlexNet for ImageNet-1K), or relatively small datasets (like CIFAR-10). In this work, we train state-of-the-art visual understanding neural networks on the ImageNet-1K dataset, with Integer operations on General Purpose (GP) hardware. In particular, we focus on Integer Fused-Multiply-and-Accumulate (FMA) operations which take two pairs of INT16 operands and accumulate results into an INT32 output.We propose a shared exponent representation of tensors and develop a Dynamic Fixed Point (DFP) scheme suitable for common neural network operations. The nuances of developing an efficient integer convolution kernel is examined, including methods to handle overflow of the INT32 accumulator. We implement CNN training for ResNet-50, GoogLeNet-v1, VGG-16 and AlexNet; and these networks achieve or exceed SOTA accuracy within the same number of iterations as their FP32 counterparts without any change in hyper-parameters and with a 1.8X improvement in end-to-end training throughput. To the best of our knowledge these results represent the first INT16 training results on GP hardware for ImageNet-1K dataset using SOTA CNNs and achieve highest reported accuracy using half-precision

This paper presents a novel mixed-precision quantization approach for speech foundation models that tightly integrates mixed-precision learning and quantized model parameter estimation into one single model compression stage. Experiments conducted on LibriSpeech dataset with fine-tuned wav2vec2.0-base and HuBERT-large models suggest the resulting mixed-precision quantized models increased the lossless compression ratio by factors up to 1.7x and 1.9x over the respective uniform-precision and two-stage mixed-precision quantized baselines that perform precision learning and model parameters quantization in separate and disjointed stages, while incurring no statistically word error rate (WER) increase over the 32-bit full-precision models. The system compression time of wav2vec2.0-base and HuBERT-large models is reduced by up to 1.9 and 1.5 times over the two-stage mixed-precision baselines, while both produce lower WERs. The best-performing 3.5-bit mixed-precision quantized HuBERT-large model produces a lossless compression ratio of 8.6x over the 32-bit full-precision system.

Training large language models (LLMs) efficiently while preserving model quality poses significant challenges, particularly with subbyte precision supported by state-of-the-art GPUs. Current mixed-precision training approaches either apply uniform precision to all GEMM operations or rely on heuristic-based methods that fail to generalize during training, leading to suboptimal convergence and instability. To address these challenges, this paper introduces SNIP, a fine-grained adaptive mixed-precision training framework for LLM pretraining that supports subbyte precision. SNIP periodically collects statistics on activations, gradients, and optimizer states to assess the precision loss impact on model quality. We define two key metrics: loss divergence in the forward pass, caused by quantization-induced increases in training loss, and weight divergence in the backward pass, which measures error propagation through gradients affecting model updates. These metrics guide an Integer Linear Programming (ILP) problem that systematically optimizes layerwise precision to minimize overall quality loss while meeting efficiency targets. Experiments on 1B, 3B, 7B and 70B Llama-like models demonstrate that SNIP consistently outperforms existing baselines, reducing FLOPs by up to 80% while preserving model quality across different model sizes and training phases with minimal computational overhead.

Quantization is a widely-used compression technology to reduce the overhead of serving large language models (LLMs) on terminal devices and in cloud data centers. However, prevalent quantization methods, such as 8-bit weight-activation or 4-bit weight-only quantization, achieve limited performance improvements due to poor support for low-precision (e.g., 4-bit) activation. This work, for the first time, realizes practical W4A4KV4 serving for LLMs, fully utilizing the INT4 tensor cores on modern GPUs and reducing the memory bottleneck caused by the KV cache. Specifically, we propose a novel fine-grained mixed-precision quantization algorithm (FMPQ) that compresses most activations into 4-bit with negligible accuracy loss. To support mixed-precision matrix multiplication for W4A4 and W4A8, we develop a highly optimized W4Ax kernel. Our approach introduces a novel mixed-precision data layout to facilitate access and fast dequantization for activation and weight tensors, utilizing the GPU's software pipeline to hide the overhead of data loading and conversion. Additionally, we propose fine-grained streaming multiprocessor (SM) scheduling to achieve load balance across different SMs. We integrate the optimized W4Ax kernel into our inference framework, COMET, and provide efficient management to support popular LLMs such as LLaMA-3-70B. Extensive evaluations demonstrate that, when running LLaMA family models on a single A100-80G-SMX4, COMET achieves a kernel-level speedup of 2.88times over cuBLAS and a 2.02 times throughput improvement compared to TensorRT-LLM from an end-to-end framework perspective.

Gravitational-wave approximants are essential for gravitational-wave astronomy, allowing the coverage binary black hole parameter space for inference or match filtering without costly numerical relativity (NR) simulations, but generally trading some accuracy for computational efficiency. To reduce this trade-off, NR surrogate models can be constructed using interpolation within NR waveform space. We present a 2-stage training approach for neural network-based NR surrogate models. Initially trained on approximant-generated waveforms and then fine-tuned with NR data, these dual-stage artificial neural surrogate (DANSur) models offer rapid and competitively accurate waveform generation, generating millions in under 20ms on a GPU while keeping mean mismatches with NR around 10^{-4}. Implemented in the bilby framework, we show they can be used for parameter estimation tasks.

Low-precision number formats are widely used in modern machine learning systems due to their efficiency. Accurate direction representation is key to the accuracy of vector operations. This work precisely explores the extent to which the direction of a vector can be represented by selecting its scalar elements from a common finite alphabet of a given size. This is standard practice in machine learning, where low-precision significands may be narrow-width floating-point or integer values. A geometric framework is introduced for analyzing the directional coverage of such product-structured codes. This work analytically quantifies the suboptimality gap between such product-structured codes and spherical codes for the vector as a whole, in both low and asymptotically high dimensions. Furthermore, within the product code class, it is proven that the standard formats of two's complement, fixed-point, and floating-point are suboptimal, again with quantified gap, pointing to the potential to develop new scalar number formats. Such scalar alphabets are numerically optimized across multiple block dimensions for directional coverage, including the dimension used in NVIDIA's NVFP4 format. Experimental results are presented comparing the performance of standard formats and the optimized alphabet. We find that for four bits, NVIDIA's choice of E2M1 closely approximates the optimized alphabet, providing a geometric explanation for its strong performance in low-precision machine learning workloads and an analytical understanding of the link between that superiority and block size. We provide open-source formal proofs in Lean for the theorems in this work, along with the experimental code and the optimized alphabets obtained.

Model size and inference speed/power have become a major challenge in the deployment of Neural Networks for many applications. A promising approach to address these problems is quantization. However, uniformly quantizing a model to ultra low precision leads to significant accuracy degradation. A novel solution for this is to use mixed-precision quantization, as some parts of the network may allow lower precision as compared to other layers. However, there is no systematic way to determine the precision of different layers. A brute force approach is not feasible for deep networks, as the search space for mixed-precision is exponential in the number of layers. Another challenge is a similar factorial complexity for determining block-wise fine-tuning order when quantizing the model to a target precision. Here, we introduce Hessian AWare Quantization (HAWQ), a novel second-order quantization method to address these problems. HAWQ allows for the automatic selection of the relative quantization precision of each layer, based on the layer's Hessian spectrum. Moreover, HAWQ provides a deterministic fine-tuning order for quantizing layers, based on second-order information. We show the results of our method on Cifar-10 using ResNet20, and on ImageNet using Inception-V3, ResNet50 and SqueezeNext models. Comparing HAWQ with state-of-the-art shows that we can achieve similar/better accuracy with 8times activation compression ratio on ResNet20, as compared to DNAS~wu2018mixed, and up to 1% higher accuracy with up to 14% smaller models on ResNet50 and Inception-V3, compared to recently proposed methods of RVQuant~park2018value and HAQ~wang2018haq. Furthermore, we show that we can quantize SqueezeNext to just 1MB model size while achieving above 68% top1 accuracy on ImageNet.

One of the main quests in quantum metrology is to attain the ultimate precision limit with given resources, where the resources are not only of the number of queries, but more importantly of the allowed strategies. With the same number of queries, the restrictions on the strategies constrain the achievable precision. In this work, we establish a systematic framework to identify the ultimate precision limit of different families of strategies, including the parallel, the sequential, and the indefinite-causal-order strategies, and provide an efficient algorithm that determines an optimal strategy within the family of strategies under consideration. With our framework, we show there exists a strict hierarchy of the precision limits for different families of strategies.

Fine-tuning is a crucial process for adapting large language models (LLMs) to diverse applications. In certain scenarios, such as multi-tenant serving, deploying multiple LLMs becomes necessary to meet complex demands. Recent studies suggest decomposing a fine-tuned LLM into a base model and corresponding delta weights, which are then compressed using low-rank or low-bit approaches to reduce costs. In this work, we observe that existing low-rank and low-bit compression methods can significantly harm the model performance for task-specific fine-tuned LLMs (e.g., WizardMath for math problems). Motivated by the long-tail distribution of singular values in the delta weights, we propose a delta quantization approach using mixed-precision. This method employs higher-bit representation for singular vectors corresponding to larger singular values. We evaluate our approach on various fine-tuned LLMs, including math LLMs, code LLMs, chat LLMs, and even VLMs. Experimental results demonstrate that our approach performs comparably to full fine-tuned LLMs, surpassing both low-rank and low-bit baselines by a considerable margin. Additionally, we show that our method is compatible with various backbone LLMs, such as Llama-2, Llama-3, and Mistral, highlighting its generalizability.

Serving Large Language Models (LLMs) is critical for AI-powered applications but demands substantial computational resources, particularly in memory bandwidth and computational throughput. Low-precision computation has emerged as a key technique to improve efficiency while reducing resource consumption. Existing approaches for generating low-precision kernels are limited to weight bit widths that are powers of two and suffer from suboptimal performance due to high-level GPU programming abstractions. These abstractions restrict critical optimizations, such as fine-grained register management and optimized memory access patterns, which are essential for efficient low-precision computations. In this paper, we introduce a virtual machine (VM) designed for General-Purpose GPU (GPGPU) computing, enabling support for low-precision data types with arbitrary bit widths while maintaining GPU programmability. The proposed VM features a thread-block-level programming model, a hierarchical memory space, a novel algebraic layout system, and extensive support for diverse low-precision data types. VM programs are compiled into highly efficient GPU programs with automatic vectorization and instruction selection. Extensive experiments demonstrate that our VM efficiently supports a full spectrum of low-precision data types, and outperforms state-of-the-art low-precision kernels on their supported types. Compared to existing compilers like Triton and Ladder, as well as hand-optimized kernels such as QuantLLM and Marlin, our VM achieves performance improvements of 1.75x, 2.61x, 1.29x and 1.03x, respectively.

