NeurIPS 2025 论文列表

We study the high-dimensional asymptotics of empirical risk minimization (ERM) in over-parametrized two-layer neural networks with quadratic activations trained on synthetic data. We derive sharp asymptotics for both training and test errors by mapping the $\ell_2$-regularized learning problem to a convex matrix sensing task with nuclear norm penalization. This reveals that capacity control in such networks emerges from a low-rank structure in the learned feature maps. Our results characterize the global minima of the loss and yield precise generalization thresholds, showing how the width of the target function governs learnability. This analysis bridges and extends ideas from spin-glass methods, matrix factorization, and convex optimization and emphasizes the deep link between low-rank matrix sensing and learning in quadratic neural networks.

How can we trust the correctness of a learned model on a particular input of interest? Model accuracy is typically measured *on average* over a distribution of inputs, giving no guarantee for any fixed input. This paper proposes a theoretically-founded solution to this problem: to train *Self-Proving models* that prove the correctness of their output to a verification algorithm $V$ via an Interactive Proof. Self-Proving models satisfy that, with high probability over an input sampled from a given distribution, the model generates a correct output *and* successfully proves its correctness to $V$. The *soundness* property of $V$ guarantees that, for *every* input, no model can convince $V$ of the correctness of an incorrect output. Thus, a Self-Proving model proves correctness of most of its outputs, while *all* incorrect outputs (of any model) are detected by $V$. We devise and analyze two generic methods for learning Self-Proving models: *Transcript Learning (TL)* which relies on access to transcripts of accepting interactions, and *Reinforcement Learning from Verifier Feedback (RLVF)* which trains a model by emulating interactions with the verifier.

What property of the data distribution determines the excess risk of principal component analysis? In this paper, we provide a precise answer to this question. We establish a central limit theorem for the error of the principal subspace estimated by PCA, and derive the asymptotic distribution of its excess risk under the reconstruction loss. We obtain a non-asymptotic upper bound on the excess risk of PCA that recovers, in the large sample limit, our asymptotic characterization. Underlying our contributions is the following result: we prove that the negative block Rayleigh quotient, defined on the Grassmannian, is generalized self-concordant along geodesics emanating from its minimizer of maximum rotation less than $\pi/4$.

Many modern learning tasks require models that can take inputs of varying sizes. Consequently, dimension-independent architectures have been proposed for domains where the inputs are graphs, sets, and point clouds. Recent work on graph neural networks has explored whether a model trained on low-dimensional data can transfer its performance to higher-dimensional inputs. We extend this body of work by introducing a general framework for transferability across dimensions. We show that transferability corresponds precisely to continuity in a limit space formed by identifying small problem instances with equivalent large ones. This identification is driven by the data and the learning task. We instantiate our framework on existing architectures, and implement the necessary changes to ensure their transferability. Finally, we provide design principles for designing new transferable models. Numerical experiments support our findings.

This paper proves a new fingerprinting method to embed the ownership information into a deep neural network (DNN) with theoretically guaranteed robustness to fine-tuning. Specifically, we prove that when the input feature of a convolutional layer only contains low-frequency components, specific frequency components of the convolutional filter will not be changed by gradient descent during the fine-tuning process, where we propose a revised Fourier transform to extract frequency components from the convolutional filter. Additionally, we also prove that these frequency components are equivariant to weight scaling and weight permutations. In this way, we design a fingerprint module to embed the fingerprint information into specific frequency components of convolutional filters. Preliminary experiments demonstrate the effectiveness of our method.

Low-precision Streaming PCA estimates the top principal component in a streaming setting under limited precision. We establish an information‐theoretic lower bound on the quantization resolution required to achieve a target accuracy for the leading eigenvector. We study Oja's algorithm for streaming PCA under linear and nonlinear stochastic quantization. The quantized variants use unbiased stochastic quantization of the weight vector and the updates. Under mild moment and spectral-gap assumptions on the data distribution, we show that a batched version achieves the lower bound up to logarithmic factors under both schemes. This leads to a nearly _dimension-free_ quantization error in the nonlinear quantization setting. Empirical evaluations on synthetic streams validate our theoretical findings and demonstrate that our low-precision methods closely track the performance of standard Oja’s algorithm.

