NeurIPS 2026 论文列表

In this work, we develop theoretical foundation for flow matching with neural-network–parameterized conditional velocity fields. We establish convergence guarantees for gradient descent in the over-parameterized 2-layered ReLU neural network regime. We derive generalization bounds for the conditional velocity-field matching objective. Building on these results, we provide Wasserstein-distance guarantees for the samples generated by the induced flow. Our analysis is based on generalization bound for multi-task representation learning with unbounded losses, which may be of independent interest beyond flow-based generative modeling. These theoretical results are validated through extensive experiments on both synthetic and real-world image benchmarks.

Scaling large language models (LLMs) requires tremendous computational resources, and recent advances in AI have gone hand in hand with massive amounts of capital expenditure. While it is established that scaling up LLMs reliably increases model quality (quantified in terms of loss or downstream evaluations), it is unclear how these quality improvements translate to potential revenue, and whether revenue increases would offset costs of larger-scale training and inference. In this work, we develop an economic model for characterizing the rational behavior of an LLM training firm by combining scaling laws with microeconomic theory. Under our model, LLM quality can be increased with more parameters and training tokens, leading to more potential adoption by consumers, who each have a quality threshold for using the LLM. On the other hand, additional parameters and training tokens both incur additional costs. We analyze the profit maximization problem for this model under compute-bound and data-bound regimes. In the compute-bound regime, optimal model size and token budget track hardware efficiency $E$ (FLOPs/dollar) at a near-linear rate; total training cost then scales sub-quadratically in $E$. Data efficiency improvements incentivize larger models and training expenditure. When we are limited to $D$ data, profit-optimal training expenditure scales as $D^2 / E$, i.e, increase with data and *decreases* with hardware efficiency (as well as data efficiency). Finally, we analyze practical trends in training expenditure: current trends in training expenditure are consistent with our most permissive model variants in the compute-bound regime, but are not profit-optimal in the data-bound regime or assuming hardware advances will stall. Overall, our results provide a theory of profit-optimal LLM training, providing a foundation for engaging critically with industry statements and supporting long-term economic decision making.

We study language generation in the limit under a global preference ordering on strings, as introduced by Kleinberg and Wei. We aim for breadth, but impose an additional requirement of timeliness: higher-ranked strings should be generated earlier. A string is then only credited if it is generated before a deadline, where its deadline is defined by a function that maps a string’s rank in the target language to the time by which it must be produced. This is in keeping with a central consideration in machine learning, where inductive bias favors "simpler" or "more plausible" outputs, all else being equal. We show that timely generation is impossible in a strong sense for eventually consistent generators—the protagonists of most prior related work. Under what is perhaps the mildest natural relaxation of consistency, a hallucination rate that vanishes over time, we show that we can circumvent our impossibility result. In particular, we can achieve optimal density with respect to any superlinear deadline function. We also show this is tight by ruling out timely generation with linear deadlines and vanishing hallucination rate.

Spatial cognition requires stable neural manifolds of orientation and location that track self-motion and sensory cues. Continuous attractors provide a natural dynamical principle, but learning such mechanisms that form and control manifolds with theoretical guarantees remains difficult. We propose Spatial Energy Attractor Learning (SEAL), a theory of neurodynamics learning for spatial cognition, where continuous spatial representations arise as learned low-energy manifolds in Hopfield energy landscapes through population-level Fourier energy learning. For head-direction and grid-cell systems, Fourier energy landscapes yield ring and torus attractors with learned normal attraction and input-driven tangent transport. We prove that Fourier construction is a special case of a general Hopfield-compatible energy extension whose autonomous dynamics preserve energy descent and whose expressive energy families approximate target attractor energies and restoring fields. Experiments on HD and grid-cell populations show that the learned energies restore perturbed states to ring and torus manifolds, stabilize velocity-driven moving bumps across multiple input regimes, and induce continuous-attractor interactive structures with local excitation and surround inhibition. Together, these results provide a stable, learnable, and biologically interpretable energy-based account of continuous spatial representation.

