ICML 2026 Papers

While *Prover-Verifier Games* (PVGs) offer a promising path toward verifiability in nonlinear classification models, they have not yet been applied to complex inputs such as high-dimensional images. Conversely, expressive *concept encodings* effectively allow to translate such data into interpretable concepts but are often utilised in the context of low-capacity linear predictors. In this work, we push towards real-world verifiability by combining the strengths of both approaches. We introduce *Neural Concept Verifier (NCV)*, a unified framework combining PVGs for formal verifiability with concept encodings to handle complex, high-dimensional inputs in an interpretable way. NCV achieves this by utilizing recent minimally supervised concept discovery models to extract structured concept encodings from raw inputs. A *prover* then selects a subset of these encodings, which a *verifier*, implemented as a nonlinear predictor, uses exclusively for decision-making. Our evaluations show that NCV outperforms classic concept-based models and pixel-based PVG classifier baselines on high-dimensional, logically complex datasets and helps mitigate shortcut behavior. Overall, we demonstrate NCV as a promising step toward concept-level, verifiable AI.

Scaling laws provide a fundamental framework for understanding the performance of Language Models (LMs), yet deriving them requires prohibitively expensive evaluations across thousands of checkpoints or millions of inference samples. To address this, we introduce Item Response Scaling Laws (IRSL), a unified framework that integrates Item Response Theory (IRT) within the scaling law framework. Unlike traditional approaches that treat each model-benchmark pair in isolation, IRSL disentangles latent model ability from question characteristics, factorizing the scaling law estimation for $M$ models and $N$ questions to significantly reduce parameter complexity from $O(M \times N)$ to $O(M + N)$. We instantiate IRSL with Beta-IRT, which leverages the empirical probability responses of LMs---such as token probabilities in pre-training and pass rates in test-time sampling---to capture richer signals than binary responses. We validate our approach across two prevalent scaling paradigms: (1) pre-training downstream scaling, using 6,612 LM checkpoints and 37,682 questions from 10 benchmarks; and (2) test-time scaling, using 12 LMs and 120 questions from 4 benchmarks with up to 2,500 samples per question. Given a one-time calibration on existing model responses, IRSL yields more reliable scaling estimates using only 50 questions per benchmark (a 99.9\% reduction), achieving comparable or superior decision accuracy to traditional approaches. Furthermore, we show that the estimated latent model abilities are generalizable, enabling accurate performance forecasting across benchmarks that share the same measurement objective.

Deeper modern architectures are costly to tune, and the base learning rate is often one of the most sensitive hyperparameters. Maximal Update Parametrization ($\mu$P) helps explain why many hyperparameters transfer across width. Yet depthwise learning-rate scaling is less understood for modern architectures with convolution, residual aggregation, and attention. To unify various non-recurrent multi-path neural networks such as CNNs, ResNets, and Transformers, we introduce an architecture-dependent notion of effective depth. Under stabilizing initializations and a maximal-update criterion, we derive a shared leading-order -3/2 law for the base learning-rate scale as effective depth grows. Here, the budget controls typical one-step representation-update energy at initialization, and effective depth counts sequential update-bearing units while absorbing fixed local structure into constants. Experiments across diverse architectures confirm the predicted slope and enable reliable zero-shot transfer of learning rates across depths and widths, turning depth scaling into a predictable hyperparameter-transfer problem.

Physical systems near singularities, interfaces, and critical points exhibit power-law scaling, yet standard neural networks leave the governing exponents implicit. We introduce physics-informed Müntz-Szász Networks (MSN-PINN), a power-law basis network that treats scaling exponents as trainable parameters. The model outputs both the solution and its scaling structure. We prove identifiability, or unique recovery, and show that, under these conditions, the squared error between learned and true exponents scales as $O(|\mu - \alpha|^2)$. Across experiments, MSN-PINN achieves single-exponent recovery with 1–5\% error under noise and sparse sampling. It recovers corner singularity exponents for the two-dimensional Laplace equation with 0.009\% error, matches the classical result of Kondrat'ev (1967), and recovers forcing-induced exponents in singular Poisson problems with 0.03\% and 0.05\% errors. On a 40-configuration wedge benchmark, it reaches a 100\% success rate with 0.022\% mean error. Constraint-aware training encodes physical requirements such as boundary condition compatibility and improves accuracy by three orders of magnitude over naive training.