Diffusion models operating in discrete state spaces have emerged as powerful approaches, demonstrating remarkable efficacy across diverse domains, including reasoning tasks and molecular design. Despite their promising applications, the theoretical foundations of these models remain substantially underdeveloped, with the existing literature predominantly focusing on continuous-state diffusion models. A critical gap persists in the theoretical understanding of discrete diffusion modeling: the absence of a rigorous framework for quantifying estimation error with finite data. Consequently, the fundamental question of how precisely one can reconstruct the true underlying distribution from a limited training set remains unresolved. In this work, we analyze the estimation error induced by a score estimation of the discrete diffusion models. One of the main difficulties in the analysis stems from the fact that the cardinality of the state space can be exponentially large with respect to its dimension, which results in an intractable error bound by a naive approach. To overcome this difficulty, we make use of a property that the state space can be smoothly embedded in a continuous Euclidean space that enables us to derive a cardinality independent bound, which is more practical in real applications. In particular, we consider a setting where the state space is structured as a hypercube graph, and another where the induced graph Laplacian can be asymptotically well approximated by the ordinary Laplacian defined on the continuous space, and then derive state space size independent bounds.

The softmax-contaminated mixture of experts (MoE) model is deployed when a large-scale pre-trained model, which plays the role of a fixed expert, is fine-tuned for learning downstream tasks by including a new contamination part, or prompt, functioning as a new, trainable expert. Despite its popularity and relevance, the theoretical properties of the softmax-contaminated MoE have remained unexplored in the literature. In the paper, we study the convergence rates of the maximum likelihood estimator of gating and prompt parameters in order to gain insights into the statistical properties and potential challenges of fine-tuning with a new prompt. We find that the estimability of these parameters is compromised when the prompt acquires overlapping knowledge with the pre-trained model, in the sense that we make precise by formulating a novel analytic notion of distinguishability. Under distinguishability of the pre-trained and prompt models, we derive minimax optimal estimation rates for all the gating and prompt parameters. By contrast, when the distinguishability condition is violated, these estimation rates become significantly slower due to their dependence on the prompt convergence rate to the pre-trained model. Finally, we empirically corroborate our theoretical findings through several numerical experiments.

Machine unlearning algorithms aim to remove the influence of specific training samples, ideally recovering the model that would have resulted from training on the remaining data alone. We study unlearning in the overparameterized setting, where many models interpolate the data, and defining the solution as any loss minimizer over the retained set—as in prior work in the underparameterized setting—is inadequate, since the original model may already interpolate the retained data and satisfy this condition. In this regime, loss gradients vanish, rendering prior methods based on gradient perturbations ineffective, motivating both new unlearning definitions and algorithms. For this setting, we define the unlearning solution as the minimum-complexity interpolator over the retained data and propose a new algorithmic framework that only requires access to model gradients on the retained set at the original solution. We minimize a regularized objective over perturbations constrained to be orthogonal to these model gradients, a first-order relaxation of the interpolation condition. For different model classes, we provide exact and approximate unlearning guarantees and demonstrate that an implementation of our framework outperforms existing baselines across various unlearning experiments.

We present an information-theoretic framework for discrete diffusion models that yields principled estimators of log-likelihood using score-matching losses. Inspired by the I-MMSE identity for the Gaussian setup, we derive analogous results for the discrete setting. Specifically, we introduce the Information–Minimum Denoising Score Entropy (I-MDSE) relation, which links mutual information between data and its diffused version to the minimum denoising score entropy (DSE) loss. We extend this theory to masked diffusion and establish the Information–Minimum Denoising Cross-Entropy (I-MDCE) relation, connecting cross-entropy losses to mutual information in discrete masked processes. These results provide a time-integral decomposition of the log-likelihood of the data in terms of optimal score-based losses, showing that commonly used losses such as DSE and DCE are not merely variational bounds but tight and principled estimators of log-likelihood. The I-MDCE decomposition further enables practical extensions, including time-free formula, conditional likelihood estimation in prompt–response tasks, and coupled Monte Carlo estimation of likelihood ratios. Experiments on synthetic and real-world data confirm the accuracy, variance stability, and utility of our estimators. The code is publicly available at https://github.com/Dongjae0324/infodis.

Expressivity theory, characterizing which graphs a GNN can distinguish, has become the predominant framework for analyzing GNNs, with new models striving for higher expressivity. However, we argue that this focus is misguided: First, higher expressivity is not necessary for most real-world tasks as these tasks rarely require expressivity beyond the basic WL test. Second, expressivity theory's binary characterization and idealized assumptions fail to reflect GNNs' practical capabilities. To overcome these limitations, we propose Message Passing Complexity (MPC): a continuous measure that quantifies the difficulty for a GNN architecture to solve a given task through message passing. MPC captures practical limitations like over-squashing while preserving the theoretical impossibility results from expressivity theory, effectively narrowing the gap between theory and practice. Through extensive validation on fundamental GNN tasks, we show that MPC's theoretical predictions correlate with empirical performance, successfully explaining architectural successes and failures. Thereby, MPC advances beyond expressivity theory to provide a more powerful framework for understanding and developing GNN architectures.