Autoregressive generation lies in the heart of the mechanism of large language models. It can be viewed as the repeated application of a next-token generator: starting from an input string (prompt), the generator is applied for $M$ steps, and the last generated token is taken as the final output. [Joshi et al., 2025] proposed a PAC model for studying the learnability of the input-output maps arising from this process. We develop an online analogue of this framework, focusing on the mistake bound of learning the final output induced by an unknown next-token generator. We distinguish between two forms of feedback. In the End-to-End model, after each round the learner observes only the final token produced after M autoregressive steps. In the Chain-of-Thought model, the learner is additionally shown the entire $M$-step trajectory. Our goal is to understand how the optimal mistake bound depends on the generation horizon $M$, and to what extent observing intermediate tokens can reduce this dependence. Our main results show that the online theory of autoregressive learning exhibits a qualitative picture analogous to the statistical one found by [Hanneke et al., 2026], but with a different scale of dependence on the generation horizon. In the End-to-End model, we prove a taxonomy of possible mistake-bound growth rates in the generation horizon $M$: subject to mild regularity conditions, every rate between constant and logarithmic can arise. We also show that this logarithmic ceiling is unavoidable for the online setting, in the sense that every class of finite Littlestone dimension exhibits at most logarithmic dependence on $M$. In the Chain-of-Thought model, the parallel is even sharper: as in the statistical setting, access to the full generated trajectory eliminates the dependence on $M$ altogether. We also analyze autoregressive linear threshold classes. For autoregressive linear thresholds in \(\mathbb{R}^d\), we prove that the optimal mistake bound is \(\Theta(d^2)\) under both End-to-End and Chain-of-Thought feedback, and for any generation length $M$. The methods developed for this analysis also yield new lower bounds in the statistical setting. Along the way, our results resolve several questions left open by [Joshi et al., 2025]. In particular, we show that even for classes of finite Littlestone dimension, the End-to-End statistical sample complexity can depend on the generation length M.

We introduce the problem of *adversary-directed* online learning, in which the learner is aware of the set of instances in advance, and an adversary adaptively determines their ordering during the learning process. Surprisingly, this formulation has not been previously studied, perhaps due to its conceptual resemblance to the traditional adversarial online learning problem, despite related models, including the transductive, self-directed, and best-order, having been extensively explored. However, by utilizing novel techniques, we demonstrate that the landscape of adversary-directed online learning significantly diverges from that of traditional adversarial online learning. In the realizable setting, we establish a trichotomy of possible rates of the minimax number of mistakes. Specifically, for a learning horizon $\operatorname{T}$, the minimax number of mistakes can only be of the orders $\Theta(\operatorname{T})$, $\Theta(\log \operatorname{T})$, or $\Theta(1)$. To prove this, we introduce a new combinatorial complexity parameter, termed the perfect Littlestone dimension, whose finiteness distinguishes the $\Theta(\log \operatorname{T})$ rate from the $\Theta(\operatorname{T})$ rate. On the other hand, in the agnostic setting, we essentially show a dichotomy of possible rates of the minimax expected regret. In particular, if the learner plays for $\operatorname{T} \in \mathbb{N}$ rounds, its minimax expected regret can only be of the orders $\Theta(\operatorname{T})$, or $\widetilde{\Theta}(\sqrt{\operatorname{T}})$, which is also characterized by the finiteness of the perfect Littlestone dimension. Technically, a key ingredient in the proof of our $\mathcal{O}(\log \operatorname{T})$ and $\widetilde{\mathcal{O}}(\sqrt{\operatorname{T}})$ upper bounds is a novel online learning algorithm that leverages a new notion of shattering based on the perfect Littlestone dimension, which exploits the adaptive adversarial nature of the problem.