Language models achieve impressive performance on a variety of knowledge, language, and reasoning tasks due to the scale and diversity of pretraining data available. The standard training recipe is a two-stage paradigm: pretraining first on the full corpus of data followed by specialization on a subset of high quality, specialized data from the full corpus. In the multi-domain setting, this involves continued pretraining of multiple models on each specialized domain, referred to as split model training. We propose a method for pretraining multiple models independently over a general pretraining corpus, and determining the optimal compute allocation between pretraining and continued pretraining using scaling laws. Our approach accurately predicts the loss of a model of size $N$ with $D$ pretraining and $D'$ specialization tokens, and extrapolates to larger model sizes and number of tokens. Applied to language model training, our approach improves performance consistently across common sense knowledge and reasoning benchmarks across different model sizes and compute budgets.

While large language models (LLMs) show promise in scientific discovery, existing research focuses on inference or feedback-driven training, leaving the direct modeling of the generative reasoning process, $P(\text{hypothesis}|\text{background})$ ($P(h|b)$), unexplored. We demonstrate that directly training $P(h|b)$ is mathematically intractable due to the combinatorial complexity ($O(N^k)$) inherent in retrieving and composing inspirations from a vast knowledge base. To break this barrier, we introduce MOOSE-Star, a unified framework that enables tractable and scalable training of $P(h|b)$, while supporting more scalable inference. In the best case, MOOSE-Star reduces complexity from exponential to logarithmic ($O(\log N)$) by (1) training on decomposed subtasks derived from the probabilistic equation of discovery, (2) employing motivation-guided hierarchical search to enable logarithmic retrieval and prune irrelevant subspaces, and (3) utilizing bounded composition for robustness against retrieval noise. To facilitate this, we release TOMATO-Star, a dataset of 108,717 decomposed papers (38,400 GPU hours) for training. Empirically, MOOSE-Star scales continuously with training data and inference budget, whereas direct brute-force sampling hits a "complexity wall."

Inference-time scaling has recently emerged as a powerful paradigm for improving the reasoning capability of large language models. Among various approaches, \emph{Sequential Monte Carlo (SMC)} has become a particularly important framework, enabling iterative generation, evaluation, rejection, and resampling of intermediate reasoning trajectories. A central component in this process is the \emph{reward model}, which evaluates partial solutions and guides the allocation of computation during inference. However, in practice, true reward models are never available. All deployed systems rely on \emph{approximate reward models}, raising a fundamental question: \emph{Why and when do approximate reward models suffice for effective inference-time scaling?} In this work, we provide a theoretical answer. We identify the \emph{Bellman error} of the approximate reward model as the key quantity governing the effectiveness of SMC-based inference-time scaling. For a reasoning process of length $T$, we show that if the Bellman error of the approximate reward model is bounded by $O(1/T)$, then combining this reward model with SMC reduces the computational complexity of reasoning from exponential in $T$ to polynomial in $T$. This yields an \emph{exponential improvement} in inference efficiency despite using only approximate rewards.