Given a noisy linear measurement $y = Ax + \xi$ of a distribution $p(x)$, and a good approximation to the prior $p(x)$, when can we sample from the posterior $p(x \mid y)$? Posterior sampling provides an accurate and fair framework for tasks such as inpainting, deblurring, and MRI reconstruction, and several heuristics attempt to approximate it. Unfortunately, approximate posterior sampling is computationally intractable in general. To sidestep this hardness, we focus on (local or global) log-concave distributions $p(x)$. In this regime, Langevin dynamics yields posterior samples when the exact scores of $p(x)$ are available, but it is brittle to score--estimation error, requiring an MGF bound (sub‑exponential error). By contrast, in the unconditional setting, diffusion models succeed with only an $L^2$ bound on the score error. We prove that combining diffusion models with an *annealed* variant of Langevin dynamics achieves conditional sampling in polynomial time using merely an $L^4$ bound on the score error.

The Median-of-Means (MoM) is a robust estimator widely used in machine learning that is known to be (minimax) optimal in scenarios where samples are i.i.d. In more grave scenarios, samples are contaminated by an adversary that can inspect and modify the data. Previous work has theoretically shown the suitability of the MoM estimator in certain contaminated settings. However, the (minimax) optimality of MoM and its limitations under adversarial contamination remain unknown beyond the Gaussian case. In this paper, we present upper and lower bounds for the error of MoM under adversarial contamination for multiple classes of distributions. In particular, we show that MoM is (minimax) optimal in the class of distributions with finite variance, as well as in the class of distributions with infinite variance and finite absolute $(1+r)$-th moment. We also provide lower bounds for MoM's error that match the order of the presented upper bounds, and show that MoM is sub-optimal for light-tailed distributions.

Agnostic learning of Boolean halfspaces is a fundamental problem in computational learning theory, but it is known to be computationally hard even for weak learning. Recent work \citep{chandrasekaran2024smoothed} proposed smoothed analysis as a way to bypass such hardness, but existing frameworks rely on additive Gaussian perturbations, making them unsuitable for discrete domains. We introduce a new smoothed agnostic learning framework for Boolean inputs, where perturbations are modeled via random bit flips. This defines a natural discrete analogue of smoothed optimality generalizing the Gaussian case. Under strictly subexponential assumptions on the input distribution, we give an efficient algorithm for learning halfspaces in this model, with runtime and sample complexity $\tilde{O}(n^{\mathrm{poly}(\frac{1}{\sigma\epsilon})})$. Previously, such algorithms were known only with strong structural assumptions for the discrete hypercube—for example, independent coordinates or symmetric distributions. Our result provides the first computationally efficient guarantee for smoothed agnostic learning of halfspaces over the Boolean hypercube, bridging the gap between worst-case intractability and practical learnability in discrete settings.

Nonlinear activation functions are widely recognized for enhancing the expressivity of neural networks, which is the primary reason for their widespread implementation. In this work, we focus on ReLU activation and reveal a novel and intriguing property of nonlinear activations. By comparing enabling and disabling the nonlinear activations in the neural network, we demonstrate their specific effects on wide neural networks: (a) *better feature separation*, i.e., a larger angle separation for similar data in the feature space of model gradient, and (b) *better NTK conditioning*, i.e., a smaller condition number of neural tangent kernel (NTK). Furthermore, we show that the network depth (i.e., with more nonlinear activation operations) further amplifies these effects; in addition, in the infinite-width-then-depth limit, all data are equally separated with a fixed angle in the model gradient feature space, regardless of how similar they are originally in the input space. Note that, without the nonlinear activation, i.e., in a linear neural network, the data separation remains the same as for the original inputs and NTK condition number is equivalent to the Gram matrix, regardless of the network depth. Due to the close connection between NTK condition number and convergence theories, our results imply that nonlinear activation helps to improve the worst-case convergence rates of gradient based methods.

We initiate the study of contextual dynamic pricing with a heterogeneous population of buyers, where a seller repeatedly posts prices (over $T$ rounds) that depend on the observable $d$-dimensional context and receives binary purchase feedback. Unlike prior work assuming homogeneous buyer types, in our setting the buyer's valuation type is drawn from an unknown distribution with finite support size $K_{\star}$. We develop a contextual pricing algorithm based on optimistic posterior sampling with regret $\widetilde{O}(K_{\star}\sqrt{dT})$, which we prove to be tight in $d$ and $T$ up to logarithmic terms. Finally, we refine our analysis for the non-contextual pricing case, proposing a variance-aware zooming algorithm that achieves the optimal dependence on $K_{\star}$.