We study the problem of estimating the time delay between two signals representing delayed, irregularly sampled and noisy versions of the same underlying pattern. We propose and demonstrate an evolutionary algorithm for the (hyper)parameter estimation of a kernel-based technique in the context of an astronomical problem, namely estimating the time delay between two gravitationally lensed signals from a distant quasar. Mixed types (integer and real) are used to represent variables within the evolutionary algorithm. We test the algorithm on several artificial data sets, and also on real astronomical observations of quasar Q0957+561. By carrying out a statistical analysis of the results we present a detailed comparison of our method with the most popular methods for time delay estimation in astrophysics. Our method yields more accurate and more stable time delay estimates: for Q0957+561, we obtain 419.6 days for the time delay between images A and B. Our methodology can be readily applied to current state-of-the-art optical monitoring data in astronomy, but can also be applied in other disciplines involving similar time series data.

As large language models continue to scale, fine-grained block-scaled low-precision formats such as NVFP4 are increasingly adopted for their substantial throughput and memory benefits. However, a single FP4 micro-format often mismatches heterogeneous block-level tensor statistics. To address this without changing the standard block-scaled MMA/GEMM execution path, we propose MixFP4, a mixed micro-format extension to NVFP4 that selects between two stored FP4 micro-formats (E2M1 and E1M2) per block. MixFP4 reuses NVFP4's scale hierarchy and encodes the format choice with zero additional metadata by repurposing the sign bit of the FP8 E4M3 block scale. By decoding both micro-formats into a unified internal E2M2 compute representation, MixFP4 avoids datapath duplication. Across representative LLM families, MixFP4 improves FP4 quantization robustness and accuracy over NVFP4 baselines with modest tensor-core overhead (3.1\% area, 1.5\% power).

Transformer-based large language models (LLMs) have achieved remarkable success as model sizes continue to grow, yet their deployment remains challenging due to significant computational and memory demands. Quantization has emerged as a promising solution, and state-of-the-art quantization algorithms for LLMs introduce the need for mixed-precision matrix multiplication (mpGEMM), where lower-precision weights are multiplied with higher-precision activations. Despite its benefits, current hardware accelerators such as GPUs and TPUs lack native support for efficient mpGEMM, leading to inefficient dequantization operations in the main sequential loop. To address this limitation, we introduce MixPE, a specialized mixed-precision processing element designed for efficient low-bit quantization in LLM inference. MixPE leverages two key innovations to minimize dequantization overhead and unlock the full potential of low-bit quantization. First, recognizing that scale and zero point are shared within each quantization group, we propose performing dequantization after per-group mpGEMM, significantly reducing dequantization overhead. Second, instead of relying on conventional multipliers, MixPE utilizes efficient shift\&add operations for multiplication, optimizing both computation and energy efficiency. Our experimental results demonstrate that MixPE surpasses the state-of-the-art quantization accelerators by 2.6times speedup and 1.4times energy reduction.

We discuss methods for modeling eclipsing binary stars using the "physical", "simplified" and "phenomenological" models. There are few realizations of the "physical" Wilson-Devinney (1971) code and its improvements, e.g. Binary Maker, Phoebe. A parameter search using the Monte-Carlo method was realized by Zola et al. (2010), which is efficient in expense of too many evaluations of the test function. We compare existing algorithms of minimization of multi-parametric functions and propose to use a "combined" algorithm, depending on if the Hessian matrix is positively determined. To study methods, a simply fast-computed function resembling the "complete" test function for the physical model. Also we adopt a simplified model of an eclipsing binary at a circular orbit assuming spherical components with an uniform brightness distribution. This model resembles more advanced models in a sense of correlated parameter estimates due to a similar topology of the test function. Such a model may be applied to detached Algol-type systems, where the tidal distortion of components is negligible.

Algorithm configuration (AC) is concerned with the automated search of the most suitable parameter configuration of a parametrized algorithm. There is currently a wide variety of AC problem variants and methods proposed in the literature. Existing reviews do not take into account all derivatives of the AC problem, nor do they offer a complete classification scheme. To this end, we introduce taxonomies to describe the AC problem and features of configuration methods, respectively. We review existing AC literature within the lens of our taxonomies, outline relevant design choices of configuration approaches, contrast methods and problem variants against each other, and describe the state of AC in industry. Finally, our review provides researchers and practitioners with a look at future research directions in the field of AC.

A fully differential calculation in perturbative quantum chromodynamics is presented for the production of massive photon pairs at hadron colliders. All next-to-leading order perturbative contributions from quark-antiquark, gluon-(anti)quark, and gluon-gluon subprocesses are included, as well as all-orders resummation of initial-state gluon radiation valid at next-to-next-to-leading logarithmic accuracy. The region of phase space is specified in which the calculation is most reliable. Good agreement is demonstrated with data from the Fermilab Tevatron, and predictions are made for more detailed tests with CDF and DO data. Predictions are shown for distributions of diphoton pairs produced at the energy of the Large Hadron Collider (LHC). Distributions of the diphoton pairs from the decay of a Higgs boson are contrasted with those produced from QCD processes at the LHC, showing that enhanced sensitivity to the signal can be obtained with judicious selection of events.

To bridge the ever increasing gap between deep neural networks' complexity and hardware capability, network quantization has attracted more and more research attention. The latest trend of mixed precision quantization takes advantage of hardware's multiple bit-width arithmetic operations to unleash the full potential of network quantization. However, this also results in a difficult integer programming formulation, and forces most existing approaches to use an extremely time-consuming search process even with various relaxations. Instead of solving a problem of the original integer programming, we propose to optimize a proxy metric, the concept of network orthogonality, which is highly correlated with the loss of the integer programming but also easy to optimize with linear programming. This approach reduces the search time and required data amount by orders of magnitude, with little compromise on quantization accuracy. Specifically, we achieve 72.08% Top-1 accuracy on ResNet-18 with 6.7Mb, which does not require any searching iterations. Given the high efficiency and low data dependency of our algorithm, we used it for the post-training quantization, which achieve 71.27% Top-1 accuracy on MobileNetV2 with only 1.5Mb. Our code is available at https://github.com/MAC-AutoML/OMPQ.

Current low-precision quantization algorithms often have the hidden cost of conversion back and forth from floating point to quantized integer values. This hidden cost limits the latency improvement realized by quantizing Neural Networks. To address this, we present HAWQV3, a novel mixed-precision integer-only quantization framework. The contributions of HAWQV3 are the following: (i) An integer-only inference where the entire computational graph is performed only with integer multiplication, addition, and bit shifting, without any floating point operations or even integer division; (ii) A novel hardware-aware mixed-precision quantization method where the bit-precision is calculated by solving an integer linear programming problem that balances the trade-off between model perturbation and other constraints, e.g., memory footprint and latency; (iii) Direct hardware deployment and open source contribution for 4-bit uniform/mixed-precision quantization in TVM, achieving an average speed up of 1.45times for uniform 4-bit, as compared to uniform 8-bit for ResNet50 on T4 GPUs; and (iv) extensive evaluation of the proposed methods on ResNet18/50 and InceptionV3, for various model compression levels with/without mixed precision. For ResNet50, our INT8 quantization achieves an accuracy of 77.58%, which is 2.68% higher than prior integer-only work, and our mixed-precision INT4/8 quantization can reduce INT8 latency by 23% and still achieve 76.73% accuracy. Our framework and the TVM implementation have been open sourced.

The rapid adaptation of data driven AI models, such as deep learning inference, training, Vision Transformers (ViTs), and other HPC applications, drives a strong need for runtime precision configurable different non linear activation functions (AF) hardware support. Existing solutions support diverse precision or runtime AF reconfigurability but fail to address both simultaneously. This work proposes a flexible and SIMD multiprecision processing element (FlexPE), which supports diverse runtime configurable AFs, including sigmoid, tanh, ReLU and softmax, and MAC operation. The proposed design achieves an improved throughput of up to 16X FxP4, 8X FxP8, 4X FxP16 and 1X FxP32 in pipeline mode with 100% time multiplexed hardware. This work proposes an area efficient multiprecision iterative mode in the SIMD systolic arrays for edge AI use cases. The design delivers superior performance with up to 62X and 371X reductions in DMA reads for input feature maps and weight filters in VGG16, with an energy efficiency of 8.42 GOPS / W within the accuracy loss of 2%. The proposed architecture supports emerging 4-bit computations for DL inference while enhancing throughput in FxP8/16 modes for transformers and other HPC applications. The proposed approach enables future energy-efficient AI accelerators in edge and cloud environments.

Model quantization is widely applied for compressing and accelerating deep neural networks (DNNs). However, conventional Quantization-Aware Training (QAT) focuses on training DNNs with uniform bit-width. The bit-width settings vary across different hardware and transmission demands, which induces considerable training and storage costs. Hence, the scheme of one-shot joint training multiple precisions is proposed to address this issue. Previous works either store a larger FP32 model to switch between different precision models for higher accuracy or store a smaller INT8 model but compromise accuracy due to using shared quantization parameters. In this paper, we introduce the Double Rounding quantization method, which fully utilizes the quantized representation range to accomplish nearly lossless bit-switching while reducing storage by using the highest integer precision instead of full precision. Furthermore, we observe a competitive interference among different precisions during one-shot joint training, primarily due to inconsistent gradients of quantization scales during backward propagation. To tackle this problem, we propose an Adaptive Learning Rate Scaling (ALRS) technique that dynamically adapts learning rates for various precisions to optimize the training process. Additionally, we extend our Double Rounding to one-shot mixed precision training and develop a Hessian-Aware Stochastic Bit-switching (HASB) strategy. Experimental results on the ImageNet-1K classification demonstrate that our methods have enough advantages to state-of-the-art one-shot joint QAT in both multi-precision and mixed-precision. We also validate the feasibility of our method on detection and segmentation tasks, as well as on LLMs task. Our codes are available at https://github.com/haiduo/Double-Rounding.

Gravitational-wave data analysis relies on accurate and efficient methods to extract physical information from noisy detector signals, yet the increasing rate and complexity of observations represent a growing challenge. Deep learning provides a powerful alternative to traditional inference, but existing neural models typically lack the flexibility to handle variations in data analysis settings. Such variations accommodate imperfect observations or are required for specialized tests, and could include changes in detector configurations, overall frequency ranges, or localized cuts. We introduce a flexible transformer-based architecture paired with a training strategy that enables adaptation to diverse analysis settings at inference time. Applied to parameter estimation, we demonstrate that a single flexible model -- called Dingo-T1 -- can (i) analyze 48 gravitational-wave events from the third LIGO-Virgo-KAGRA Observing Run under a wide range of analysis configurations, (ii) enable systematic studies of how detector and frequency configurations impact inferred posteriors, and (iii) perform inspiral-merger-ringdown consistency tests probing general relativity. Dingo-T1 also improves median sample efficiency on real events from a baseline of 1.4% to 4.2%. Our approach thus demonstrates flexible and scalable inference with a principled framework for handling missing or incomplete data -- key capabilities for current and next-generation observatories.

