NeurIPS 2026 Papers

Scaling up model parameters has long been a prevalent training paradigm driven by the assumption that larger models yield superior generation capabilities. However, under lossy context compression in a compressor--decoder setup, we find a \textbf{\textit{Size-Fidelity Paradox}}: increasing compressor size can lessen the faithfulness of reconstructed contexts though reconstruction error decreases. Across 27 compressor setups spanning model families, scales, and compression rates, we coin this paradox arising from two dominant factors: 1) \textit{knowledge overwriting}: larger models increasingly replace source facts with their own prior beliefs, \textit{e.g.}, ``the white strawberry`` $\to$ ``the red strawberry``; and 2) \textit{semantic drift}: larger models tend to paraphrase or restructure content instead of reproducing it verbatim, \textit{e.g.}, ``Alice hit Bob`` $\to$ ``Bob hit Alice``. Interestingly, this paradox persists across varied settings, with mid-sized compressors often outperforming larger ones in faithful recovery. By analyzing the compressed memory via embedding geometry and reconstruction determinacy, we further reveal that compressors tend to organize memory across broader semantic subspaces, yielding more ambiguous representations prone to overwriting, drift, and weakened recovery. These findings complement existing evaluations of context compression and expose a breakdown of scaling laws when the objective shifts from plausible generation to faithful preservation.

Large language models are increasingly deployed with test-time strategies: sample $N$ responses, score them with a reward model or verifier, and return the best. This deployment rule exposes a mismatch in post-training: standard objectives optimize the mean reward of a single response, whereas best-of-$N$ performance is governed by the upper tail of the reward distribution. Recent test-time-aware objectives partly address this mismatch, but typically assume that training can use the same per-prompt rollout budget as deployment, which is impractical when post-training must cover many prompts while deployment can allocate much larger per-prompt test-time compute. We study this budget-mismatch regime, where only $m\ll N$ per-prompt rollouts are available during training but the target objective is best-of-$N$ deployment. Under structural assumptions on the reward tails, we show that the policy gradient of the best-of-$N$ objective can be approximated from a much smaller rollout group by extrapolating upper-tail statistics. This yields a family of Tail-Extrapolated estimators for best-of-$N$-oriented post-training: a simple direct estimator, Tail-Extrapolated Advantage (TEA), and a fixed-order debiased Prefix-TEA estimator based on moment cancellation. Experiments on instruction-following tasks show that TEA and Prefix-TEA improve best-of-$N$ performance across different language models, reward models and datasets under various training and test-time budget settings.

We introduce Quokka, the first comprehensive scaling law for diffusion language models (DLMs), encompassing both compute- and data-constrained regimes alongside key modeling and optimization designs. Under compute constraints, we find that optimal parameter and dataset sizes scale equally with compute ($N_{\mathrm{opt}} \propto C^{0.5}$, $D_{\mathrm{opt}} \propto C^{0.5}$); however, DLMs are 2--5$\times$ more data-hungry than autoregressive (AR) models, requiring larger corpora and proportionally smaller models for a given FLOP budget. Under data constraints, validation loss follows a U-shaped curve across epochs, with the onset of overfitting scaling as $e_{\mathrm{opt}} \propto U_D^{0.39}/N^{0.55}$. This indicates that optimally leveraging a fixed unique data budget ($U_D$) requires allocating both modestly larger models and more training epochs. Beyond data and parameter allocation, we provide actionable guidance on DLM design choices: absorbing-mask transitions outperform uniform diffusion, linear noise schedules are the most stable and performant, and principled diffusion ELBO objectives ultimately surpass MaskGIT losses despite slower initial convergence. Furthermore, we demonstrate that easy-to-hard noise curricula accelerate early learning, established AR scaling laws for batch size and learning rate transfer seamlessly to DLMs, and weight decay, while unhelpful for single-epoch runs, is essential for multi-epoch regularization.