Self-play, a type of training algorithm that enables a model to self-improve, has recently shown promising empirical results in the context of formal theorem proving using Large Language Models (LLMs). (Dong & Ma, 2025) instantiate self-play with two cooperating agents: a prover, which proves theorems, and a conjecturer, which generates new theorems as a curriculum to the prover. In this paper, we provide a theoretical framework for understanding the self-improvement capabilities of self-play algorithms for theorem proving. First, we formalize the set of theorems as a graph, with nodes as theorems and edges between pairs of theorems with similar semantics. We introduce a set of primitive assumptions that characterize the guarantees of a trained prover and how a conjecturer can access the structure of the graph. Second, we show that if the underlying graph of theorems is well-connected, then a prover-conjecturer system, where the conjecturing algorithm is based on a reversible random walk, is sufficient to grow the set of proved theorems exponentially. Third, we formulate an issue of diversity encountered empirically by self-play algorithms in our framework, where the conjecturer tends to generate artificially complex and non-fundamental theorems. We propose a diversity measure for a training distribution of theorems generated by a conjecturer and an improved conjecturing algorithm that locally maximizes this diversity measure, by computing the diffusion similarity between neighboring theorems in the theorem graph. Finally, we describe a method to compute the diffusion similarity by first using contrastive learning to embed nodes into Euclidean space and second computing the inner-product between the embeddings.

We theoretically and empirically establish a previously unexplored connection between Long-Tailed Recognition (LTR) and Continual Learning (CL). Specifically, we show that training on a long-tailed dataset drives model parameters into an $\mathcal{O}(1/\sqrt{\mathrm{IF}})$ neighborhood of the dominant-class solution, under both uniform and exponentially decaying cardinality distributions. Building on this result, we prove that the CL objective upper-bounds the balanced LTR loss, revealing that LTR can be reformulated as a sequential learning problem. Motivated by this connection, we introduce Continual Learning for Long-Tailed Recognition (CLTR), a principled framework that leverages standard off-the-shelf CL methods to sequentially learn Head and Tail classes while mitigating catastrophic forgetting. Extensive experiments on CIFAR100-LT, CIFAR10-LT, ImageNet-LT, and Caltech256 validate our theoretical predictions and demonstrate strong performance across diverse LTR benchmarks. In the foundation-model regime, CLTR with DualPrompt on a CLIP backbone outperforms standard adaptation baselines and is competitive with specialized methods using external semantic supervision. Our work bridges LTR and CL both theoretically and empirically, providing a principled approach for addressing long-tailed learning using standard CL strategies.

Autoregressive language models (ARMs) suffer from the reversal curse: after learning "$A$ is $B$," they often fail on the reverse query "$B$ is $A$." Masked diffusion language models (MDMs) exhibit this failure in a much weaker form, but the underlying reason has remained unclear. A common explanation attributes this mitigation to their any-order masked training objective. However, observing "$[\textnormal{\textbf{M}}]$ is $B$" during training teaches recovery of $A$ from $B$ in one positional configuration, and does not by itself explain why the learned evidence should transfer to the reverse prompt "$B$ is $[\textnormal{\textbf{M}}]$." We provide a theoretical analysis showing that this transfer arises from a parameter-level coupling between forward and reverse positional conditionals: shared Transformer parameters store token-pair evidence, while relative positional encodings route attention through queries and keys without changing the value-side evidence being retrieved. In a one-layer MDM, we prove that forward masked training strengthens evidence that is reusable in reverse queries, induces correlated forward--reverse attention routes, and yields a positively aligned shared-storage gradient component that decreases the reverse loss to first order. Controlled one-layer experiments and large-scale LLaDA/Dream experiments verify these signatures and show that they translate into improved reverse prediction.