Deploying neural networks on microcontroller units (MCUs) presents substantial challenges due to their constrained computation and memory resources. Previous researches have explored patch-based inference as a strategy to conserve memory without sacrificing model accuracy. However, this technique suffers from severe redundant computation overhead, leading to a substantial increase in execution latency. A feasible solution to address this issue is mixed-precision quantization, but it faces the challenges of accuracy degradation and a time-consuming search time. In this paper, we propose QuantMCU, a novel patch-based inference method that utilizes value-driven mixed-precision quantization to reduce redundant computation. We first utilize value-driven patch classification (VDPC) to maintain the model accuracy. VDPC classifies patches into two classes based on whether they contain outlier values. For patches containing outlier values, we apply 8-bit quantization to the feature maps on the dataflow branches that follow. In addition, for patches without outlier values, we utilize value-driven quantization search (VDQS) on the feature maps of their following dataflow branches to reduce search time. Specifically, VDQS introduces a novel quantization search metric that takes into account both computation and accuracy, and it employs entropy as an accuracy representation to avoid additional training. VDQS also adopts an iterative approach to determine the bitwidth of each feature map to further accelerate the search process. Experimental results on real-world MCU devices show that QuantMCU can reduce computation by 2.2x on average while maintaining comparable model accuracy compared to the state-of-the-art patch-based inference methods.

We present Loom, a computer architecture that executes programs compiled from C inside a looped transformer whose weights are derived analytically. The architecture implements a 22-opcode instruction set in 8 transformer layers. Each forward pass executes one instruction; the model is applied iteratively until the program counter reaches zero. The full machine state resides in a single tensor X in R^{d times n} of fixed size, and every step has fixed cost for fixed d and n, independent of program length or execution history. The default configuration uses d = 155 and n = 1024, yielding 4.7 million parameters and 928 instruction slots. A compact configuration at d = 146 and n = 512 suffices for a 9times9 Sudoku solver (284 instructions). The weights are program-independent: programs live in the state tensor, and the same fixed-weight model executes any compiled program. We make Loom source code publicly available at https://github.com/mkturkcan/Loom.

Training models for Natural Language Processing (NLP) requires substantial computational resources and time, posing significant challenges, especially for NLP development in Bangla, where access to high-end hardware is often limited. In this work, we explore automatic mixed precision (AMP) training as a means to improve computational efficiency without sacrificing model performance. By leveraging a dynamic mix of 16-bit and 32-bit floating-point computations, AMP lowers GPU memory requirements and speeds up training without degrading model performance. We evaluate AMP across four standard Bangla NLP tasks, namely sentiment analysis, named entity recognition, error classification, and question answering, using four transformer-based models: BanglaBERT, BanglishBERT, XLM-R, and mBERT. Our results demonstrate that AMP accelerates training by 44.5% and reduces memory consumption by 17.6%, while maintaining F-1 score within 99.7% of the full-precision baselines. This empirical study highlights AMP's potential to democratize access to state-of-the-art NLP capabilities in hardware-constrained settings by lowering computational barriers.

We present an open-source database of superconducting quantum device designs that may be used as the starting point for customized devices. Each design can be generated programmatically using the open-source Qiskit Metal package, and simulated using finite-element electromagnetic solvers. We present a robust workflow for achieving high accuracy on design simulations. Many designs in the database are experimentally validated, showing excellent agreement between simulated and measured parameters. Our database includes a front-end interface that allows users to generate ``best-guess'' designs based on desired circuit parameters. This project lowers the barrier to entry for research groups seeking to make a new class of devices by providing them a well-characterized starting point from which to refine their designs.

Recently, using automatic configuration tuning to improve the performance of modern database management systems (DBMSs) has attracted increasing interest from the database community. This is embodied with a number of systems featuring advanced tuning capabilities being developed. However, it remains a challenge to select the best solution for database configuration tuning, considering the large body of algorithm choices. In addition, beyond the applications on database systems, we could find more potential algorithms designed for configuration tuning. To this end, this paper provides a comprehensive evaluation of configuration tuning techniques from a broader perspective, hoping to better benefit the database community. In particular, we summarize three key modules of database configuration tuning systems and conduct extensive ablation studies using various challenging cases. Our evaluation demonstrates that the hyper-parameter optimization algorithms can be borrowed to further enhance the database configuration tuning. Moreover, we identify the best algorithm choices for different modules. Beyond the comprehensive evaluations, we offer an efficient and unified database configuration tuning benchmark via surrogates that reduces the evaluation cost to a minimum, allowing for extensive runs and analysis of new techniques.

Stochastic rounding (SR) is a probabilistic method used to round numbers to floating-point and fixed-point representations. In length n summation, the worst-case error of SR grows as n with high probability, unlike for standard modes, like round-to-nearest (RN), which grows as n. For this reason, the former is increasingly employed in large-scale, low-precision computations as an RN alternative. Additionally, SR alleviates stagnation, whereby relatively small summands are completely rounded off and do not contribute to the sum. We provide an update to [Croci et al., Roy. Soc. Open Sci. 9.3 (2022), pp. 1-25], a survey which discusses the development and use of SR between 1949 and 2022, citing over 100 references. Since then, there has been a surge of new research, and this update covers almost four years of further progress in applying, analysing, and implementing SR. Our main focus is limited-precision stochastic rounding, a new variant that fixes the precision of the random numbers used. We provide insights into industrial and numerical analysis activities surrounding SR, highlighting the next possible steps in making this rounding mode more widely available in hardware.

Post-training quantization of Large Language Models (LLMs) has proven effective in reducing the computational requirements for running inference on these models. In this study, we focus on a straightforward question: When aiming for a specific accuracy or perplexity target for low-precision quantization, how many high-precision numbers or calculations are required to preserve as we scale LLMs to larger sizes? We first introduce a critical metric named the quantization ratio, which compares the number of parameters quantized to low-precision arithmetic against the total parameter count. Through extensive and carefully controlled experiments across different model families, arithmetic types, and quantization granularities (e.g. layer-wise, matmul-wise), we identify two central phenomenons. 1) The larger the models, the better they can preserve performance with an increased quantization ratio, as measured by perplexity in pre-training tasks or accuracy in downstream tasks. 2) The finer the granularity of mixed-precision quantization (e.g., matmul-wise), the more the model can increase the quantization ratio. We believe these observed phenomena offer valuable insights for future AI hardware design and the development of advanced Efficient AI algorithms.

Reinforcement learning (RL) fine-tuning of large language models (LLMs) often suffers from instability due to the numerical mismatch between the training and inference policies. While prior work has attempted to mitigate this issue through algorithmic corrections or engineering alignments, we show that its root cause lies in the floating point precision itself. The widely adopted BF16, despite its large dynamic range, introduces large rounding errors that breaks the consistency between training and inference. In this work, we demonstrate that simply reverting to FP16 effectively eliminates this mismatch. The change is simple, fully supported by modern frameworks with only a few lines of code change, and requires no modification to the model architecture or learning algorithm. Our results suggest that using FP16 uniformly yields more stable optimization, faster convergence, and stronger performance across diverse tasks, algorithms and frameworks. We hope these findings motivate a broader reconsideration of precision trade-offs in RL fine-tuning.

Hyperparameter tuning almost always means search: fit the model at every value on a grid, score each by cross-validation, and keep the winner. For spline regression that search is unnecessary. The optimal resolution can be solved for in closed form, to the accuracy an exhaustive search reaches, at a fraction of the compute. Three ingredients make this possible: classical approximation theory pins the squared bias to a known power of the resolution G, exactly the Kolmogorov n-width of the smoothness class; the basis dimension is an explicit polynomial in G; and leave-one-out error follows from a single fit via the PRESS identity. Balancing the two known curves gives the minimizer analytically. We extend this calculus to many coordinates by replacing ambient input dimension with interaction order, the number of active low-order components in an ANOVA decomposition, yielding a scaling law in which the optimal resolution and error are power functions of the effective density (sample size per active component), with input dimension absent from the exponent. The law becomes an algorithm. KORE (Kolmogorov-optimal Order-aware Resolution Estimation) fits two pilot resolutions, solves a leverage-calibrated 2x2 system for the bias and noise scales, and evaluates the closed-form plug-in resolution with a tiny leave-one-out certificate: about a dozen fits instead of a full grid sweep, with a consistency guarantee as the sample grows. Across additive and sparse pairwise targets up to 80 input dimensions, KORE matches exhaustive 3-fold cross-validation and the full classical ladder (GCV, Mallows' Cp, AIC, BIC) while fitting roughly 8x fewer models; on 36 real tabular datasets it ranks first among 21 methods in accuracy per unit of compute, ahead of tuned boosters and kernel machines. When complexity lives in low interaction order, solving for the resolution beats searching for it.

Post-training quantization (PTQ) of large language models (LLMs) holds the promise in reducing the prohibitive computational cost at inference time. Quantization of all weight, activation and key-value (KV) cache tensors to 4-bit without significantly degrading generalizability is challenging, due to the high quantization error caused by extreme outliers in activations. To tackle this problem, we propose ResQ, a PTQ method that pushes further the state-of-the-art. By means of principal component analysis (PCA), it identifies a low-rank subspace (in practice 1/8 of the hidden dimension) in which activation variances are highest, and keep the coefficients within this subspace in high precision, e.g. 8-bit, while quantizing the rest to 4-bit. Within each subspace, invariant random rotation is applied to further suppress outliers. We show that this is a provably optimal mixed precision quantization scheme that minimizes error. With the Llama and Qwen2.5 families of models, we demonstrate that ResQ outperforms recent uniform and mixed precision PTQ methods on a variety of benchmarks, achieving up to 33\% lower perplexity on Wikitext than the next best method SpinQuant, and upto 3\times speedup over 16-bit baseline. Code is available at https://github.com/utkarsh-dmx/project-resq.