Neural scaling laws approximate a language model's loss as a power-law function of parameter count $N$ and token count $D$. Following Chinchilla-style compute-optimal training, many studies fit scaling laws from runs performed under a fixed tokens-per-parameter (TPP) ratio $k$ and set $D = kN$. We show that this collinear design, combined with the empirically common near-equality of the exponents governing $N$ and $D$, induces an inherent ill-conditioning in the Gauss-Newton least-squares problem: the condition number of the design grows as the inverse square of the gap between the $N$ and $D$-exponents. The scale coefficients become practically unidentifiable, with confidence intervals inflating by an order of magnitude or more, yielding a ``sloppy'' model whose extrapolations degrade sharply off the training ray. We prove this for four scaling-law formalisms and derive a closed-form TPP-diversity threshold that is necessary and sufficient for well-conditioned estimation. Empirically, non-collinear designs outperform collinear ones on held-out splits with a 97.3\% win rate across four laws, five corpora, multiple floating point precision modes. We further show the degeneracy is rooted in Jacobian geometry and is not an artifact of the loss function: any smooth estimation objective whose curvature involves the Jacobian inherits the same ill-conditioning.

Setting the learning rate for a deep learning model is a critical part of successful training, yet choosing this hyperparameter is often done empirically with trial and error. In this work, we explore a solvable model of optimal learning rate schedules for a powerlaw random feature model trained with stochastic gradient descent (SGD). We consider the optimal schedule $\eta_T^\star(t)$ where $t$ is the current iterate and $T$ is the total training horizon. This schedule is computed both as a numerical optimization problem and also analytically (when possible) using optimal control theory. Our analysis reveals two regimes which we term the easy phase and hard phase. In the easy phase the optimal schedule is a polynomial decay $\eta_T^\star(t) \simeq T^{-\xi} (1-t/T)^{\delta}$ where $\xi$ and $\delta$ depend on the properties of the features and task. In the hard phase, the optimal schedule resembles warmup-stable-decay with constant (in $T$) initial learning rate and annealing performed over a vanishing (in $T$) fraction of training steps. We investigate joint optimization of learning rate and batch size, identifying a degenerate optimality condition and showing that batch ramps can improve the scaling in wall-clock time in the easy phase. Beyond SGD, we derive optimal schedules for the momentum parameter $\beta(t)$ and show that momentum achieves a better loss-scaling exponent in the hard phase. We compare our optimal schedule to various benchmarks in our task including (1) optimal constant learning rates $\eta_T(t) \sim T^{-\xi}$ (2) optimal power laws $\eta_T(t) \sim T^{-\xi} t^{-\chi}$, finding that our schedule achieves better rates than either of these. Our theory suggests that learning rate transfer across training horizon depends on the structure of the model and task. For ResNet image classification on CIFAR-5M, the learning curves exhibit hard-phase behavior where optimal base learning rates are constant under sufficient annealing. GPT-2 style transformers trained in language modeling exhibit easy-phase behavior where optimal learning rates shift even under annealing.

As agent systems scale, skills accumulate into large reusable libraries, yet their scaling laws remain poorly understood. Across 15 frontier LLMs, 1,141 real-world skills, and over 3M routing or execution decisions, we identify two coupled laws. Routing law: single-step routing accuracy decays logarithmically with library size ($R^2{>}0.97$ for all models), with errors progressing from local skill competition to cross-family drift and capture by overly general ``black-hole skills''. Execution law: before state realization, joint routing is approximately multiplicative, whereas correct execution can improve difficult downstream decisions by about $4{\times}$. A single parameter, the routing logarithmic decay slope $b$, couples the two laws: routing-side fits predict execution-side rescue across models, showing that the same library property controls both pre-execution collapse and downstream recoverability. The laws are actionable: law-guided optimization raises held-out routing accuracy from 71.3% to 91.7%, reduces hijack from 22.4% to 4.1%, and transfers directionally to downstream ClawBench and ClawMark execution settings, improving mean pass rate from 49.3% to 61.6% on ClawBench and from 28.4% to 34.5% on ClawMark. These results show that agent performance depends not only on model capability, but also on the structure, granularity, and exposure policy of the skill library.

