ICML 2026 Papers

MARS (Yuan et al., 2025) has recently emerged as a strong optimizer for large language model (LLM) training by scaling the correction term in momentum-based variance reduction (MVR). However, existing theory does not explain why this modification can improve convergence over the unscaled MVR choice $\gamma=1$. In this paper, we provide a theoretical explanation for this phenomenon. We introduce **$\gamma$-similarity**, a refined similarity condition that captures how the scaling coefficient interacts with the stochastic gradient-difference structure. This condition recovers standard similarity at $\gamma=1$ and smoothness at $\gamma=0$. Using $\gamma$-similarity, we derive convergence guarantees for fixed-$\gamma$ MARS whose complexity depends explicitly on $\gamma$ and the corresponding $\gamma$-similarity constant. The bound reveals why small values of $\gamma$ can be beneficial: they may reduce the similarity term enough to outweigh the penalty from deviating from MVR. We prove that optimizing $\gamma$ gives MARS a lower complexity guarantee than MVR. Experiments with MARS-AdamW on GPT-style LLM pretraining corroborate the theory, showing that properly chosen small values of $\gamma$ improve token efficiency over $\gamma=1$ and AdamW under a fixed training protocol.

Recent research has established empirical scaling laws to predict model performance on multi-domain data mixtures. However, a theoretical understanding of these model loss behaviors remains absent. In this work, we propose a unified framework to explain the underlying mechanics of data mixing. Our approach extends theoretical perspectives originally developed for standard neural scaling laws (e.g., Kaplan and Chinchilla) to the multi-domain setting. Based on the distributional assumption that domains overlap on fundamental skills while diverging on specialized skills, we identify two key factors that govern the domain losses of models trained on different data mixtures: (1) Capacity Competition, where the allocation of finite model capacity couples domain losses globally, and (2) Noise Reduction, where optimal weights shift toward harder-to-learn domains to minimize overall noise. Empirical evaluations show that our framework outperforms existing baselines by fitting the loss landscape with a lower Mean Relative Error and identifying higher-performing training mixtures. Most importantly, our model successfully extrapolates across scales, predicting highly effective mixtures for large, unseen scales using parameters fitted on smaller ones. In addition, our model achieves these results using significantly fewer parameters compared to previous empirical laws.

Reasoning is a core capability of language models (LMs), yet it remains unclear how much model capacity is necessary to support reasoning during pretraining. In this work, we study the minimal parameter budget required for implicit reasoning, defined as the ability to infer new facts from learned knowledge without explicit chain-of-thought supervision. To isolate this phenomenon, we pretrain LMs from scratch in a controlled synthetic environment that mimics the structure and distribution of real-world knowledge graphs, and evaluate their ability to complete missing edges via multi-hop inference. From both a theoretical and an empirical perspective, we identify a scaling law linking this optimal parameter budget to a graph search entropy measure. Across a wide range of model sizes, training steps, and graph complexities, we show that an optimally sized language model can reliably reason over approximately 0.008 bits of information per parameter at most. Our results characterize the minimal sufficient capacity for implicit reasoning during pretraining. Our findings provide principled guidance for matching model size to data complexity and offer new insights into the scaling behavior of reasoning in large language models.

Scaling factors in residual branches have emerged as a prevalent method for boosting neural network performance, especially in normalization-free architectures. While prior work has primarily examined scaling effects from an optimization perspective, this paper investigates their role in residual architectures through the lens of generalization theory. Specifically, we establish that wide residual networks (ResNets) with constant scaling factors become asymptotically unlearnable as depth increases. In contrast, when the scaling factor exhibits rapid depth-wise decay combined with early stopping, over-parameterized ResNets achieve minimax-optimal generalization rates. To establish this, we demonstrate that the generalization capability of wide ResNets can be approximated by the kernel regression associated with a specific kernel. Our theoretical findings are validated through experiments on synthetic data and real-world classification tasks, including MNIST and CIFAR-100.

Large language models (LLMs) demand substantial computational and memory resources, creating deployment challenges. Quantization-aware training (QAT) addresses these challenges by reducing model precision while maintaining performance. However, the scaling behavior of QAT, especially at 4-bit precision (W4A4), is not well understood. Existing QAT scaling laws often ignore key factors such as the number of training tokens and quantization granularity, which limits their applicability. This paper proposes a unified scaling law for QAT that models quantization error as a function of model size, training data volume, and quantization group size. Through 268 QAT experiments, we show that quantization error decreases as model size increases, but rises with more training tokens and coarser quantization granularity. To identify the sources of W4A4 quantization error, we decompose it into weight and activation components. Both components follow the overall trend of W4A4 quantization error, but with different sensitivities. Specifically, weight quantization error increases more rapidly with more training tokens. Further analysis shows that the activation quantization error in the FC2 layer, caused by outliers, is the primary bottleneck of W4A4 QAT quantization error. By applying mixed-precision quantization to address this bottleneck, we demonstrate that weight and activation quantization errors can converge to similar levels. Additionally, with more training data, weight quantization error eventually exceeds activation quantization error, suggesting that reducing weight quantization error is also important in such scenarios. These findings offer key insights for improving QAT research and development.