The rapid advancement of large language models (LLMs) has been paralleled by unprecedented increases in computational demands, with training costs for state-of-the-art models doubling every few months. Training models directly in low-precision arithmetic offers a solution, by improving both computational throughput and energy efficiency. Specifically, NVIDIA's recent Blackwell architecture facilitates extremely low-precision operations, specifically FP4 variants, promising substantial efficiency gains. Yet, current algorithms for training LLMs in FP4 precision face significant accuracy degradation and often rely on mixed-precision fallbacks. In this paper, we systematically investigate hardware-supported FP4 training and introduce Quartet, a new approach enabling accurate, end-to-end FP4 training with all the major computations (in e.g. linear layers) being performed in low precision. Through extensive evaluations on Llama-type models, we reveal a new low-precision scaling law that quantifies performance trade-offs across varying bit-widths and allows us to identify a "near-optimal" low-precision training technique in terms of accuracy-vs-computation, called Quartet. We implement Quartet using optimized CUDA kernels tailored for NVIDIA Blackwell GPUs, and show that it can achieve state-of-the-art accuracy for FP4 precision, successfully training billion-scale models. Our method demonstrates that fully FP4-based training is a competitive alternative to standard-precision and FP8 training. Our code is available at https://github.com/IST-DASLab/Quartet.

Kronecker adapters have emerged as a promising approach for fine-tuning large-scale models, enabling high-rank updates through tunable component structures. However, existing work largely treats the component structure as a fixed or heuristic design choice, leaving the dimensions and number of Kronecker components underexplored. In this paper, we identify component structure as a key factor governing the capacity of Kronecker adapters. We perform a fine-grained analysis of both the dimensions and number of Kronecker components. In particular, we show that the alignment between Kronecker adapters and full fine-tuning depends on component configurations. Guided by these insights, we propose Component Designed Kronecker Adapters (CDKA). We further provide parameter-budget-aware configuration guidelines and a tailored training stabilization strategy for practical deployment. Experiments across various architectures and modalities demonstrate the effectiveness of CDKA. Code is available at https://github.com/rainstonee/CDKA.

Motivated by the growing demand for low-precision arithmetic in computational science, we exploit lower-precision emulation in Python -- widely regarded as the dominant programming language for numerical analysis and machine learning. Low-precision training has revolutionized deep learning by enabling more efficient computation and reduced memory and energy consumption while maintaining model fidelity. To better enable numerical experimentation with and exploration of low precision computation, we developed the Pychop library, which supports customizable floating-point formats and a comprehensive set of rounding modes in Python, allowing users to benefit from fast, low-precision emulation in numerous applications. Pychop also introduces interfaces for both PyTorch and JAX, enabling efficient low-precision emulation on GPUs for neural network training and inference with unparalleled flexibility. In this paper, we offer a comprehensive exposition of the design, implementation, validation, and practical application of Pychop, establishing it as a foundational tool for advancing efficient mixed-precision algorithms. Furthermore, we present empirical results on low-precision emulation for image classification and object detection using published datasets, illustrating the sensitivity of the use of low precision and offering valuable insights into its impact. Pychop enables in-depth investigations into the effects of numerical precision, facilitates the development of novel hardware accelerators, and integrates seamlessly into existing deep learning workflows. Software and experimental code are publicly available at https://github.com/inEXASCALE/pychop.

4-bit quantization significantly reduces the memory footprint and accelerates the inference of large language models (LLMs). However, its limited bit-width representation struggles to faithfully capture both dense common values (inliers) and rare large-magnitude values (outliers), causing substantial accuracy degradation. Existing mixed-precision methods mitigate this by retaining outliers in high precision, but at the cost of breaking the uniformity of low-bit execution, introducing precision conversion and extra data movement that undermine practical speedup. We propose MosaicQuant, a unified 4-bit LLM quantization paradigm built on a novel principle of inlier--outlier disaggregation. Rather than elevating outlier precision, MosaicQuant quantizes the full weight matrix into a dense 4-bit base component, where inliers are captured faithfully while outlier are inevitably quantized. A sparse 4-bit residual component is then introduced to compensate for these quantization errors, selectively targeting the most error-critical weight blocks where output distortion is shown to be concentrated. However, a unified representation alone is insufficient, as naïvely executing the sparse residual as a separate kernel still breaks the unified low-bit inference pipeline. To bridge this gap, we introduce ZipperEngine, which fuses sparse block computation into the dense 4-bit GEMM kernel via an overlapped pipeline, unifying not only the representation but also the execution into a single coherent low-bit inference pipeline. Extensive experiments on LLaMA3 and Qwen3 demonstrate that MosaicQuant preserves near-FP16 accuracy while achieving up to 1.24times speedup over the W16A16 baseline.

The quadratic computational complexity of the standard attention mechanism constitutes a fundamental bottleneck for large language models in long-context inference. While existing KV cache compression methods alleviate memory pressure, they often sacrifice generation quality and fail to address the high overhead of floating-point arithmetic. This paper introduces DASH-KV, an innovative acceleration framework that reformulates attention as approximate nearest-neighbor search via asymmetric deep hashing. Under this paradigm, we design an asymmetric encoding architecture that differentially maps queries and keys to account for their distinctions in precision and reuse characteristics. To balance efficiency and accuracy, we further introduce a dynamic mixed-precision mechanism that adaptively retains full-precision computation for critical tokens. Extensive experiments on LongBench demonstrate that DASH-KV significantly outperforms state-of-the-art baseline methods while matching the performance of full attention, all while reducing inference complexity from O(N^2) to linear O(N). The code is available at https://github.com/Zhihan-Zh/DASH-KV

Quantizing neural networks is one of the most effective methods for achieving efficient inference on mobile and embedded devices. In particular, mixed precision quantized (MPQ) networks, whose layers can be quantized to different bitwidths, achieve better task performance for the same resource constraint compared to networks with homogeneous bitwidths. However, finding the optimal bitwidth allocation is a challenging problem as the search space grows exponentially with the number of layers in the network. In this paper, we propose QBitOpt, a novel algorithm for updating bitwidths during quantization-aware training (QAT). We formulate the bitwidth allocation problem as a constraint optimization problem. By combining fast-to-compute sensitivities with efficient solvers during QAT, QBitOpt can produce mixed-precision networks with high task performance guaranteed to satisfy strict resource constraints. This contrasts with existing mixed-precision methods that learn bitwidths using gradients and cannot provide such guarantees. We evaluate QBitOpt on ImageNet and confirm that we outperform existing fixed and mixed-precision methods under average bitwidth constraints commonly found in the literature.

Post-selecting output states in measurements can effectively amplify weak signals and improve precision. However, post-selection effects may also introduce unintended biases in precision measurements. Here, we investigate the influence of post-selection in the precision spectroscopy of the 2^3S - 2^3P transition of helium (^4He) using an atomic beam. We directly observe that post-selection based on atomic positions causes a shift in the measured transition frequency, amounting to approximately -55 kHz. After accounting for this post-selection shift, we obtain a corrected frequency of 276,764,094,712.45 pm 0.86 kHz for the 2^3S_1 - 2^3P_0 transition. Combining this result with existing data for ^3He, we derive a new value for the difference in squared nuclear charge radii, δr^2 [r_{h}^{2} - r_α^{2}] = 1.0733 pm 0.0021 fm^2. This value shows a 2.8σ deviation from measurements of muonic helium ions, potentially pointing to new physics that challenges lepton universality in quantum electrodynamics.

This paper presents a comprehensive analysis of quantization techniques for optimizing Large Language Models (LLMs), specifically focusing on Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT). Through empirical evaluation across models ranging from 10M to 1B parameters, we demonstrate that quantization can achieve up to 68% reduction in model size while maintaining performance within 6% of full-precision baselines when utilizing our proposed scaling factor {\gamma}. Our experiments show that INT8 quantization delivers a 40% reduction in computational cost and power consumption, while INT4 quantization further improves these metrics by 60%. We introduce a novel theoretical framework for mixed-precision quantization, deriving optimal bit allocation strategies based on layer sensitivity and weight variance. Hardware efficiency evaluations on edge devices reveal that our quantization approach enables up to 2.4x throughput improvement for INT8 and 3x for INT4, with 60% power reduction compared to full-precision models.

Database knob tuning is a critical challenge in the database community, aiming to optimize knob values to enhance database performance for specific workloads. DBMS often feature hundreds of tunable knobs, posing a significant challenge for DBAs to recommend optimal configurations. Consequently, many machine learning-based tuning methods have been developed to automate this process. Despite the introduction of various optimizers, practical applications have unveiled a new problem: they typically require numerous workload runs to achieve satisfactory performance, a process that is both time-consuming and resource-intensive. This inefficiency largely stems from the optimal configuration often being substantially different from the default setting, necessitating multiple iterations during tuning. Recognizing this, we argue that an effective starting point could significantly reduce redundant exploration in less efficient areas, thereby potentially speeding up the tuning process for the optimizers. Based on this assumption, we introduce LLMTune, a large language model-based configuration generator designed to produce an initial, high-quality configuration for new workloads. These generated configurations can then serve as starting points for various base optimizers, accelerating their tuning processes. To obtain training data for LLMTune's supervised fine-tuning, we have devised a new automatic data generation framework capable of efficiently creating a large number of <workload, configuration> pairs. We have conducted thorough experiments to evaluate LLMTune's effectiveness with different workloads, such as TPC-H and JOB. In comparison to leading methods, LLMTune demonstrates a quicker ability to identify superior configurations. For instance, with the challenging TPC-H workload, our LLMTune achieves a significant 15.6x speed-up ratio in finding the best-performing configurations.

We present anyakrakusuma, an open-source Python library that solves the discrete static Schrödinger bridge problem, the entropically regularized counterpart of optimal transport, through a log-domain Sinkhorn--Knopp iteration and reconstructs the entropic interpolation between two empirical point clouds. The solver is paired with a diagnostic pipeline that characterizes the optimal coupling and the intermediate distributions through information-theoretic and geometric measures. We exercise the library on four idealized planar cases spanning a circle-to-circle dilation, a spiral-to-mixture fragmentation, a rigid reorientation of two moons, and a Lissajous-to-trefoil deformation. The log-domain formulation is necessary rather than merely convenient at the parameters studied, where the cost-to-regularization ratio reaches four hundred and the Gibbs kernel underflows double precision across most of its range; the iteration nonetheless attains a marginal residual of 10^{-9} and unit marginal fidelity in every case. Residual histories decay geometrically over approximately eight decades at per-iteration contraction factors between 0.966 and 0.976, which are local rates near the fixed point that lie many orders of magnitude below the worst-case Hilbert-metric bound. The covariance analysis recovers an imposed ninety-degree reorientation to within 0.07^circ, roughly forty times smaller than its uncertainty, across a masked interval of near-isotropy on which the principal axis is unobservable. The diagnostics are reported with explicit attention to the regimes in which each is well defined, including the differential entropy, which is meaningful only on the open interpolation interval. The presented cases are constructed rather than measured; quantitative application to empirical point clouds requires further study.