Low-precision training is critical for optimizing the trade-off between model quality and training costs, necessitating the joint allocation of model size, dataset size, and numerical precision. While empirical scaling laws suggest that quantization impacts effective model and data capacities or acts as an additive error, the theoretical mechanisms governing these effects remain largely unexplored. In this work, we initiate a theoretical study of scaling laws for low-precision training within a high-dimensional sketched linear regression framework. By analyzing multiplicative (signal-dependent) and additive (signal-independent) quantization, we identify a critical dichotomy in their scaling behaviors. Our analysis reveals that in the worst case, while both schemes introduce an additive error and degrade the effective data size, they exhibit distinct effects on effective model size: multiplicative quantization maintains the full-precision model size, whereas additive quantization reduces the effective model size. Numerical experiments validate our theoretical findings. By rigorously characterizing the complex interplay among model scale, dataset size, and quantization error, our work provides a principled theoretical basis for optimizing training protocols under practical hardware constraints.

Language models famously improve under a smooth scaling law, but some specific capabilities exhibit sudden breakthroughs in performance. Advocates of "emergence" view these capabilities as unlocked at a specific scale, but others attribute breakthroughs to superficial metric thresholding effects. We propose that breakthroughs are instead driven by continuous changes in the probability distribution of training outcomes when performance is bimodally distributed across random seeds. We show that different random seeds can produce either smooth or emergent scaling trends in synthetic length generalization tasks, multiple choice question answering, and grammatical generalization. We reveal that sharp breakthroughs in metrics are produced by underlying continuous changes in their distribution across seeds. These distributions may become abruptly bimodal at a capacity threshold—but this threshold appears at scales well before most seeds achieve breakthrough. Our observations hold true even under continuous loss metrics, confirming that random variation must be considered when predicting a model's performance from its scale.

Verification of model outputs is rapidly emerging as a key primitive for both training and real-world deployment of large language models (LLMs). In practice, this often involves using imperfect LLM judges and reward models since ground truth acquisition can be time-consuming and expensive. We introduce Fully Unsupervised Score Ensembling (FUSE), a method for improving verification quality by ensembling verifiers without access to ground truth correctness labels. The key idea behind FUSE is to control conditional dependencies between verifiers in a manner that improves the unsupervised performance of a class of spectral algorithms from the ensembling literature. Despite requiring zero ground truth labels, FUSE typically matches or improves upon semi-supervised alternatives in test-time scaling experiments with diverse sets of generator models, verifiers, and benchmarks. In particular, we validate our method on both conventional academic benchmarks such as GPQA Diamond and on frontier, unsaturated benchmarks such as Humanity's Last Exam and IMO Shortlist questions.

Large and diverse datasets are needed for training generalist robot policies that can control a variety of robot embodiments--robot arm and gripper combinations--across diverse tasks and environments. As re-collecting demonstrations and retraining for each new embodiment are prohibitively costly, we study whether existing robot data can be augmented to improve transfer and generalization across embodiments. The Open X-Embodiment (OXE) dataset, which aggregates demonstrations from over 60 robot datasets, has been widely used for training generalist policies. However, it is highly imbalanced: the top four robot types account for over 85% of its real data, which risks overfitting to robot--scene combinations. We present AugE-Toolkit, a scalable robot augmentation pipeline, and OXE-AugE, a high-quality open-source dataset that augments OXE with 9 different robot embodiments. OXE-AugE provides over 4.4 million trajectories, more than triple the size of the original OXE. We conduct a systematic study of how scaling robot augmentation impacts cross-embodiment learning. Results suggest that augmenting datasets with diverse arms and grippers improves policy performance not only on the augmented robots, but also on unseen robots and even the original robots under distribution shifts. In physical experiments, fine-tuning generalist policies such as OpenVLA and $\pi_0$ on OXE-AugE improves success rates by 24-45% on unseen robot-gripper combinations across four real-world manipulation tasks. Project website: https://OXE-AugE.github.io/.