Upweighting high-quality data in LLM pretraining often improves performance, but in data-limited regimes, especially under overtraining, stronger upweighting increases repetition and can degrade performance. However, standard scaling laws do not reliably extrapolate across mixture recipes or under repetitions, making the selection for optimal data recipes at scaling underdetermined. To solve this, we introduce \textbf{InfoLaw} (Information Scaling Laws), a data-aware scaling framework that predicts loss from consumed tokens, model size, data mixture weights, and repetition. The key idea is to model pretraining as information accumulation, where quality controls information density and repetition induces scale-dependent diminishing returns. We first collect the model performance after training on datasets that vary in scale, quality distribution, and repetition level. Then we build up the modeling for information so that information accurately predicts those model performance. InfoLaw predicts performance on unseen data recipes and larger-scale runs (up to 7B, 425B tokens) with 0.15\% mean and 0.96\% max absolute error in loss, and it extrapolates reliably across overtraining levels, enabling efficient data-recipe selection under varying compute budgets.

Data-driven models are revolutionizing weather forecasting. To optimize training efficiency and model performance, this paper analyzes empirical scaling laws within this domain. We investigate the relationship between model performance (validation loss) and three key factors: model size ($N$), dataset size ($D$), and compute budget ($C$). Across a range of models, we find that Aurora exhibits the strongest data-scaling behavior: increasing the training dataset by 10× reduces validation loss by up to 3.2×. GraphCast demonstrates the highest parameter efficiency, yet suffers from limited hardware utilization. Our compute-optimal analysis indicates that, under fixed compute budgets, allocating resources to more total training data yields greater performance gains than increasing model size. Furthermore, we analyze model shape and uncover scaling behaviors that differ fundamentally from those observed in language models: weather forecasting models consistently favor increased width over depth. These findings suggest that future weather models should prioritize wider architectures and larger effective training datasets to maximize predictive performance.

Efficient and stable training of large language models (LLMs) remains a core challenge in modern machine learning systems. To address this challenge, Reparameterized Orthogonal Equivalence Training (POET), a spectrum-preserving framework that optimizes each weight matrix through orthogonal equivalence transformation, has been proposed. Although POET provides strong training stability, its original implementation incurs high memory consumption and computational overhead due to intensive matrix multiplications. To overcome these limitations, we introduce POET-X, a scalable and memory-efficient variant that performs orthogonal equivalence transformations with significantly reduced computational cost. POET-X maintains the generalization and stability benefits of POET while achieving substantial improvements in throughput and memory efficiency. In our experiments, POET-X enables the pretraining of billion-parameter LLMs on a single Nvidia H100 GPU, and in contrast, standard optimizers such as AdamW run out of memory under the same settings.

Temperature scaling is widely used to control generation behavior and improve reasoning performance in discrete sequence models. However, temperature scaling at the sequence level is particularly challenging since the partition function of the model is typically not tractable and naive *token-wise* temperature scaling induces bias in the maximum a posteriori (MAP) estimates. Discrete diffusion settings exacerbate this issue: enhanced flexibility due to multiple potential orderings for generating tokens complicates likelihood computation and incurs high variance. We propose and investigate fine-tuning objectives for sequence-level temperature scaling, along with metrics to quantify the success of a given temperature scaling procedure. Specifically, our contributions are threefold. First, we introduce our objective across a wide class of sequence models through a novel formalism that unifies autoregressive and diffusion-based language models. Second, we design two metrics that quantify temperature scaling based on likelihood *ratios* and thus obviate access to the partition function.  Finally, we introduce fine-tuning objectives that reliably achieve the desired change in model temperature. Our experiments with language models (GPT2, BD3LM) show that the proposed approach leads to more consistent generation with lower perplexity.  Furthermore, we provide empirical evidence that it can enhance the reasoning performance in language models. Code is available at [github.com/Aalto-QuML/temp_lm](https://github.com/Aalto-QuML/temp_lm).

Understanding gradient descent dynamics is key to explaining the success of over-parameterized models, where implicit bias manifests through conservation laws in gradient flow. While such laws are well understood for linear and ReLU networks, they remain largely unexplored for modern architectures. This work develops a unified framework to characterize conservation laws for contemporary models, including feedforward networks with GELU, SiLU, and SwiGLU activations, multihead attention with sinusoidal and rotary positional encodings, and Mixture-of-Experts architectures under diverse gating designs. Our theoretical findings are supported by experiments that validate the predicted invariants.