Large language models (LLMs) have achieved impressive performance across various domains. However, the substantial hardware resources required for their training present a significant barrier to efficiency and scalability. To mitigate this challenge, low-precision training techniques have been widely adopted, leading to notable advancements in training efficiency. Despite these gains, low-precision training involves several componentsx2013such as weights, activations, and gradientsx2013each of which can be represented in different numerical formats. The resulting diversity has created a fragmented landscape in low-precision training research, making it difficult for researchers to gain a unified overview of the field. This survey provides a comprehensive review of existing low-precision training methods. To systematically organize these approaches, we categorize them into three primary groups based on their underlying numerical formats, which is a key factor influencing hardware compatibility, computational efficiency, and ease of reference for readers. The categories are: (1) fixed-point and integer-based methods, (2) floating-point-based methods, and (3) customized format-based methods. Additionally, we discuss quantization-aware training approaches, which share key similarities with low-precision training during forward propagation. Finally, we highlight several promising research directions to advance this field. A collection of papers discussed in this survey is provided in https://github.com/Hao840/Awesome-Low-Precision-Training.

This paper introduces bucket calculus, a novel mathematical framework that fundamentally transforms the computational complexity landscape of parallel machine scheduling optimization. We address the strongly NP-hard problem P2|r_j|C_{max} through an innovative adaptive temporal discretization methodology that achieves exponential complexity reduction from O(T^n) to O(B^n) where B ll T, while maintaining near-optimal solution quality. Our bucket-indexed mixed-integer linear programming (MILP) formulation exploits dimensional complexity heterogeneity through precision-aware discretization, reducing decision variables by 94.4\% and achieving a theoretical speedup factor 2.75 times 10^{37} for 20-job instances. Theoretical contributions include partial discretization theory, fractional bucket calculus operators, and quantum-inspired mechanisms for temporal constraint modeling. Empirical validation on instances with 20--400 jobs demonstrates 97.6\% resource utilization, near-perfect load balancing (σ/μ= 0.006), and sustained performance across problem scales with optimality gaps below 5.1\%. This work represents a paradigm shift from fine-grained temporal discretization to multi-resolution precision allocation, bridging the fundamental gap between exact optimization and computational tractability for industrial-scale NP-hard scheduling problems.

Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below O(10^{-5}) even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. Across successive stages, the residue magnitudes decreases substantially and follows an inverse power-law relationship with the residue frequencies. The multi-stage neural networks effectively mitigate the spectral biases associated with regular neural networks, enabling them to capture the high frequency feature of target functions. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision O(10^{-16}) of double-floating point within a finite number of iterations. Such levels of accuracy are rarely attainable using single neural networks alone.

We propose a modified moment matching algorithm to avoid catastrophic failures for sources with a low signal to noise ratio (SNR). The proposed modifications include a method to eliminate non-physical negative pixel values and a forced single iteration with an initial guess derived from co-add measurements when iterative methods are unstable. We correct for all biases in measurements introduced by the method. We find that the proposed modifications allow the algorithm to avoid catastrophic failures in nearly 100\% of the cases, especially at low signal to noise ratio. Additionally, with a reasonable guess from co-add measurements, the algorithm measures the flux, centroid, size, shape and ellipticity with bias statistically consistent with zero. We show the proposed method allows us to measure sources seven times fainter than traditional methods when applied to images obtained from WIYN-ODI. We also present a scheme to find uncertainties in measurements when using the new method to measure astronomical sources.

This paper presents the first comprehensive empirical study demonstrating the efficacy of the Brain Floating Point (BFLOAT16) half-precision format for Deep Learning training across image classification, speech recognition, language modeling, generative networks and industrial recommendation systems. BFLOAT16 is attractive for Deep Learning training for two reasons: the range of values it can represent is the same as that of IEEE 754 floating-point format (FP32) and conversion to/from FP32 is simple. Maintaining the same range as FP32 is important to ensure that no hyper-parameter tuning is required for convergence; e.g., IEEE 754 compliant half-precision floating point (FP16) requires hyper-parameter tuning. In this paper, we discuss the flow of tensors and various key operations in mixed precision training, and delve into details of operations, such as the rounding modes for converting FP32 tensors to BFLOAT16. We have implemented a method to emulate BFLOAT16 operations in Tensorflow, Caffe2, IntelCaffe, and Neon for our experiments. Our results show that deep learning training using BFLOAT16 tensors achieves the same state-of-the-art (SOTA) results across domains as FP32 tensors in the same number of iterations and with no changes to hyper-parameters.

Key-Value (KV) Caching has become an essential technique for accelerating the inference speed and throughput of generative Large Language Models~(LLMs). However, the memory footprint of the KV cache poses a critical bottleneck in LLM deployment as the cache size grows with batch size and sequence length, often surpassing even the size of the model itself. Although recent methods were proposed to select and evict unimportant KV pairs from the cache to reduce memory consumption, the potential ramifications of eviction on the generative process are yet to be thoroughly examined. In this paper, we examine the detrimental impact of cache eviction and observe that unforeseen risks arise as the information contained in the KV pairs is exhaustively discarded, resulting in safety breaches, hallucinations, and context loss. Surprisingly, we find that preserving even a small amount of information contained in the evicted KV pairs via reduced precision quantization substantially recovers the incurred degradation. On the other hand, we observe that the important KV pairs must be kept at a relatively higher precision to safeguard the generation quality. Motivated by these observations, we propose Mixed-precision KV cache~(MiKV), a reliable cache compression method that simultaneously preserves the context details by retaining the evicted KV pairs in low-precision and ensure generation quality by keeping the important KV pairs in high-precision. Experiments on diverse benchmarks and LLM backbones show that our proposed method offers a state-of-the-art trade-off between compression ratio and performance, compared to other baselines.

When finetuning large language models for specialized tasks such as mathematical reasoning, models exhibit catastrophic forgetting, losing previously learned capabilities. We investigate this by finetuning Flan-T5-Base (250M parameters) on the DeepMind Mathematics dataset and measuring forgetting on MultiNLI. Math-only training improves mathematical accuracy from 3.1\% to 12.0\% but causes NLI accuracy to collapse from 81.0\% to 16.5\%--a 64.5 percentage point drop occurring within the first 1,000 training steps. We propose mixed training strategies that interleave mathematical and NLI examples during training. Our results demonstrate that mixed training completely eliminates catastrophic forgetting while maintaining equivalent mathematical performance: the balanced 1:1 ratio achieves 12.0\% math accuracy (matching math-only) while preserving 86.2\% NLI accuracy. We systematically explore mixing ratios from 1:1 to 15:1, finding that even minimal NLI exposure (6.2\%) provides effective regularization. These findings demonstrate that specialization need not require forgetting general capabilities, with implications for scaling to larger models where mixed training may confer additional benefits beyond forgetting prevention.

We present new methods and results for constructing optimal Kobon triangle arrangements. First, we introduce a compact table notation for describing arrangements of pseudolines, enabling the representation and analysis of complex cases, including symmetrical arrangements, arrangements with parallel lines, and arrangements with multiple-line intersection points. Building on this, we provide a simple heuristic method and tools for recovering straight-line arrangements from a given table, with the ability to enforce additional properties such as symmetries. The tool successfully recovers arrangements for many previously known optimal solutions. Additionally, we develop a tool that transforms the search for optimal Kobon arrangement tables into a SAT problem, allowing us to leverage modern SAT solvers (specifically Kissat) to efficiently find new solutions or to show that no other solutions exist (for example, confirming that no optimal solution exists in the 11-line case). Using these techniques, we find new optimal Kobon arrangements for 23 and 27 lines, along with several other new results.

Quantization addresses the high resource demand for large language models (LLMs) by alleviating memory pressure and bandwidth congestion and providing significantly scaled compute power with a tolerable impact on accuracy. Four-bit floating point (FP4), the lowest-precision format that preserves essential numerical properties such as exponent and sign, has begun to be adopted in cutting-edge architectures, including Blackwell and AMD CDNA, to support LLM quantization and reduce deployment costs. Although aggressive quantization can yield efficiency gains, the quantization sensitivity of within-transformer layers and whether these sensitivities generalize across existing FP4 formats and model scales remain underexplored. To elucidate quantization sensitivity, this study conducts a systematic analysis of two FP4 formats, MXFP4 and NVFP4, across three Qwen2.5 model scales (0.5B, 7B, and 14B), using controlled component-wise and block-wise isolation methodologies. We observe that MLP up- and down-projection layers consistently dominate in terms of sensitivity, while gate and attention projections are moderately and substantially less sensitive to FP4 quantization, respectively. We further find that sensitivity does not universally localize to the final blocks, but early blocks can be highly sensitive, particularly under MXFP4. Our results provide a diagnostic characterization of the inference behavior of FP4 across components, depths, and FP4 formats.

In this paper, we propose a unified approach to harness quantum conformal methods for multi-output distributions, with a particular emphasis on two experimental paradigms: (i) a standard 2-qubit circuit scenario producing a four-dimensional outcome distribution, and (ii) a multi-basis measurement setting that concatenates measurement probabilities in different bases (Z, X, Y) into a twelve-dimensional output space. By combining a multioutput regression model (e.g., random forests) with distributional conformal prediction, we validate coverage and interval-set sizes on both simulated quantum data and multi-basis measurement data. Our results confirm that classical conformal prediction can effectively provide coverage guarantees even when the target probabilities derive from inherently quantum processes. Such synergy opens the door to next-generation quantum-classical hybrid frameworks, providing both improved interpretability and rigorous coverage for quantum machine learning tasks. All codes and full reproducible Colab notebooks are made available at https://github.com/detasar/QECMMOU.

Datasets often have their intrinsic symmetries, and particular deep-learning models called equivariant or invariant models have been developed to exploit these symmetries. However, if some or all of these symmetries are only approximate, which frequently happens in practice, these models may be suboptimal due to the architectural restrictions imposed on them. We tackle this issue of approximate symmetries in a setup where symmetries are mixed, i.e., they are symmetries of not single but multiple different types and the degree of approximation varies across these types. Instead of proposing a new architectural restriction as in most of the previous approaches, we present a regularizer-based method for building a model for a dataset with mixed approximate symmetries. The key component of our method is what we call equivariance regularizer for a given type of symmetries, which measures how much a model is equivariant with respect to the symmetries of the type. Our method is trained with these regularizers, one per each symmetry type, and the strength of the regularizers is automatically tuned during training, leading to the discovery of the approximation levels of some candidate symmetry types without explicit supervision. Using synthetic function approximation and motion forecasting tasks, we demonstrate that our method achieves better accuracy than prior approaches while discovering the approximate symmetry levels correctly.