We characterize the minimax rate of estimating the second-order calibration error for binary classification, which quantifies whether a higher-order predictor's epistemic-uncertainty estimate matches the conditional variance of the label probability on its level sets. Our key observation is that the sech perturbation kernel, previously used only to enforce smoothness of calibration functions, in fact makes them \emph{analytic} in a strip of half-width $h\pi/2$. Polynomial regression then estimates the calibration error at rate $\tilde{O}(1/\sqrt{n})$, with explicit constants, a qualitative improvement over the $O(n^{-1/4})$ rate achievable by bucketing or kernel smoothing. A matching $\Omega(1/\sqrt{n})$ lower bound establishes minimax optimality up to logarithmic factors. As a corollary, we give the first finite-sample guarantee for second-order Platt scaling, yielding a post-hoc procedure that recalibrates both the mean prediction and the epistemic-variance estimate of any higher-order predictor. Along the way, we provide a bucket-free definition of second-order calibration and relate it quantitatively to the bucketed formulation of Ahdritz et al. [2025]. Our experiments confirm the predicted rate and the quality of the recalibrated uncertainties.

In Low-Rank Adaptation (LoRA), the scaling factor $\alpha$ is often treated as a mere complement to the learning rate, yet its role in optimization remains poorly understood. In this paper, we reveal that the scaling factor $\alpha$ and the learning rate function differently, with $\alpha$ emerging as the dominant driver of effective optimization, delivering gains that cannot be replicated by learning rate scaling alone. Through the synergy of extensive empirical analysis and a theoretical Signal-Drift framework, we uncover three findings into LoRA’s scaling mechanism: First, LoRA’s spectral suppression smooths the optimization landscape, rendering standard hyperparameters overly conservative and creating an optimization gap. Second, when leveraging this smoothness to accelerate convergence, $\alpha$ outperforms the learning rate by amplifying the task signal without increasing the drift ratio. Third, the optimal scaling factor follows a sublinear relationship with the rank, well characterized by a square-root law with an unexpectedly large coefficient, revealing the insufficient scaling of existing rank-tied heuristics. Based on these insights, we propose LoRA-$\alpha$, a minimalist framework that restores $\alpha$ to its principled regime, making LoRA compatible with standard small learning rates. Extensive evaluations across diverse tasks demonstrate that LoRA-$\alpha$ consistently improves performance while streamlining hyperparameter search, unleashing the learning potential of LoRA.

Sparse autoencoders (SAEs) operationalise the linear representation hypothesis: they reconstruct model activations as sparse linear combinations of interpretable dictionary atoms, on the implicit assumption that activation space is well approximated by a globally linear structure. Their reconstruction error varies sharply across layers in ways that existing scaling laws, fitted at single layers, do not explain. We argue that this variation is the empirical trace of a geometric mismatch: where the activation manifold is curved and its intrinsic dimension varies across layers, no sparse linear dictionary can match it uniformly, and the SAE's width-sparsity scaling becomes a layer-dependent function of manifold structure rather than a single universal law. We conduct the first cross-layer SAE scaling study, fitting and regressing on 844 residual-stream Gemma Scope SAE checkpoints across 68 layers of Gemma 2 2B and 9B. Stage 1 fits a per-layer scaling-law surface; Stage 2 regresses the fitted parameters and the derived per-layer width exponents on four layerwise geometric summaries. We find that manifold geometry predicts the per-layer width exponent in both models, and that the same regression coefficients learnt on one model predict the other model's per-layer exponents under cross-model transfer, indicating a transferable geometric law. At the showcase layers where richer width grids permit identification of the asymptotic floor, we find that the fitted floor tracks the layerwise geometric ordering: higher curvature and intrinsic dimension correspond to higher floor, consistent with the irreducible second-order residual that any sparse linear approximation of a curved manifold must leave behind. SAEs thus encounter not a finite-resource ceiling but a geometry-dependent wall, set by the manifold they are trying to reconstruct.

A common recipe to improve diffusion models at test-time so that samples score highly against a user-specified reward is to introduce the gradient of the reward into the dynamics of the diffusion itself. This procedure is often ill posed, as user-specified rewards are usually only well defined on the data distribution at the end of generation. While common workarounds to this problem are to use an approximate denoiser to estimate what a sample would have been at the end of generation, we propose a simple solution to this problem by working directly with a flow map. By exploiting a relationship between the flow map and velocity field governing the instantaneous transport, we construct an algorithm, Flow Map Trajectory Tilting (FMTT), which leverages a fast yet precise flow map look-ahead and provably performs better ascent on the reward than standard test-time methods involving the gradient of the reward. The approach can be used to either perform exact sampling via importance weighting or principled search that identifies local maximizers of the reward-tilted distribution. We demonstrate the efficacy of our approach against other look-ahead techniques, and show how the flow map enables engagement with complicated reward functions that make possible new forms of image editing, e.g. by interfacing with vision language models.