Typical reinforcement learning (RL) methods for LLM reasoning waste compute on hard problems, where correct on-policy traces are rare and policy gradients vanish. To bootstrap more efficient RL, we consider reusing old sampling FLOPs (from prior inference or RL training) in the form of off-policy traces. We introduce PrefixRL, where we condition on the prefix of successful off-policy traces and run on-policy RL to complete them, side-stepping instabilities from using off-policy data as supervision targets. PrefixRL boosts the learning signal on hard problems by modulating the difficulty of the problem through the off-policy prefix length. We prove that the PrefixRL objective is not only consistent with the standard RL objective but also more sample efficient. Empirically, we discover back-generalization: training only on prefixed problems generalizes to out-of-distribution unprefixed performance, with learned strategies often differing from those in the prefix. In our experiments, we source the off-policy traces by rejection sampling with the base model, creating a self-improvement loop. On hard reasoning problems, PrefixRL reaches the same training reward 2x faster than the strongest baseline (SFT on off-policy data then RL), even after accounting for the compute spent on the initial rejection sampling, and increases the final reward by 3x.

As the cost of pretraining large language models grows, there is continued interest in strategies to improve learning efficiency during this core training stage. Motivated by cognitive development, where humans gradually build knowledge as their brains mature, we propose Curriculum-Guided Layer Scaling (CGLS), a framework for compute-efficient pretraining that synchronizes increasing data difficulty with model growth through progressive layer stacking (i.e. gradually adding layers during training). At the 100M parameter scale, using a curriculum transitioning from synthetic short stories to general web data, CGLS outperforms baseline methods on the question-answering benchmarks PIQA and ARC. Pretraining at the 1.2B scale, we stratify the DataComp-LM corpus with a DistilBERT-based classifier and progress from general text to highly technical or specialized content. Our results show that progressively increasing model depth alongside sample difficulty leads to better generalization and zero-shot performance on various downstream benchmarks. Altogether, our findings demonstrate that CGLS unlocks the potential of progressive stacking, offering a simple yet effective strategy for improving generalization on knowledge-intensive and reasoning tasks.

Conventional wisdom holds that large-batch training is fundamentally incompatible with Reinforcement Learning (RL) — beyond a modest threshold, increasing batch sizes typically yields diminishing returns or performance degradation due to the inherent non-stationarity of the data distribution. We challenge this view by observing that non-stationarity is not a fixed property of RL, but evolves throughout training: early stages exhibit rapid behavioral shifts that demand small batches for plasticity, whereas late stages approach a quasi-stationary regime where large batches enable precise convergence. Motivated by this observation, we propose Adaptive Batch Scaling (ABS), that dynamically adjusts the effective batch size according to the stability of the learning policy. Central to ABS is Behavioral Divergence, a novel metric that quantifies policy non-stationarity by measuring action-level shifts between consecutive updates, which we use to scale batch size inversely to policy volatility. Integrated with the Parallelised Q-Network (PQN) algorithm and evaluated on the ALE benchmark, ABS seamlessly reconciles early-stage plasticity with late-stage stable convergence. Strikingly, contrary to conventional wisdom, our results reveal that the combination of larger networks and larger batch sizes achieves the best performance — a scaling behavior previously thought to be unattainable in RL, now unlocked through adaptive batch control.

We develop a theory of generalization and scaling for Mixture-of-Experts (MoE) Transformers that cleanly separates active per-input capacity from routing combinatorics. By conditioning on fixed routing patterns and union-bounding across them, we derive a sup-norm covering-number bound whose metric entropy scales with the active parameter budget and incurs a MoE-specific routing overhead. Combined with a standard ERM analysis for squared loss, this yields a generalization bound under a $d$-dimensional manifold data model and $C^\beta$ targets, showing that approximation and estimation trade off as in dense networks once active parameters are accounted for appropriately. We further prove a constructive approximation theorem for MoE architectures, showing that, under the approximation construction, error can decrease either by scaling active capacity or by increasing the number of experts, depending on the dominant bottleneck. From these results we derive neural scaling laws for model size, data size, and compute-optimal tradeoffs. Overall, our results provide a transparent statistical reference point for reasoning about MoE scaling, clarifying which behaviors are certified by worst-case theory and which must arise from data-dependent routing structure or optimization dynamics.