We study charm production in charged-current deep-inelastic scattering (DIS) using the xFitter framework. Recent results from the LHC have focused renewed attention on the determination of the strange-quark parton distribution function (PDF) and the DIS charm process provides important complementary constraints on this quantity. We examine the current PDF uncertainty, and use LHeC pseudodata to estimate the potential improvement from this proposed facility. As xFitter implements both fixed-flavor- and variable-flavor-number schemes, we can compare the impact of these different theoretical choices; this highlights some interesting aspects of multi-scale calculations. We find that the high-statistics LHeC data covering a wide kinematic range could substantially reduce the strange PDF uncertainty.

Quantization has become a crucial step for the efficient deployment of deep neural networks, where floating point operations are converted to simpler fixed point operations. In its most naive form, it simply consists in a combination of scaling and rounding transformations, leading to either a limited compression rate or a significant accuracy drop. Recently, Gradient-based post-training quantization (GPTQ) methods appears to be constitute a suitable trade-off between such simple methods and more powerful, yet expensive Quantization-Aware Training (QAT) approaches, particularly when attempting to quantize LLMs, where scalability of the quantization process is of paramount importance. GPTQ essentially consists in learning the rounding operation using a small calibration set. In this work, we challenge common choices in GPTQ methods. In particular, we show that the process is, to a certain extent, robust to a number of variables (weight selection, feature augmentation, choice of calibration set). More importantly, we derive a number of best practices for designing more efficient and scalable GPTQ methods, regarding the problem formulation (loss, degrees of freedom, use of non-uniform quantization schemes) or optimization process (choice of variable and optimizer). Lastly, we propose a novel importance-based mixed-precision technique. Those guidelines lead to significant performance improvements on all the tested state-of-the-art GPTQ methods and networks (e.g. +6.819 points on ViT for 4-bit quantization), paving the way for the design of scalable, yet effective quantization methods.

There are now several comprehensive web applications, stand-alone computer programs and computer algebra functions that, given a floating point number such as 6.518670730718491, can return concise nonfloat constants such as 3 arctan 2 + ln 9 + 1, that closely approximate the float. Examples include AskConstants, Inverse Symbolic Calculator, the Maple identify function, MESearch, OEIS, RIES, and WolframAlpha. Usefully often such a result is the exact limit as the float is computed with increasing precision. Therefore these program results are candidates for proving an exact result that you could not otherwise compute or conjecture without the program. Moreover, candidates that are not the exact limit can be provable bounds, or convey qualitative insight, or suggest series that they truncate, or provide sufficiently close efficient approximations for subsequent computation. This article describes some of these programs, how they work, and how best to use each of them. Almost everyone who uses or should use mathematical software can benefit from acquaintance with several such programs, because these programs differ in the sets of constants that they can return.

Long-context prefill in large language models (LLMs) incurs substantial computation and memory traffic because dense self-attention computes quadratic query-key scores. Existing methods either use a uniform low-precision path or select token interactions, leaving spatial precision routing over hardware-aligned score tiles outside fused dense attention. We introduce TileMix, a tile-centric precision-routing kernel that makes numerical precision an executable spatial decision over score-tile groups within fused dense attention. TileMix partitions the attention matrix into hardware-aligned score tiles, packs routing decisions into compact bitmasks, and dispatches each tile group through FP16 or INT8 score computation while both paths update a shared online-softmax state. Scalable precision grouping lets each routing bit govern multiple adjacent key tiles, preserving hardware-aligned compute tiles and compact metadata at long contexts. By routing all legal tile groups, TileMix preserves dense token connectivity, requires no training, and supports grouped-query attention, variable-length batches, and INT8 key/value caches. Across LongEval, LV-Eval, and A100 prefill benchmarks on LLaMA, Qwen, and Vicuna, TileMix recovers long-context quality lost under uniform INT8 and improves prefill throughput over FP16, yielding a controllable accuracy-efficiency frontier across model families. The implementation is available at https://github.com/HanzhiZhang-Ulrica/TileMix.

Deep neural networks have enabled progress in a wide variety of applications. Growing the size of the neural network typically results in improved accuracy. As model sizes grow, the memory and compute requirements for training these models also increases. We introduce a technique to train deep neural networks using half precision floating point numbers. In our technique, weights, activations and gradients are stored in IEEE half-precision format. Half-precision floating numbers have limited numerical range compared to single-precision numbers. We propose two techniques to handle this loss of information. Firstly, we recommend maintaining a single-precision copy of the weights that accumulates the gradients after each optimizer step. This single-precision copy is rounded to half-precision format during training. Secondly, we propose scaling the loss appropriately to handle the loss of information with half-precision gradients. We demonstrate that this approach works for a wide variety of models including convolution neural networks, recurrent neural networks and generative adversarial networks. This technique works for large scale models with more than 100 million parameters trained on large datasets. Using this approach, we can reduce the memory consumption of deep learning models by nearly 2x. In future processors, we can also expect a significant computation speedup using half-precision hardware units.

This paper presents a practical investigation into fine-tuning model parameters for mathematical reasoning tasks through experimenting with various configurations including randomness control, reasoning depth, and sampling strategies, careful tuning demonstrates substantial improvements in efficiency as well as performance. A holistically optimized framework is introduced for five state-of-the-art models on mathematical reasoning tasks, exhibiting significant performance boosts while maintaining solution correctness. Through systematic parameter optimization across Qwen2.5-72B, Llama-3.1-70B, DeepSeek-V3, Mixtral-8x22B, and Yi-Lightning, consistent efficiency gains are demonstrated with 100% optimization success rate. The methodology achieves an average 29.4% reduction in computational cost and 23.9% improvement in inference speed across all tested models. This framework systematically searches parameter spaces including temperature (0.1-0.5), reasoning steps (4-12), planning periods (1-4), and nucleus sampling (0.85-0.98), determining optimal configurations through testing on mathematical reasoning benchmarks. Critical findings show that lower temperature regimes (0.1-0.4) and reduced reasoning steps (4-6) consistently enhance efficiency without compromising accuracy. DeepSeek-V3 achieves the highest accuracy at 98%, while Mixtral-8x22B delivers the most cost-effective performance at 361.5 tokens per accurate response. Key contributions include: (1) the first comprehensive optimization study for five diverse SOTA models in mathematical reasoning, (2) a standardized production-oriented parameter optimization framework, (3) discovery of universal optimization trends applicable across model architectures, and (4) production-ready configurations with extensive performance characterization.

An observable in an Effective Field Theory (EFT) is an expansion in a set of small parameters, no different than any other perturbation series. Truncating such a series expansion leaves the leading dropped term as the dominant source of error, and that term itself contains a calculable portion. We explore the partial calculation of this next-order error, available at dim-6^2 in SMEFT with no additional tools needed to simulate it, and explore how to efficiently use that partial calculation to model the next-order uncertainty in full using a minimal number of nuisance parameters. This estimate of the uncertainty of the EFT signal rate naturally imposes EFT validity by ensuring that bounds are driven by kinematic regions where truncation uncertainties are parametrically smaller than the signal. We incorporate the calculable dim-6^2 piece into the signal and cover the remaining higher-dimension uncertainty with a small set of nuisance parameters derived from a scan over Wilson-coefficient values. Our algorithm to determine the relevant nuisance parameters and distributions is automation-friendly and applies to arbitrary truncation choices. We then provide multiple examples of its implementation in high-energy collider processes focused on SMEFT truncated at O(Λ^{-2}), where the dim-6^2 piece is used to estimate the full mathcal O(Λ^{-4}) dependence. In the examples considered here the reduction of nuisance parameters is appreciable, generically reducing the nuisance parameter count by an order of magnitude compared to naïve estimates.

Precision and Recall are two prominent metrics of generative performance, which were proposed to separately measure the fidelity and diversity of generative models. Given their central role in comparing and improving generative models, understanding their limitations are crucially important. To that end, in this work, we identify a critical flaw in the common approximation of these metrics using k-nearest-neighbors, namely, that the very interpretations of fidelity and diversity that are assigned to Precision and Recall can fail in high dimensions, resulting in very misleading conclusions. Specifically, we empirically and theoretically show that as the number of dimensions grows, two model distributions with supports at equal point-wise distance from the support of the real distribution, can have vastly different Precision and Recall regardless of their respective distributions, hence an emergent asymmetry in high dimensions. Based on our theoretical insights, we then provide simple yet effective modifications to these metrics to construct symmetric metrics regardless of the number of dimensions. Finally, we provide experiments on real-world datasets to illustrate that the identified flaw is not merely a pathological case, and that our proposed metrics are effective in alleviating its impact.

The current best bounds on the matrix multiplication exponent ω are obtained through a refinement of the laser method called combination loss analysis (Duan et al., 2022; Williams et al., 2024; Alman et al., 2025). In this note, we address the optimization problem at the core of this approach and propose several improvements. First, we reformulate the optimization problem allowing us to solve it in a larger setting than was previously possible. Second, we leverage recent advances in machine learning to design a new optimization algorithm for this problem. Finally, we refine the resulting optimization algorithm with AlphaEvolve. Our combined approach yields an upper bound of ω < 2.371177, improving the previous best bound of 2.371339.

We present the new parton distribution functions (PDFs) from the CTEQ-TEA collaboration, obtained using a wide variety of high-precision Large Hadron Collider (LHC) data, in addition to the combined HERA I+II deep-inelastic scattering data set, along with the data sets present in the CT14 global QCD analysis. New LHC measurements in single-inclusive jet production with the full rapidity coverage, as well as production of Drell-Yan pairs, top-quark pairs, and high-p_T Z bosons, are included to achieve the greatest sensitivity to the PDFs. The parton distributions are determined at NLO and NNLO, with each of these PDFs accompanied by error sets determined using the Hessian method. Fast PDF survey techniques, based on the Hessian representation and the Lagrange Multiplier method, are used to quantify the preference of each data set to quantities such as α_s(m_Z), and the gluon and strange quark distributions. We designate the main resulting PDF set as CT18. The ATLAS 7 TeV precision W/Z data are not included in CT18, due to their tension with other data sets in the global fit. Alternate PDF sets are generated including the ATLAS precision 7 TeV W/Z data (CT18A), a new scale choice for low-x DIS data (CT18X), or all of the above with a slightly higher choice for the charm mass (CT18Z). Theoretical calculations of standard candle cross sections at the LHC (such as the gg fusion Higgs boson cross section) are presented.