Reasoning with large language models often benefits from generating multiple chains-of-thought, but existing aggregation strategies are typically trajectory-level (e.g., selecting the best trace or voting on the final answer), discarding useful intermediate work from partial or “nearly correct” attempts. We propose Stitching Noisy Diffusion Thoughts, a self-consistency framework that turns cheap diffusion-sampled reasoning into a reusable pool of step-level candidates. Given a problem, we (i) sample many diverse, low-cost reasoning trajectories using a masked diffusion language model, (ii) score every intermediate step with an off-the-shelf process reward model (PRM), and (iii) stitch these highest-quality steps across trajectories into a composite rationale. This rationale is then used to recompute only the final answer. This modular pipeline separates exploration (diffusion) from evaluation and solution synthesis, avoiding monolithic unified hybrids while preserving broad search. Across math reasoning benchmarks, we find that step-level recombination is most beneficial on harder problems, and ablations highlight the importance of the final solver in converting stitched but imperfect rationales into accurate answers. Using low-confidence diffusion sampling with parallel, independent rollouts, our training-free framework improves average accuracy by up to 23.8% across six math and coding tasks. At the same time, it achieves up to a 1.8× latency reduction relative to both traditional diffusion models (e.g., Dream, LLaDA) and unified architectures (e.g., TiDAR). The code will be publicly available.

Scaling laws are increasingly used to pick which expensive training run to scale up. The practical question — which candidate is best at a fixed future compute budget $C^\star$ — fits neither classical best-arm identification nor minimum-oriented functional bandits. We formulate it as the *target-budget best-function identification* (TB-BFI) problem, with two natural flavors: TB-BFI-FC asks for a $\delta$-correct certificate of the winner at $C^\star$, and TB-BFI-FB asks for low target-budget simple regret under a hard exploration budget. For the shared-exponent power-law class we prove uniform target-budget validity, a finite-time stopping bound for the cost-aware TB-LUCB rule, and a per-arm-exponent perturbation extension. We then close two structural gaps. **(a)** We extend the theory *beyond* shared-exponent to the joint $(N,D)$ Chinchilla law with shared but *unknown* exponents, via a two-stage protocol with explicit pilot sample complexity and a quadratic-in-$\epsilon$ centred-design perturbation theorem. **(b)** We give the first positive TB-BFI-FB result: a cost-aware Sequential-Halving algorithm whose cost-weighted simple-regret upper bound matches a new information-theoretic lower bound up to the standard $K \log K$ adaptive-vs-oracle gap. Empirically, theorem-aligned synthetic experiments certify the $C^\star$-winner in $95$–$100\%$ of trials; on a char-level transformer learning-rate-crossover our extrapolator picks the right arm in $3/3$ seeds where Successive Halving picks $0/3$; and on a flip-rank Chinchilla instance the joint-law procedure reaches $70$–$87\%$ accuracy where the univariate $c = N \cdot D$ baseline scores $0\%$.

Best-of-N sampling is a simple inference-time scaling strategy for large language models, yet its behavior under incomplete reward models remains poorly understood. We study Best-of-N as a competition between the extreme reward tails of correct and incorrect responses. This view shows that success depends not only on the probability of sampling a correct response, but also on whether correct responses dominate incorrect ones in the upper tail. Using extreme value theory, we characterize three asymptotic regimes: correct-tail-dominant, incorrect-tail-dominant, and critical explaining both monotonic gains and reward-hacking-induced degradation. We further derive finite-sample scaling laws for representative reward distributions, showing that, under Gaussian rewards, incorrect responses with sufficiently large variance can dominate selection even when their mean reward is lower. Unlike oracle selection, convergence with an incomplete selector can be polynomial rather than exponential. Finally, we propose a lightweight tail-tension criterion to estimate when additional sampling may become harmful. Experiments on synthetic data and real reward model scores validate the predicted regimes, scaling laws, and diagnostic behavior, suggesting that tail competition provides a unified lens on Best-of-N sampling for verifiable problems.