As context windows in large language models continue to expand, it is essential to characterize how attention behaves at extreme sequence lengths. We introduce token sample complexity: the rate at which attention computed on $n$ tokens converges to its infinite-token limit. We estimate finite-$n$ convergence bounds at two levels: pointwise uniform convergence of the attention map, and convergence of moments for the transformed token distribution. For compactly supported (and more generally sub-Gaussian) distributions, our first result shows that the attention map converges uniformly on a ball of radius $R$ at rate $C(R)/\sqrt{n}$, where $C(R)$ grows exponentially with $R$. For large $R$, this estimate loses practical value, and our second result addresses this issue by establishing convergence rates for the moments of the transformed distribution (the token output of the attention layer). In this case, the rate is $C'(R)/n^{\beta}$ with $\beta<\tfrac{1}{2}$, and $C'(R)$ depends polynomially on the size of the support of the distribution. The exponent $\beta$ depends on the attention geometry and the spectral properties of the token distribution. We also examine the regime in which the attention parameter tends to infinity and the softmax approaches a hardmax, and in this setting, we establish a logarithmic rate of convergence. Experiments on synthetic and real data support our predictions and show that the predicted slowdown is reflected in downstream accuracy.

Foundation models for genomics have the potential to revolutionize therapeutic design, yet the optimal architectural choices for modeling the vast and diverse distribution of metagenomic data remain under-explored. In this work, we present the machine learning methodology behind EDEN, a family of metagenomic foundation models scaled up to 28 billion parameters and trained on 9.7 trillion nucleotide tokens. We provide a systematic empirical study of architectural trade-offs between autoregressive Transformers (Llama-style), State-Space Models (Mamba), and Long-convolutional architectures (Hyena) for nucleotide-level modeling. Contrary to recent trends favoring linear-time sequence models for long-range biological data, we demonstrate that the Llama architecture exhibits superior scaling efficiency and semantic retrieval capabilities as the model capacity grows. We derive a set of quality-aware scaling laws for metagenomics, showing how model performance follows predictable power-law behavior across three orders of magnitude in parameters and data. Through extensive benchmarking, spanning unsupervised zeroshot fitness prediction, semantic completion, and gene recovery, we establish a blueprint for scaling biological foundation models and provide empirical evidence demonstrating why Transformer-based architectures define the current frontier.

Large Language Models (LLMs) represent a promising frontier for recommender systems, yet their development has been impeded by the absence of predictable scaling laws, which are crucial for guiding research and optimizing resource allocation. We hypothesize that this may be attributed to the inherent noise, bias, and incompleteness of raw user interaction data in prior continual pre-training (CPT) efforts. This paper introduces a novel, layered framework for generating high-quality synthetic data that circumvents such issues by creating a curated, pedagogical curriculum for the LLM. We provide powerful, direct evidence for the utility of our curriculum by showing that standard sequential models trained on our principled synthetic data significantly outperform ($+130\%$ on recall@100 for SasRec) models trained on real data in downstream ranking tasks, demonstrating its superiority for learning generalizable user preference patterns. Building on this, we empirically demonstrate, for the first time, robust power-law scaling for an LLM that is continually pre-trained on our high-quality, recommendation-specific data. Our experiments reveal consistent and predictable perplexity reduction across multiple synthetic data modalities. These findings establish a foundational methodology for reliable scaling LLM capabilities in the recommendation domain, thereby shifting the research focus from mitigating data deficiencies to leveraging high-quality, structured information.

Machine learning model performance arises from competition and application. For deployment, we consider the prescriptive scaling laws: given a pre-training compute budget, what downstream accuracy is attainable with contemporary post-training practice, and how stable is that mapping as the field evolves? Using large-scale observational evaluations with 5k observational and 2k newly sampled data on model performance, we estimate capability boundaries—high conditional quantiles of benchmark scores as a function of log pre-training FLOPs, via smoothed quantile regression with a monotone, saturating sigmoid parameterization. We validate the temporal reliability by fitting on earlier model generations and evaluating on later releases. Across various tasks, the estimated boundaries are mostly stable, with the exception of math reasoning that exhibits a consistently advancing boundary over time. We then extend our approach to analyze task-dependent saturation and to probe contamination-related shifts on math reasoning tasks. Finally, we introduce an efficient algorithm that recovers near-full-data frontiers using roughly 20% of evaluation budget. Together, our work releases the Proteus-2k, the latest model performance evaluation dataset, and introduces a practical methodology for translating compute budgets into reliable performance expectations and for monitoring when capability boundaries move.