Reduced numerical precision is a common technique to reduce computational cost in many Deep Neural Networks (DNNs). While it has been observed that DNNs are resilient to small errors and noise, no general result exists that is capable of predicting a given DNN system architecture's sensitivity to reduced precision. In this project, we emulate arbitrary bit-width using a specified floating-point representation with a truncation method, which is applied to the neural network after each batch. We explore the impact of several model parameters on the network's training accuracy and show results on the MNIST dataset. We then present a preliminary theoretical investigation of the error scaling in both forward and backward propagations. We end with a discussion of the implications of these results as well as the potential for generalization to other network architectures.

Generating high-fidelity 3D geometries under explicit parameter constraints is central to engineering design, yet current methods often require large datasets and fail to provide reliable control beyond the training distribution. We introduce LAMP, a data-efficient framework for controllable and interpretable 3D generation that aligns signed distance function (SDF) decoders by overfitting each exemplar from a shared initialization, then generates new designs by solving a parameter-constrained affine mixing problem in the aligned weight space. To improve reliability, we propose a linearity-mismatch safety metric that detects when mixed decoders leave the valid local regime. We evaluate LAMP on DrivAerNet++, BlendedNet, and additional industry-level vehicle families, including sports cars, SUVs, and convertibles. LAMP enables controlled interpolation with as few as 50 samples, safe extrapolation up to 100% beyond training ranges, and performance-guided optimization under fixed parameters, outperforming conditional autoencoder and Deep Network Interpolation (DNI) baselines in extrapolation, data efficiency, and parameter fidelity. Our results demonstrate that LAMP advances controllable, data-efficient, and safe 3D generation for design exploration, dataset generation, and performance-driven optimization.

As large language models have grown larger, interest has grown in low-precision numerical formats such as NVFP4 as a way to improve speed and reduce memory usage. However, quantizing models to NVFP4 remains difficult as the lack of precision generally degrades model performance. In this work, we address this issue with Four Over Six (4/6), a modification to the block-scaled NVFP4 quantization algorithm that yields reduced quantization error. Unlike integer formats, floating point formats have non-uniform step sizes which create larger quantization error on larger values. 4/6 takes advantage of this by adaptively scaling some blocks to smaller FP4 values, making the distribution of representable values more uniform and reducing quantization error for near-maximal values. We show that 4/6 can be implemented efficiently on NVIDIA Blackwell GPUs, resulting in performance gains during both pre-training and inference with minimal computational overhead. In pre-training experiments with the Nemotron 3 Nano 30B-A3B model architecture, we find that 4/6 brings training loss closer to BF16 compared to models trained with current state-of-the-art NVFP4 training recipes. Our code is available at http://github.com/mit-han-lab/fouroversix.

Serving many task-specialized LLM variants is often limited by the large size of fine-tuned checkpoints and the resulting cold-start latency. Since fine-tuned weights differ from their base model by relatively small structured residuals, a natural approach is to represent them as compressed deltas. We propose a simple 1-bit delta scheme that stores only the sign of the weight difference together with lightweight per-axis (row/column) FP16 scaling factors, learned from a small calibration set. This design preserves the compactness of 1-bit deltas while more accurately capturing variation across weight dimensions, leading to improved reconstruction quality over scalar alternatives. From a systems perspective, a streamlined loader that transfers packed deltas in a single operation per module reduces cold-start latency and storage overhead, with artifacts several times smaller than a full FP16 checkpoint. The method is drop-in, requires minimal calibration data, and maintains inference efficiency by avoiding dense reconstruction. Our experimental setup and source code are available at https://github.com/kuiumdjiev/Per-Axis-Weight-Deltas-for-Frequent-Model-Updates.

In aerodynamics, accurately modeling subsonic compressible flow over airfoils is critical for aircraft design. However, solving the governing nonlinear perturbation velocity potential equation presents computational challenges. Traditional approaches often rely on linearized equations or finite, truncated domains, which introduce non-negligible errors and limit applicability in real-world scenarios. In this study, we propose a novel framework utilizing Physics-Informed Neural Networks (PINNs) to solve the full nonlinear compressible potential equation in an unbounded (infinite) domain. We address the unbounded-domain and convergence challenges inherent in standard PINNs by incorporating a coordinate transformation and embedding physical asymptotic constraints directly into the network architecture. Furthermore, we employ a Multi-Stage PINN (MS-PINN) approach to iteratively minimize residuals, achieving solution accuracy approaching machine precision. We validate this framework by simulating flow over circular and elliptical geometries, comparing our results against traditional finite-domain and linearized solutions. Our findings quantify the noticeable discrepancies introduced by domain truncation and linearization, particularly at higher Mach numbers, and demonstrate that this new framework is a robust, high-fidelity tool for computational fluid dynamics.

The study emphasizes the challenge of finding the optimal trade-off between bias and variance, especially as hyperparameter optimization increases in complexity. Through empirical analysis, three hyperparameter tuning algorithms Tree-structured Parzen Estimator (TPE), Genetic Search, and Random Search are evaluated across regression and classification tasks. The results show that nonlinear models, with properly tuned hyperparameters, significantly outperform linear models. Interestingly, Random Search excelled in regression tasks, while TPE was more effective for classification tasks. This suggests that there is no one-size-fits-all solution, as different algorithms perform better depending on the task and model type. The findings underscore the importance of selecting the appropriate tuning method and highlight the computational challenges involved in optimizing machine learning models, particularly as search spaces expand.

Numerical simulations in climate, chemistry, or astrophysics are computationally too expensive for uncertainty quantification or parameter-exploration at high-resolution. Reduced-order or surrogate models are multiple orders of magnitude faster, but traditional surrogates are inflexible or inaccurate and pure machine learning (ML)-based surrogates too data-hungry. We propose a hybrid, flexible surrogate model that exploits known physics for simulating large-scale dynamics and limits learning to the hard-to-model term, which is called parametrization or closure and captures the effect of fine- onto large-scale dynamics. Leveraging neural operators, we are the first to learn grid-independent, non-local, and flexible parametrizations. Our multiscale neural operator is motivated by a rich literature in multiscale modeling, has quasilinear runtime complexity, is more accurate or flexible than state-of-the-art parametrizations and demonstrated on the chaotic equation multiscale Lorenz96.

The performance of an algorithm often critically depends on its parameter configuration. While a variety of automated algorithm configuration methods have been proposed to relieve users from the tedious and error-prone task of manually tuning parameters, there is still a lot of untapped potential as the learned configuration is static, i.e., parameter settings remain fixed throughout the run. However, it has been shown that some algorithm parameters are best adjusted dynamically during execution, e.g., to adapt to the current part of the optimization landscape. Thus far, this is most commonly achieved through hand-crafted heuristics. A promising recent alternative is to automatically learn such dynamic parameter adaptation policies from data. In this article, we give the first comprehensive account of this new field of automated dynamic algorithm configuration (DAC), present a series of recent advances, and provide a solid foundation for future research in this field. Specifically, we (i) situate DAC in the broader historical context of AI research; (ii) formalize DAC as a computational problem; (iii) identify the methods used in prior-art to tackle this problem; (iv) conduct empirical case studies for using DAC in evolutionary optimization, AI planning, and machine learning.

Slater-type orbitals (STOs) provide the physically correct description of atomic wavefunctions but have been largely replaced by Gaussian-type orbitals in computational chemistry due to the lack of closed-form multi-center integrals. We present a systematic study of amplitude encoding of STOs on quantum computers using matrix product states (MPS). For one-dimensional orbital functions of the form p_d(x) e^{-ζx}, we derive analytical MPS constructions with constant bond dimension χ= d + 1, requiring O(n) classical and quantum resources for n-qubit registers with no grid sampling. We demonstrate a complete one-electron integral pipeline -- overlap, kinetic energy, and nuclear attraction -- in one dimension, validating the overlap and kinetic energy on IBM Heron processors at 5~qubits with 0.67\% hardware-induced error using Zero-Noise Extrapolation. In three dimensions, we compute multi-center overlap integrals between 1s and 2s orbitals in Cartesian coordinates with 0.02\% discretization error at 18~qubits. A systematic entanglement analysis reveals that the MPS bond dimension of three-dimensional STOs in Cartesian coordinates saturates with increasing grid resolution -- reaching sim138 for the hydrogen 1s orbital at 12~qubits per coordinate -- establishing bounded encoding complexity rather than the exponential scaling initially expected. The SVD truncation threshold provides a practical resource parameter, reducing the bond dimension to 39 at threshold 10^{-6} with negligible accuracy loss. These results map the entanglement landscape for amplitude encoding of atomic orbitals and establish MPS-based state preparation as a viable path toward exact STO basis sets on quantum computers.

Fine-tuning a large language model on new data degrades what it previously learned. We present Omega-S, a drop-in penalty computed from the weight matrix alone: it needs no previous-task data, no Fisher matrix and no stored copy of the old weights. It is three lines in an existing training loop and adds under 4% to the cost of a step. Retention. On Llama-3-8B with LoRA, fine-tuned from code to prose and measured by HumanEval over ten seeds, Omega-S retains more of the original capability than no regularisation on 9 of 10 seeds (0.173 -> 0.238 absolute pass@1; sign test one-sided p=0.011, Wilcoxon p=0.006), as a retention ratio, 62.9% -> 84.1%. It also beats tuned weight decay on 10 of 10 seeds (p=0.002) and tuned EWC on 8 of 10 (p=0.014), every arm re-measured in the same session. Mechanism, measured rather than asserted. Omega-S is topological by construction, its objective built from Tr(A^3), but we measured which of its four factors actually moves and three do not: their elasticity with respect to the weights is at or below 1e-4, against 9e-3 for the degree-variance term. As implemented, the composite reduces to a penalty on the variance of node degrees, which means row magnitude in square modules and directional alignment in non-square ones. We report this because a method whose name promises one thing and whose gradient does another should say so. We also enumerate the open design choices, including a contrast-preserving construction that does what it was designed to do and makes retention worse on all ten seeds. Repeating an identical configuration, same seed and same hardware, gives a standard deviation of 0.104 in retention ratio. We have not found this quantified for low-rank fine-tuning of language models, and it bounds every seed-paired comparison in this literature, ours included. Code, per-seed results and the full record of negative results are available.