A function \( f : \mathbb{F}_2^n \to \mathbb{R} \) is said to be \( s \)-Fourier sparse if its Fourier expansion contains at most \( s \) nonzero coefficients. In general, the existence of a sparse representation in the Fourier basis serves as a key enabler for the design of efficient learning algorithms. However, most existing techniques assume prior knowledge of the function’s Fourier sparsity, with algorithmic parameters carefully tuned to this value. This motivates the following decision problem: given \( s > 0 \), determine whether a function is \( s \)-Fourier sparse. In this work, we study the problem of tolerant testing of Fourier Sparsity for real-valued functions over \( \mathbb{F}_2^n \), accessed via oracle queries. The goal is to decide whether a given function is close to being \( s \)-Fourier sparse or far from every \( s \)-Fourier sparse function. Our algorithm provides an estimator that, given oracle access to the function, estimates its distance to the nearest \( s \)-Fourier sparse function with query complexity \( \widetilde{O}(s) \), for constant accuracy and confidence parameters. A key structural ingredient in our analysis is a new spectral concentration result for real-valued functions over \( \mathbb{F}_2^n \) when restricted to small-dimensional random affine subspaces. We further complement our upper bound with a matching lower bound of \( \Omega(s) \), establishing that our tester is optimal up to logarithmic factors. The lower bound exploits spectral properties of a class of cryptographically hard functions, namely, the Maiorana--McFarland family, in a novel way.

Binary classification in the classic PAC model exhibits a curious phenomenon: Empirical Risk Minimization (ERM) learners are suboptimal in the realizable case yet optimal in the agnostic case. Roughly speaking, this owes itself to the fact that non-realizable distributions $\\mathcal{D}$ are more difficult to learn than realizable distributions -- even when one discounts a learner's error by $\\mathrm{err}(h^\\ast_\\mathcal{D})$, i.e., the error of the best hypothesis in $\\mathcal{H}$. Thus, optimal agnostic learners are permitted to incur excess error on (easier-to-learn) distributions $\\mathcal{D}$ for which $\\tau = \\mathrm{err}(h^\\ast_\\mathcal{D})$ is small. Recent work of Hanneke, Larsen, and Zhivotovskiy (FOCS '24) addresses this shortcoming by including $\\tau$ itself as a parameter in the agnostic error term. In this more fine-grained model, they demonstrate tightness of the error lower bound $\\tau + \\Omega \\left(\\sqrt{\\frac{\\tau (d + \\log(1 / \\delta))}{m}} + \\frac{d + \\log(1 / \\delta)}{m} \\right)$ in a regime where $\\tau > d/m$, and leave open the question of whether there may be a higher lower bound when $\\tau \\approx d/m$, with $d$ denoting $\\mathrm{VC}(\\mathcal{H})$. In this work, we resolve this question by exhibiting a learner which achieves error $c \\cdot \\tau + O \\left(\\sqrt{\\frac{\\tau (d + \\log(1 / \\delta))}{m}} + \\frac{d + \\log(1 / \\delta)}{m} \\right)$ for a constant $c \\leq 2.1$, matching the lower bound and demonstrating optimality when $\\tau =O( d/m)$. Further, our learner is computationally efficient and is based upon careful aggregations of ERM classifiers, making progress on two other questions of Hanneke, Larsen, and Zhivotovskiy (FOCS '24). We leave open the interesting question of whether our approach can be refined to lower the constant from 2.1 to 1, which would completely settle the complexity of agnostic learning.

Hypergraphs effectively model higher-order relationships in natural phenomena, capturing complex interactions beyond pairwise connections. We introduce a novel hypergraph message passing framework inspired by interacting particle systems, where hyperedges act as fields inducing shared node dynamics. By incorporating attraction, repulsion, and Allen-Cahn forcing terms, particles of varying classes and features achieve class-dependent equilibrium, enabling separability through the particle-driven message passing. We investigate both first-order and second-order particle system equations for modeling these dynamics, which mitigate over-smoothing and heterophily thus can capture complete interactions. The more stable second-order system permits deeper message passing. Furthermore, we enhance deterministic message passing with stochastic element to account for interaction uncertainties. We prove theoretically that our approach mitigates over-smoothing by maintaining a positive lower bound on the hypergraph Dirichlet energy during propagation and thus to enable hypergraph message passing to go deep. Empirically, our models demonstrate competitive performance on diverse real-world hypergraph node classification tasks, excelling on both homophilic and heterophilic datasets. Source code is available at \href{https://github.com/Xuan-Elfin/HAMP}{the link}.