Model depth is a double-edged sword in deep learning: deeper models achieve higher accuracy but require higher computational cost. To efficiently train models at scale, progressive training (also known as model expansion) scales up model capacity during training and significantly reduces computation with little performance degradation. In this work, we study the depth expansion of large-scale models through the lens of optimization theory and feature learning, offering insights on the initialization of new layers, hyperparameter transfer, learning rate schedule, and timing of model expansion. Specifically, we propose zero/one-layer progressive training to achieve an optimal tradeoff between computation and loss, with a comprehensive ablations on our expansion strategy. For example, zero/one-layer progressive training on GPT2 can save $\approx 80\%$ compute, or equivalently achieve an $\approx 5\times$ acceleration, while attaining a loss comparable to that of a fully trained 60-layer model with 7B parameters, thus demonstrating a mixing behavior in terms of loss. Furthermore, scaling laws on LLAMA3 and DeepSeekV3 models show a $3\sim 5\times$ improvement in compute efficiency, with an increasing advantage at larger scales.

Many reinforcement learning (RL) problems admit multiple terminal solutions of comparable quality, where the goal is not to identify a single optimum but to represent a diverse set of high-quality outcomes. Nevertheless, policies trained by standard expected-return maximization routinely collapse onto a small subset of outcomes, a phenomenon commonly attributed to insufficient exploration or weak regularization. We show that this explanation is incomplete: outcome-level mode collapse is a structural consequence of the expected-return objective itself. Under idealized learning dynamics, the log-probability ratio between any two outcomes evolves linearly in their reward difference, implying exponential ratio divergence and inevitable collapse—independent of the exploration strategy, entropy regularization, or optimization algorithm. We identify the source of this pathology as the probability multiplier inside the expectation and propose a minimal correction: inverse probability scaling, which removes outcome-frequency amplification from the learning signal, fundamentally changes the learning dynamics, and provably yields reward-proportional terminal distributions, preventing collapse in multimodal settings. We instantiate this principle in Group Relative Policy Optimization (GRPO) as a drop-in modification, IPS-GRPO, requiring no auxiliary models or architectural changes. Across different reasoning and molecular generation tasks, IPS-GRPO consistently reduces outcome-level mode collapse while matching or exceeding baseline performance, suggesting that correcting the objective rather than adding exploration heuristics is key to reliable multimodal policy optimization.

We study empirical scaling laws for language model merging measured by cross-entropy. Despite its wide practical use, merging lacks a quantitative rule that predicts returns as we add experts or scale the model size. We identify a compact power law that links model size and expert number: the size-dependent floor decreases with model capacity, while the merging tail exhibits clear diminishing returns in the number of experts. The law holds in-domain and cross-domain, tightly fits measured curves across diverse architectures and methods (Average, TA, TIES, DARE), and explains two robust regularities: most gains arrive early, and variability shrinks as more experts are included. Building on this, we present a simple theory that explains why gains fall roughly as \(1/k\) and links the floor and tail to properties of the base model and the diversity across domains. This law enables \emph{predictive planning}: estimate how many experts are needed to reach a target loss, decide when to stop adding experts, and trade off scaling the base model versus adding experts under a fixed budget—turning merging from heuristic practice into a computationally efficient, planable alternative to multitask training. This suggests a scaling principle for distributed generative AI: predictable gains can be achieved by composing specialists, offering a complementary path toward AGI-level systems.

Despite the fact that experimental neural scaling laws have substantially guided empirical progress in large-scale machine learning, no existing theory can quantitatively predict the exponents of these important laws for any modern LLM trained on any natural language dataset. We provide the first such theory in the case of data-limited scaling laws. We isolate two key statistical properties of language that _alone_ can predict neural scaling exponents: (i) the decay of pairwise token correlations with time separation between token pairs, and (ii) the decay of the next-token conditional entropy with the length of the conditioning context. We further derive a simple formula in terms of these statistics that predicts data-limited neural scaling exponents from first principles _without any_ free parameters or synthetic data models. Our theory exhibits a remarkable match with experimentally measured neural scaling laws obtained from training GPT-2 and LLaMA style models from scratch on two qualitatively different benchmarks, TinyStories and WikiText.

Spiking Neural Networks (SNNs) have garnered attention for their biological plausibility, energy efficiency, and temporal modeling capability. Due to the non-differentiability of spike generation, a widely used training method for SNNs is backpropagation through time with surrogate gradients, achieving competitive performance with few timesteps. Intuitively, scaling timesteps should improve performance by enriching temporal dynamics. However, we observe timestep scaling paradox (TSP), a counter-intuitive accuracy degradation when scaling timesteps. We investigate TSP and link it to long-term temporal gradient vanishing and weakened cross-timestep dependencies. To address this, we propose the Timestep-Scalable (TS) neuron model. It introduces long-term memory reconsolidation to enhance cross-timestep information flow and enable effective learning with more timesteps. In parallel, a temporal forgetting mechanism periodically truncates the accumulation path, suppressing excessive temporal buildup and improving training stability. Supported by theoretical analysis and extensive experiments, TS consistently improves performance when scaling timesteps. In addition, it attains state-of-the-art results on time-series and event-based tasks, while remaining strong on static image classification and object detection.