Large language models remain vulnerable to jailbreak attacks, yet we still lack a systematic understanding of how jailbreak success scales with attacker effort across methods, model families, and harm types. We initiate a scaling-law framework for jailbreaks by treating each attack as a compute-bounded optimization procedure and measuring progress on a shared FLOPs axis. Our systematic evaluation spans four representative jailbreak paradigms, covering optimization-based attacks, self-refinement prompting, sampling-based selection, and genetic optimization, across multiple model families and scales on a diverse set of harmful goals. We investigate scaling laws that relate attacker budget to attack success score by fitting a simple saturating exponential function to FLOPs--success trajectories, and we derive comparable efficiency summaries from the fitted curves. Empirically, prompting-based paradigms tend to be the most compute-efficient compared to optimization-based methods. To explain this gap, we cast prompt-based updates into an optimization view and show via a same-state comparison that prompt-based attacks more effectively optimize in prompt space. We also show that attacks occupy distinct success--stealthiness operating points with prompting-based methods occupying the high-success, high-stealth region. Finally, we find that vulnerability is strongly goal-dependent: harms involving misinformation are typically easier to elicit than other non-misinformation harms.

Predicting the maximum stable learning rate of a neural network from its architecture alone has remained outside the reach of current theory, except for deep linear and shallow scalar models, where exact sharpness expressions are available. We close this gap for deep polynomial networks by proving the first computable, architecture-explicit lower bound on the largest Hessian eigenvalue at any interpolating minimum. The bound factorizes into three independently computable terms: a data-geometry factor, an architectural-capacity factor, and a label-energy factor. Combining our bound with the dynamical framework of Chemnitz and Engel [2025] and the stability conditions of Cohen et al. [2022], we derive critical step-size upper bounds for a range of optimizers from vanilla gradient descent to adaptive versions like Adam. The maximum stable learning rate for deep polynomial networks scales as $(\text{activation degree})^{(-2 \times \text{depth})}$, placing realistic polynomial network training in the Edge of Stability regime. Our derivation shows that this scaling is a Hessian-level consequence of the network's algebraic structure, not a property of any specific optimizer or dataset. With identity activation, our bound recovers the exact deep linear sharpness of Mulayoff and Michaeli [2020].

We study the information-theoretic limits of learning a one-hidden-layer teacher network with hierarchical features from noisy queries, in the context of knowledge transfer to a smaller student model. We work in the high-dimensional regime where the teacher width $k$ scales linearly with the input dimension $d$ -- a setting that captures large-but-finite-width networks and has only recently become analytically tractable. Using a heuristic leave-one-out decoupling argument, validated numerically throughout, we derive asymptotically sharp characterizations of the Bayes-optimal generalization error and individual feature overlaps via a system of closed fixed-point equations. These equations reveal that feature learnability is governed by a sequence of sharp phase transitions: as data grows, teacher features become recoverable sequentially, each through a discontinuous jump in overlap. This sequential acquisition underlies a precise notion of effective width $k_c$ -- the number of learnable features at a given data budget $n$ -- which unifies two distinct scaling regimes: a feature-learning regime in which the Bayes-optimal generalization error $\varepsilon^{\rm BO}$ scales as $ n^{1/(2\beta)-1}$, and a refinement regime in which it scales as $n^{-1}$, where $\beta>1/2$ is the exponent of the power-law feature hierarchy. Both laws collapse to the single relation $\varepsilon^{\rm BO}=\Theta(k_c d/n)$. We further show empirically that a student trained with \textsc{Adam} near the effective width $k_c$ achieves these optimal scaling laws (up to a small algorithmic gap), and provide an information-theoretic account of the associated scaling in model size.