Backdoor attacks implant a trigger-target association into a model, causing malicious behavior at test time while largely preserving clean performance. Despite extensive empirical study, a unified explanation for why standard training dynamics learn backdoors so effectively remains largely missing. We argue that backdoor learning can be understood as a consequence of simplicity-biased optimization dynamics: trigger features are dynamically simpler than semantic features under stochastic gradient descent (SGD) because they induce more coherent gradient alignment, more stable gating behavior, and stronger early-stage amplification. To formalize this perspective, we introduce a directional data model that separates semantic and trigger features and study a one-hidden-layer ReLU network trained by SGD. Our analysis identifies a frozen-gate signal that governs group-wise early-stage loss decrease up to controlled gate-drift and gate-flip errors, yielding a quantitative explanation for faster poisoned-sample fitting and its dependence on the poisoning ratio. Beyond optimization dynamics, we show that this early-stage bias can induce trigger-dominant neurons with selective activation on poisoned inputs, explaining why vanilla clean fine-tuning can attenuate backdoor behavior without necessarily erasing the underlying trigger-related representation. Empirically, we validate the predicted signatures of early-stage optimization bias, representation-level trigger dominance, and post-training residual behavior using ResNet-18 on standard image classification benchmarks.

Small-molecule foundation models are typically pretrained on standalone molecular data, unlike vision and language models that often benefit from cross-modal or relational supervision. Protein-ligand co-folding provides a molecular analogue of such supervision by exposing models to atom-level ligand-protein interactions, raising the question of whether co-folding models can yield strong small-molecule representations. We study this question using Boltz2, a modern co-folding model, by transferring its atom-level ligand representations to standalone small-molecule tasks. Through systematic probing and distillation, we show that Boltz2 representations match or outperform existing models on the ADMET benchmark, accelerate molecular generative modeling, and improve sample efficiency in structure-guided ligand optimization. We further find that Boltz2 representations are complementary to those learned from conventional standalone molecular supervision, including 3D conformers, bioassay labels, and quantum-chemical properties. Finally, we extend representation alignment to reinforcement learning, showing that dense representation-level supervision can complement scalar rewards in molecular discovery. These results identify protein-ligand co-folding as a promising pretraining paradigm for small-molecule representation learning and position Boltz2 as a strong, off-the-shelf molecular foundation model.

We present a systematic benchmark of out-of-distribution (OOD) detection CSFs through a representation-centric lens. Our study spans CNN and ViT backbones, multiple training paradigms, four image-classification source datasets (CIFAR-10, CIFAR-100, SuperCIFAR-100, and TinyImageNet), and OOD datasets grouped into near, mid, and far regimes using CLIP-derived semantic distances. To compare CSFs across these settings, we employ a multiple-comparison-controlled rank pipeline that identifies top cliques of statistically indistinguishable winners under threshold-free ranking metrics (AURC and AUGRC). The main empirical finding is that the competitive detector family depends more on the learned representation than on score design alone. For both CNNs and ViTs, simple probabilistic scores dominate misclassification detection. On CNNs, margin-based scores are strongest in near-OOD regimes, while geometry-aware scores such as NNGuide, fDBD, and CTM become more competitive as shift severity increases. On fine-tuned ViTs, the top cliques are led mainly by reconstruction- and residual-based scores. To interpret these ranking shifts, we analyze the last-layer representation using Neural Collapse (NC) metrics. The resulting picture is consistent across architectures: prototype- and boundary-aware scores become stronger when the representation is more collapsed and better aligned with classifier weights, whereas weaker-collapse regimes favor gradient- and manifold-based scores. Building on these insights, we propose two contributions: a simple PCA-based projection-filtering procedure that improves detector performance, and an approach that uses NC measurements computed from a trained classifier to predict its competitive out-of-distribution detector shortlist, without requiring any additional OOD data.