Deep neural networks frequently exhibit overconfidence, undermining reliability in safety-critical applications. Existing adaptive methods rely on indirectly learned proxies of sample difficulty. We establish the logit margin as a direct and principled hardness indicator. We prove that margin constrains the feasible temperature range for a target confidence. Empirically, margin strongly correlates with decision boundary proximity and reveals systematic calibration patterns across difficulty levels. We further identify a fundamental flaw in NLL-based optimization: minimizing NLL can paradoxically worsen calibration. To address this, we introduce Charbonnier-SoftECE, a smooth objective that provably upper-bounds the smooth calibration error (smCE). Building on these insights, we propose SMART (Sample Margin-Aware Recalibration of Temperature), a lightweight method that learns a sample-wise margin-to-temperature mapping guided by our calibration-centric objective. Experiments demonstrate state-of-the-art calibration across CNNs and ViTs on standard, long-tailed, and distribution-shifted benchmarks, with minimal inference-time overhead. Code is available at: https://github.com/Misakaaaaaz/ICML2026-SMART.

We introduce a predictive model that estimates the pre-training loss of large models from model size ($N$), batch size ($B$) and number of weight updates ($K$). This is the first loss prediction model that can handle changing batch size. The model outperforms Chinchilla's loss model, a model of the test loss using the batch size and number of tokens, in terms of projecting the loss at extrapolated compute budgets (up to 1000 folds). A natural use of the model is to find optimal $N,B,K$ configurations under explicit and compound resource constraints like time, memory and compute. In our experiments, the model-selected configurations are close to ground-truth optimal. Our work advocates for loss prediction as a better alternative to heuristic-based laws, which are growing in complexity. The implementation is available on \href{https://github.com/chuningxdy/Noisy-Quadratic-System}{GitHub}.

Quantum hardware suffers from intrinsic device heterogeneity and environmental drift, forcing practitioners to choose between suboptimal non-adaptive controllers or costly per-device recalibration. We derive a scaling law lower bound for meta-learning showing that the adaptation gain (expected fidelity improvement from task-specific gradient steps) saturates exponentially with gradient steps and scales linearly with task variance, providing a quantitative criterion for when adaptation justifies its overhead. Validation on quantum gate calibration shows negligible benefits for low-variance tasks but > 40% fidelity gains on two-qubit gates under extreme out-of-distribution conditions (10× the training noise), with implications for reducing per-device calibration time on cloud quantum processors. Further validation on classical linear-quadratic control confirms these laws emerge from general optimization geometry rather than quantum-specific physics. We further introduce a few-shot pre-adaptation protocol that estimates the optimal adaptation budget from $N =3–5$ probe steps within 3–19% relative error across out-of-distribution regimes.

Vision Transformer (ViT) tokenizers offer a scal- able alternative to convolutional auto-encoders, yet current architectures have two key limitations: their performance degrades when images vary in aspect ratio or resolution, and their reliance on adversarial losses makes them harder to train at scale. To address this, we introduce ViTok-v2, a ViT tokenizer building on ViTok. We add native resolution support via NaFlex with 2D RoPE and stabilize training by replacing the standard LPIPS-plus-discriminator objective with our novel DINO perceptual loss. We scale our model to 5B parameters, training the largest ViT-based image compression auto- encoder to date and demonstrate continued im- provements with scale. In downstream generation experiments with flow matching models, we find that smaller generators perform best with aggres- sive channel compression while larger generators effectively leverage higher channel counts. ViTok- v2 matches state-of-the-art reconstruction at 256p and outperforms across benchmarks at 512p and higher resolutons, while remaining compatible with any pipeline requiring flexible aspect ratios.