Spectral optimizers such as Muon have recently shown strong empirical performance in large-scale language model training, but the source and extent of their advantage remain poorly understood. We study this question through the linear associative memory problem, a tractable model for factual recall in transformer-based models. In particular, we go beyond orthogonal embeddings and consider Gaussian inputs and outputs, which allows the number of stored associations to greatly exceed the embedding dimension. Our main result sharply characterizes the recovery rates of one step of Muon, SGD, and Newton's method on the logistic regression loss under a power law frequency distribution. We show that the storage capacity of Muon significantly exceeds that of SGD, and even matches Newton's method while only using first-order information. Moreover, Muon saturates at a larger critical batch size. We further analyze the multi-step dynamics under a thresholded gradient approximation and show that Muon achieves a substantially faster initial recovery rate than SGD, while both methods eventually converge to the information-theoretic limit at comparable speeds. Experiments on synthetic tasks validate the predicted scaling laws. Our analysis provides a quantitative understanding of the signal amplification of spectral preconditioners and lays the groundwork for establishing scaling laws across more practical language modeling tasks and optimizers.

Today, verifier-guided best-of-$N$ inference is a common way to spend test-time compute in language modeling, code generation, and alignment: generate $N$ candidates, score them with a verifier or reward model, and deploy the highest-scoring output. This paper argues that the relevant reliability object is not average verifier accuracy but \emph{selected-tail reliability}, namely the behavior of the output chosen after optimizing over $N$ noisy public scores; we derive exact selected-rank exposure laws for best-of-$N$ selection and finite-verifier resource laws showing that selected public optimism scales at the $\sqrt{\log N/m}$ regime under $m$ units of verifier evidence. We then separate public optimism from hidden utility harm, showing that public-score inflation alone is not a harm claim and that harm or uncertifiability requires public-hidden mismatch, tail dominance, count imbalance, or an unresolved selected-tail regime. Building on these results, we introduce Tail-Certified Compute Caps, a conservative procedure that certifies candidate budgets or refuses certification using selected-tail intervals, verifier-resource penalties, mismatch envelopes, and feasibility gates. In exact-verifier/public-hidden experiments, the theory predicts selected false-positive exposure and supports resource-aware cap decisions over an all-attempted denominator, while non-code candidate tables with pre-existing public human annotations serve as secondary scoped $K\ge16$ consistency checks under proxy/model-prior/verifier-style scores across Arena55K and Stanford SHP. Overall, the paper provides a reliability framework for verifier-guided inference-time scaling: candidate budgets should be justified by evidence about the selected tail that is actually deployed, not only by average verifier validation.

Kilometer-scale convection shapes precipitation extremes, tropical organization, and cloud feedbacks, but most global atmospheric models approximate these processes at 25--100\,km resolution. Global storm-resolving physics models resolve convective systems explicitly, but at a cost---roughly one MWh per simulated day on exascale supercomputers---that limits their use for long-duration atmospheric simulation. We introduce STRATA (Storm-resolving Tile-based autoRegressive Atmosphere Transformer Architecture), the first autoregressive AI emulator for global storm-resolving atmospheric dynamics. STRATA is trained on the highest-resolution atmospheric dataset yet used for global AI emulation: 17 days of output from the SCREAM physics model at 4.9-km resolution (${\sim}25$ million grid cells) sampled every 10 minutes. Our central premise is that since on 10-minute timescales atmospheric dynamics are predominantly local, training on small spatial tiles trades scarce global temporal samples for abundant local spatial samples and enables global rollout via overlapping-tile blending. STRATA combines 3D patch embedding and local 3D neighborhood attention for tractable modeling of high-resolution atmospheric tiles, a novel Stereographic Rotary Position Embedding (StereoRoPE) for grid-invariant positional encoding, and a pixel-space de-aliasing decoder that suppresses patch-scale rollout artifacts. An iso-FLOP scaling study reveals that km-scale emulation requires ${\sim}10\times$ more FLOPs per horizontal grid point than coarse-resolution AI weather models, consistent with the higher information density of convective-scale dynamics. Despite training on only 17 days of SCREAM output, STRATA produces stable 24-hour global rollouts with realistic km-scale dynamics across diverse weather regimes, though large-scale biases develop with lead time. STRATA achieves 48 simulation days per megawatt-hour---about 50 times better energy efficiency than the underlying SCREAM physics model---and 741 simulated days per wall-clock day at 512 H100 GPUs. Code and dataset will be publicly released.