Deep Q-Networks (DQNs) approximate value iteration by freezing a target network, forming Bellman-optimality targets, and running an inner regression loop that minimizes a smooth \emph{surrogate loss} (e.g., MSE/Huber loss) on data from a replay buffer. Yet, control is governed by the Bellman optimality operator's $\gamma$-contraction in $\ell_\infty$, whereas common objectives (MSE/Huber loss) measure progress in an averaged $\ell_2/\ell_1$ geometry; thus, improving the surrogate need not reduce the worst-case Bellman error that drives performance. To bridge this gap we connect the progress in the inner loop for each surrogate to the Bellman residual under the $\ell_\infty$ norm. Our analysis provides explicit thresholds denoted by $\alpha$ under which sufficient progress guarantees a $\ell_\infty$ contraction up to a standard approximation floor. This yields new convergence guarantees for DQN with fixed targets trained using MSE or Huber-type losses, and motivates a novel soft-$\ell_\infty$ surrogate that smoothly approximates the sup-norm and better matches the $\ell_\infty$ geometry resulting in the least stringent inner loop accuracy requirements. Across Atari benchmarks, soft-$\ell_\infty$ consistently reduces the worst-case Bellman residual and matches or improves returns relative to standard objectives, providing a practical, geometry-aligned recipe for stable DQN training.

Material synthesis planning (MSP) remains a fundamental and underexplored bottleneck in AI-driven materials discovery, as it requires not only identifying suitable precursor materials but also designing coherent sequences of synthesis operations to realize a target material. Although several AI-based approaches have been proposed to address isolated subtasks of MSP, a methodology for solving the entire MSP task has yet to be established. We propose MSP-LLM, a structured LLM-based framework that formulates MSP as a two-stage process composed of two constituent subproblems: precursor prediction (PP) and synthesis operation prediction (SOP). Our approach introduces a discrete material class as an intermediate decision variable that organizes both tasks into a chemically consistent decision chain. For SOP, we further incorporate hierarchical precursor types as synthesis-relevant inductive biases and employ precursor constraint factorization that preserves precursor-related information in the autoregressive decoding state. Extensive experiments show that MSP-LLM consistently outperforms existing methods on both PP and SOP, as well as on the MSP task, demonstrating an effective and scalable framework with practical potential for autonomous MSP.

The Objective Structured Clinical Examination (OSCE) is widely used to assess clinical communication skills in medical education, requiring trainees to take histories and reason clinically with standardized patients (SPs). Large language model (LLM)-based virtual standardized patients (VSPs) offer a practical alternative to human SPs. These are typically built using persona-injected prompting, which casts a model into a clinical role via a single system prompt. However, existing evaluations of such VSPs often rely on a single backbone LLM, small human rater panels, or LLM-as-a-judge scoring without an utterance-aligned human-expert reference. Consequently, they are limited in their ability to identify the specific shortcomings of individual models. In this study, we evaluated four LLMs (GPT-4o, Claude 3.5 Sonnet, Llama-3.1 70B, and HyperCLOVA X), spanning commercial, open-source, and Korean-developed models, against a human-expert reference across three Korean OSCE cases (hematuria, fever in pregnancy, and chronic cough). In a double-blind, fully crossed design, 17 senior medical-student raters scored every model utterance on three sentence-level and six encounter-level Likert metrics, yielding 7,480 data points. We analyzed the scores using a stratified linear mixed-effects model based on an 11-category history-taking and 3-level linguistic-form taxonomy. At the encounter level, three of the four LLMs fell 0.08 to 0.47 Likert points below the human-expert reference on Comprehensibility, Consistency and Hallucination-freeness. The difference was not statistically significant for Hallucination-freeness in Claude 3.5 Sonnet and GPT-4o, Consistency in GPT-4o, and Engagement in HyperCLOVA X. Llama-3.1 70B fell significantly below the reference on all six metrics. Furthermore, we introduce a web-based OSCE platform that supports virtual practice across diverse clinical cases. This platform continuously collects student–VSP dialogues under research and privacy consent; the resulting dataset is available at https://huggingface.co/datasets/neurips-2026-cpx/neurips-2026-cpx

