NeurIPS 2026 Papers

How diffusion models circumvent the curse of dimensionality to learn complex distributions over high dimensional spaces from a finite training set, instead of memorizing it, remains a fundamental mystery. To address this, we introduce analytically tractable Bayesian information restricted diffusion (BIRD) models, in which each pixel observes restricted information about noisy data. A BIRD model time-reverses diffusion by inferring which past training sample produced its current restricted observation using the Bayesian posterior. This model class generalizes existing analytical diffusion models that use spatially local information restriction. We show that spatially local BIRD models closely approximate trained diffusion models early in training, across different architectures such as UNets and DiTs. Under minimal assumptions on the data distribution, we identify an information-theoretic phase boundary between memorization and generalization in the joint space of amount of training data, time in the reverse generative process, and amount of information restriction: a BIRD model memorizes when the mutual information between its restricted noisy observations and the training data exceeds the log number of training points, and it generalizes otherwise. Experiments across a range of datasets confirm our theoretically predicted location for the transition. We find that generation proceeds near the edge of memorization: both spatially local BIRD models and early-training diffusion models track the memorization-generalization phase boundary by increasingly restricting information over time. Overall, our results reveal a fundamental role for information restriction in generative AI to circumvent the curse of dimensionality.

Making calibrated online predictions is a central challenge in modern AI systems. Much of the existing literature focuses on fully adversarial environments where outcomes may be arbitrary, leading to conservative algorithms that can perform suboptimally in more benign settings, such as when outcomes are nearly stationary. This gap raises a natural question: can we design online prediction algorithms whose calibration error automatically adapts to the degree of non-stationarity in the environment, smoothly interpolating between i.i.d. and adversarial regimes? We answer this question in the affirmative and develop a suite of algorithms that achieve adaptive calibration guarantees under multiple calibration measures. Specifically, with $T$ being the number of rounds and $C\in[0,T]$ being an unknown non-stationary measure defined as the minimal $\ell_1$ deviation of the mean outcomes, our algorithms attain $\widetilde{O}(\sqrt{T}+(TC)^{\frac{1}{3}})$ for $\ell_1$ calibration error and $\widetilde{O}((1+C)^{\frac{1}{3}})$ for both $\ell_2$ and pseudo KL calibration error. These bounds match the optimal rates in the stationary case ($C=0$) and recover known guarantees in the fully adversarial regime ($C=T$). Our approach builds on and extends prior work [Hu et al., 2025, Luo et al., 2025], introducing an epoch-based scheduling together with a novel non-uniform partition of the prediction space that allocates finer resolution near the underlying ground truth.

We derive closed-form ODEs for the learning dynamics of a linear recurrent policy network trained with REINFORCE on sparse-reward tasks in a high-dimensional teacher–student setting. Under alignment assumptions between student and teacher networks, the dynamics reduce to a finite system of coupled order-parameter ODEs whose reward terms are governed by trajectory-level orthant probabilities of a structured correlated Gaussian process. The theory predicts that recurrence can accelerate escape from sparse-reward plateaus by accumulating task-relevant memory, and that each episode length admits an optimal recurrent scale that minimizes learning time. Our results provide a quantitative theory of how recurrence, and sparse reward interact during learning.

Modern vision--language models are trained with very different objectives: CLIP-style contrastive alignment, SigLIP-style match/no-match prediction, caption-style token generation (Flamingo, LLaVA), and latent-space prediction (VL-JEPA). Each has been claimed as best-in-class for its target task. Yet practitioners lack clear guidelines for choosing between them, and existing identifiability theory analyzes only the contrastive case. In this work, we unveil the core statistical mechanisms that distinguish each objective family using a unified spectral analysis. By leveraging closed-form solutions for the linearized losses under a latent structure in which modality-specific nuisances may spuriously carry shared semantic content, we characterize what each objective recovers from the data. We demonstrate that contrastive alignment, match/no-match prediction, and latent-space prediction collapse to the same cross-modal solution, while directed token-space generation recovers a different one. Our findings indicate that in scenarios where one modality is heavily contaminated by spurious nuisance, token-space generation is preferable because it imposes a strictly weaker, one-sided alignment condition. In scenarios where two modalities are both clean, retrieval is maximized by the latent-prediction objective whose target matches the query modality. These results clarify the trade-offs between the four objective families. We validate our theoretical predictions on numerical, controlled synthetic, and real (Flickr30k, COCO, CC3M) data.

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.

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.

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-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.

We study the complexity of smoothed agnostic learning of halfspaces on $\{\pm 1\}^n$ under uniform marginals in the model of \citet{KM25} where each input coordinate is independently flipped with probability $\sigma \in (0, {1}/{2})$. We show that $L^1$ polynomial regression achieves runtime and sample complexity $\tilde{O}(n^{O(\log(1/\varepsilon)/\sigma)})$, and prove a nearly matching Statistical Query complexity lower bound of $n^{\Omega(\log(1+\sigma/\varepsilon^2)/\sigma)}$. This complements the recent work of \citet{DK26}, which established analogous bounds in the continuous setting under Gaussian marginals.

Large language models successfully solve classic theory-of-mind tasks adapted from cognitive science, but the mechanisms underlying this ability remain unclear. Do they deploy circuitry specialized for reasoning about other minds, memorize common false-belief vignette structures and outputs, or rely on more domain-general computations? We investigate this question in Qwen2.5-14B-Instruct using causal mediation and representational similarity analyses across matched prompt sets that systematically vary an agent’s beliefs, the state of the physical world, and the correct answer. We identify a population of mid-layer attention heads that tracks divergence between an initial representation and the current state of the world. These heads are causally relevant regardless of whether the initial representation is an agent’s belief or a photograph, despite the photograph condition involving no agent or perspective-taking. A distinct population of later-layer heads retrieves the answer token. The two populations are functionally dissociable yet combine compositionally. Together, these findings argue against both mentalizing-specific and memorization accounts, suggesting instead that LLMs solve classic false-belief problems by reusing a domain-general mechanism for detecting divergence between representations and reality.