Benchmarks for LLM-generated GPU kernels (KernelBench, TritonBench, GEAK) score correctness through fixed-shape, small-sample allclose-style checks. The number of inputs varies between benchmarks. The shape, dtype, and tolerance are fixed for each kernel. We test that oracle empirically. We construct a controlled corpus of 24 Triton and CPU stand-in kernels (15 correct controls and 9 LLM-style buggy variants seeded with documented transcription errors) and re-evaluate it under op-schema-aware seeded fuzzing with a high-precision (fp64) CPU reference and per-(op, dtype) absolute tolerances. The seeded oracle flags 9 of 9 buggy kernels and passes 15 of 15 correct controls, at zero precision cost on controls. We extend the corpus to 26 ops (adding a flash-attention pair) and re-run the same protocol on five GPU classes (RTX 3060, A10, L40S, A100 SXM4, H100 NVL). The verdicts are identical across all five GPUs: 10 of 10 illusions caught and 16 of 16 controls clean. The corpus result is about LLM-style transcription bugs that the allclose-on-one-shape oracle certifies as correct, not about the bug rate of any specific deployed LLM. Every flagged failure replays byte-for-byte from a stored seed.

The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon's efficiency relies on fast, approximate orthogonalization, yet all prior theoretical work analyzes an idealized, computationally intractable version assuming exact SVD-based updates. This work moves beyond the ideal by providing the first analysis of the inexact orthogonalized update at Muon's core. We develop our analysis within the general framework of Linear Minimization Oracle (LMO)-based optimization, introducing a realistic additive error model to capture the inexactness of practical approximation schemes. Our analysis yields explicit bounds that quantify performance degradation as a function of the LMO inexactness/error. We reveal a fundamental coupling between this inexactness and the optimal step size and momentum: lower oracle precision requires a smaller step size but larger momentum parameter. These findings elevate the approximation procedure (e.g., the number of Newton-Schulz steps) from an implementation detail to a critical parameter that must be co-tuned with the learning schedule. NanoGPT experiments directly confirm the predicted coupling, with optimal learning rates clearly shifting as approximation precision changes.

Large Language Models (LLMs) are now integral across various domains and have demonstrated impressive performance. Progress, however, rests on the premise that benchmark scores are both accurate and reproducible. We demonstrate that the reproducibility of LLM performance is fragile: changing system configuration such as evaluation batch size, GPU count, and GPU version can introduce significant difference in the generated responses. This issue is especially pronounced in reasoning models, where minor rounding differences in early tokens can cascade into divergent chains of thought, ultimately affecting accuracy. For instance, under bfloat16 precision with greedy decoding, a reasoning model like DeepSeek-R1-Distill-Qwen-7B can exhibit up to 9% variation in accuracy and 9,000 tokens difference in response length due to differences in GPU count, type, and evaluation batch size. We trace the root cause of this variability to the non-associative nature of floating-point arithmetic under limited numerical precision. This work presents the first systematic investigation into how numerical precision affects reproducibility in LLM inference. Through carefully controlled experiments across various hardware, software, and precision settings, we quantify when and how model outputs diverge. Our analysis reveals that floating-point precision -- while critical for reproducibility -- is often neglected in evaluation practices. Inspired by this, we develop a lightweight inference pipeline, dubbed LayerCast, that stores weights in 16-bit precision but performs all computations in FP32, balancing memory efficiency with numerical stability. Code is available at https://github.com/nanomaoli/llm_reproducibility.

We present an end-to-end pipeline for estimating stellar parameters from Sloan Digital Sky Survey Data Release 12 spectra using a fully connected multitask neural network with residual blocks, whose hyperparameters are tuned via Bayesian optimization. The preprocessing pipeline includes per-spectrum standardization, RobustScaler normalization of the target variables -- effective temperature T_{eff}, metallicity [Fe/H], and surface gravity log g -- and data augmentation via Gaussian noise injection. On a held-out test set, the model achieved Mean Absolute Errors (MAE) of 59.76~K for T_{eff}, 0.103~dex for [Fe/H], and 0.130~dex for log g. Normalized against the full-scale range of each parameter, these results represent range-normalized errors between 1% and 3%, achieved with a highly efficient model complexity of approximately 540,000 trainable parameters. These results demonstrate that a compact residual multitask architecture, combined with principled signal preprocessing, provides a parameter-efficient solution for nonlinear parameter estimation in large-scale spectral datasets. In particular, the proposed model achieves competitive performance with substantially lower complexity than deeper neural network baselines.

We tackle the problem of estimating a Manhattan frame, i.e. three orthogonal vanishing points, and the unknown focal length of the camera, leveraging a prior vertical direction. The direction can come from an Inertial Measurement Unit that is a standard component of recent consumer devices, e.g., smartphones. We provide an exhaustive analysis of minimal line configurations and derive two new 2-line solvers, one of which does not suffer from singularities affecting existing solvers. Additionally, we design a new non-minimal method, running on an arbitrary number of lines, to boost the performance in local optimization. Combining all solvers in a hybrid robust estimator, our method achieves increased accuracy even with a rough prior. Experiments on synthetic and real-world datasets demonstrate the superior accuracy of our method compared to the state of the art, while having comparable runtimes. We further demonstrate the applicability of our solvers for relative rotation estimation. The code is available at https://github.com/cvg/VP-Estimation-with-Prior-Gravity.

We introduce featom, an open source code that implements a high-order finite element solver for the radial Schr\"odinger, Dirac, and Kohn-Sham equations. The formulation accommodates various mesh types, such as uniform or exponential, and the convergence can be systematically controlled by increasing the number and/or polynomial order of the finite element basis functions. The Dirac equation is solved using a squared Hamiltonian approach to eliminate spurious states. To address the slow convergence of the kappa=pm1 states due to divergent derivatives at the origin, we incorporate known asymptotic forms into the solutions. We achieve a high level of accuracy (10^{-8} Hartree) for total energies and eigenvalues of heavy atoms such as uranium in both Schr\"odinger and Dirac Kohn-Sham solutions. We provide detailed convergence studies and computational parameters required to attain commonly required accuracies. Finally, we compare our results with known analytic results as well as the results of other methods. In particular, we calculate benchmark results for atomic numbers (Z) from 1 to 92, verifying current benchmarks. We demonstrate significant speedup compared to the state-of-the-art shooting solver dftatom. An efficient, modular Fortran 2008 implementation, is provided under an open source, permissive license, including examples and tests, wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines.

The massive computational costs associated with large language model (LLM) pretraining have spurred great interest in reduced-precision floating-point representations to accelerate the process. As a result, the BrainFloat16 (BF16) precision has become the de facto standard for LLM training, with hardware support included in recent accelerators. This trend has gone even further in the latest processors, where FP8 has recently been introduced. However, prior experience with FP16, which was found to be less stable than BF16, raises concerns as to whether FP8, with even fewer bits than FP16, can be a cost-effective option for LLM training. We argue that reduced-precision training schemes must have similar training stability and hyperparameter sensitivities to their higher-precision counterparts in order to be cost-effective. However, we find that currently available methods for FP8 training are not robust enough to allow their use as economical replacements. This prompts us to investigate the stability of reduced-precision LLM training in terms of robustness across random seeds and learning rates. To this end, we propose new evaluation techniques and a new metric for quantifying loss landscape sharpness in autoregressive language models. By simulating incremental bit reductions in floating-point representations, we analyze the relationship between representational power and training stability with the intent of aiding future research into the field.

Sampling is a basic operation in many inference-time algorithms of large language models (LLMs). To scale up inference efficiently with a limited compute, it is crucial to find an optimal allocation for sample compute budgets: Which sampling configurations (model, temperature, language, etc.) do we use? How many samples do we generate in each configuration? We formulate these choices as a learning problem and propose OSCA, an algorithm that Optimizes Sample Compute Allocation by finding an optimal mix of different inference configurations. Our experiments show that with our learned mixed allocation, we can achieve accuracy better than the best single configuration with 128x less compute on code generation and 25x less compute on 4 reasoning tasks. OSCA is also shown to be effective in agentic workflows beyond single-turn tasks, achieving a better accuracy on SWE-Bench with 3x less compute than the default configuration. Our code and generations are released at https://github.com/LeiLiLab/OSCA.

High-precision modeling of systems is one of the main areas of industrial data analysis. Models of systems, their digital twins, are used to predict their behavior under various conditions. We have developed several models of a storage system using machine learning-based generative models. The system consists of several components: hard disk drive (HDD) and solid-state drive (SSD) storage pools with different RAID schemes and cache. Each storage component is represented by a probabilistic model that describes the probability distribution of the component performance in terms of IOPS and latency, depending on their configuration and external data load parameters. The results of the experiments demonstrate the errors of 4-10 % for IOPS and 3-16 % for latency predictions depending on the components and models of the system. The predictions show up to 0.99 Pearson correlation with Little's law, which can be used for unsupervised reliability checks of the models. In addition, we present novel data sets that can be used for benchmarking regression algorithms, conditional generative models, and uncertainty estimation methods in machine learning.

Generating 2-by-2 unitary matrices in floating-precision arithmetic is a delicate task. One way to reduce the accumulation error is to use less floating-point operations to compute each of the entries in the 2-by-2 unitary matrix. This paper shows an algorithm that reduces the number of operations to compute the entries of a Givens rotation. Overall, the new algorithm has more operations in total when compared to algorithms in different releases of LAPACK, but less operations per entry. Numerical tests show that the new algorithm is more accurate on average.

We present a hardware-oriented description of GoldenFloat (GF), a static-split floating-point family generated by a single closed rule, and three concrete artefacts: (i) an open multi-width RTL generator covering GF4-GF256 with a continuous-integration differential sweep against a correctly-rounded reference; (ii) an integer-backed Lucas-exact accumulator path verified at 500-digit precision for n = 1, ..., 256; and (iii) a GF16 FPGA codec passing a 35-of-35 testbench at 323 MHz on Artix-7 (Xilinx XC7A35T). For each total width N >= 4, the exponent width is e = round((N-1)/phi^2) with fraction f = N-1-e and phi = (1+sqrt(5))/2. The rule reproduces the realised exponent widths of nine formats (9/9) and extends consistently to GF128, GF512, GF1024. The rule is positioned alongside posit, takum, OCP-MX, and the IEEE P3109 multi-width float draft. We make no per-rung accuracy or superiority claim against any of them. The breadth/toolchain-coherence framing is recorded as an open conjecture with a pre-registered falsification path. A falsification ledger (FL-002) records open questions and the experiments that would settle them. An RTL-correctness erratum dated 2026-05-31 is reported; the fabricated TTSKY26b dies carry the defective multiplier portfolio, and the corrected generator is the regeneration baseline.