Nonrigid registration is vital to medical image analysis but remains challenging for diffusion MRI (dMRI) due to its high-dimensional, spatio-angular dependence. We present a novel, geometric deep learning framework for {\it model-free}, nonrigid registration of raw dMRI data. The dMRI registration problem is formulated in the native spatio-angular acquisition space, which exhibits a natural symmetry to the group of 3D roto-translations, denoted by $\mathrm{SE}(3)$. A by-product of this design choice is freedom from having to augment the data with roto-translated versions of itself. Our second novelty is the loss function formulation, based on the maximum mean discrepancy (MMD) loss used to compare two probability density functions. We apply this loss in the Fourier space, where it becomes the well-known weighted sum-of-squared differences (SSD) loss with the weights being the Fourier transform of a reproducing kernel Hilbert space (RKHS) kernel. Experimental results on HCP and OASIS-3 clinical-grade dMRI data demonstrate competitive performance compared to SOTA approaches, with the added advantage of bypassing the overhead for estimating derived representations. This work establishes a foundation for data-driven, geometry-aware dMRI registration directly in the acquisition space.

Self-attention has greatly contributed to the success of the widely used Transformer architecture by enabling learning from data with long-range dependencies. In an effort to improve performance, a gated attention model that leverages a gating mechanism within the multi-head self-attention has recently been proposed as a promising alternative. Gated attention has been empirically demonstrated to increase the expressiveness of low-rank mapping in standard attention and even to eliminate the attention sink phenomenon. Despite its efficacy, a clear theoretical understanding of gated attention's benefits remains lacking in the literature. To close this gap, we rigorously show that each entry in a gated attention matrix or a multi-head self-attention matrix can be written as a hierarchical mixture of experts. By recasting learning as an expert estimation problem, we demonstrate that gated attention is more sample-efficient than multi-head self-attention. In particular, while the former needs only a polynomial number of data points to estimate an expert, the latter requires exponentially many data points to achieve the same estimation error. Furthermore, our analysis also provides a theoretical justification for why gated attention yields higher performance when a gate is placed at the output of the scaled dot product attention or the value map rather than at other positions in the multi-head self-attention architecture.

As learning systems increasingly shape everyday decisions, Algorithmic Collective Action (ACA), i.e., users coordinating changes to shared data to steer model behavior, offers a complement to regulator-side policy and corporate model design. Real-world collective actions have traditionally been decentralized and fragmented into multiple collectives, despite sharing overarching objectives, with each collective differing in size, strategy, and actionable goals. However, most of the ACA literature focuses on single collective settings. To address this, we propose the first comprehensive statistical framework for ACA with multiple collectives acting on the same system. In particular, we focus on collective action in classification, studying how multiple collectives can influence a classifier's behavior. We provide quantitative statistical bounds on the success of the collectives, considering the role and the interplay of the collectives' sizes and the alignment of their goals. We make such bounds computable by each collective with only partial knowledge of other collectives' sizes and strategies. Finally, we numerically illustrate our framework on simulations inspired by interventions for climate adaptation in smart cities, demonstrating the usefulness of our bounds.

Low-rank plus sparse decompositions, such as robust PCA, typically model the low-rank component as dense and the sparse component as corruption. This viewpoint is poorly suited to graph adjacency matrices, where sparsity is structural and dense low-rank factors destroy the combinatorial and computational properties of the graph. We propose signed biclique decomposition (SBD), a sparse low-rank decomposition for integer-weighted graphs that represents the structured component as a sum of sparse signed binary rank-one factors, together with a sparse signed residual. The resulting representation remains discrete, interpretable, sparse, and compatible with standard linear algebra. We formulate fixed-rank and greedy SBD objectives and derive a gain-based greedy algorithm with theoretical guarantees. Empirically, SBD recovers block structure in noisy graphs, while its factorized representation enables efficient sparse matrix-matrix multiplication (SpMM). On real graphs with strong shared-neighborhood structure, SBD achieves compression factors above $10$ and SpMM speedups up to $6\times$ over cuSPARSE, suggesting a practical alternative to dense low-rank decompositions for graphs.