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新论文 · v1arXiv:2610.38212

Scalable Bayesian Quadrature via Operator-Adapted Features

Huanyan Zhu, Cheng Li

原文摘要

Kernel methods face two recurrent bottlenecks: the cubic cost of inverting an $n\times n$ kernel matrix, as in kernel ridge regression, and the evaluation of kernel integrals, as required in kernel quadrature. Existing low-rank kernel and Gaussian process approximations primarily address the former bottleneck, but do not control the downstream quadrature error when both the kernel matrix and kernel mean are approximated, particularly when the integrand lies outside the kernel reproducing kernel Hilbert space. We address both challenges through an operator-adapted finite-feature approximation. Specifically, we construct a finite-dimensional working kernel from the eigenvalues and eigenfunctions of a second-order differential operator. We establish convergence guarantees for the resulting finite-feature estimators in kernel quadrature, with extension to other related kernel methods. Our results show that, with $m \lesssim n$ basis functions, the truncated estimator attains the corresponding benchmark statistical rates. The theory covers fixed/random designs, noisy/noiseless observations, oversmoothing/undersmoothing regimes, and Matérn/Gaussian-type weights. We provide explicit error bounds that quantify the roles of the spectral cutoff, regularization parameter, lengthscale, sampling geometry, and operator boundary condition. We verify the convergence results in numerical experiments.

math.ST · math.NA · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.38206

Averaged Annealed Markov Chains Achieve the Algorithmic Threshold for Tensor PCA, Part II: Langevin Dynamics

Aniket Das, Tselil Schramm

原文摘要

We consider the tensor principal components analysis (tensor PCA) problem on the sphere, in which we are asked to recover an unknown vector in $\mathcal{S}_N=\{x\in\mathbb{R}^N:\lVert x\rVert^2=N\}$ from a noisy observation of its $k$-th tensor power, for a fixed $k\ge3$. Tensor PCA exhibits a well-studied information-computation gap: global algorithms such as spectral methods and sum-of-squares work above an algorithmic threshold, below which we know matching lower bounds (against these and some other restricted models of computation). In contrast, local algorithms such as Markov chains were only known to succeed from a random initialization at a polynomially larger signal-to-noise ratio, and were believed to be inferior to global algorithms for tensor PCA. Continuing the program initiated in Part I [arXiv:2610.16705], we show that spherical Langevin dynamics, when supplemented with annealing and ergodic averaging, achieves polynomial-time strong recovery above the algorithmic threshold from a uniformly random initialization. Together with Part I, our results demonstrate that local Markov chains can succeed all the way down to the algorithmic threshold from an uninformative cold start.

cs.LG · cs.DS · math.PR · math.ST

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.37866

Finite-Sample Optimal Conformal Prediction Sets under Covariate Shift

Zhongyang Li

原文摘要

We study conformal prediction under covariate shift with a known covariate density ratio, seeking prediction sets with minimum expected length under the target distribution. For every calibration sample size, we prove that the rule combining the inverse conditional-density score with randomized weighted conformal calibration minimizes this expected length among all rules valid over the full weighted-exchangeability class. We approximate the oracle using quantile and kernel histograms, including forest-weighted estimators, and select estimators and tuning parameters by resampling to estimate calibrated target length. On a bounded response domain, we bound excess length linearly in $L^1$ density error at fixed calibration size, without requiring a uniform positive lower bound on the true density. We also bound selection-pool and resampling errors. Under explicit growth conditions, the selected procedure approaches the same oracle while the calibration size remains fixed. Experiments on 36 known-density configurations and four public regression datasets compare coverage, prediction-set length, and runtime with existing methods. In targeted low-coverage and strong-shift comparisons, inverse-density selection yields shorter mean sets in most cases and lower mean runtime than the evaluated weighted baselines, with some differences in achieved coverage.

stat.ML · cs.LG · math.CO · math.PR · math.ST

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2602.02491

New explanations and inference for least angle regression

Karl B. Gregory, Daniel J. Nordman

原文摘要

Efron et al. (2004) introduced least angle regression (LAR) as an algorithm for linear predictions, intended as an alternative to forward selection with connections to penalized regression. However, LAR has remained somewhat of a "black box," where some basic behavioral properties of LAR output are not well understood, including an appropriate termination point for the algorithm. We provide a novel framework for inference with LAR, which also allows LAR to be understood from new perspectives with several newly developed mathematical properties. The LAR algorithm at a data level can viewed as estimating a population counterpart "path" that organizes a response mean along regressor variables which are ordered according to a decreasing series of population "correlation" parameters; such parameters are shown to have meaningful interpretations for explaining variable contributions whereby zero correlations denote unimportant variables. In the output of LAR, estimates of all non-zero population correlations turn out to have independent normal distributions for use in inference, while estimates of zero-valued population correlations have a certain non-normal joint distribution. These properties help to provide a formal rule for stopping the LAR algorithm. While the standard bootstrap for regression can fail for LAR, a modified bootstrap provides a practical and formally justified tool for interpreting the entrance of variables and quantifying uncertainty in estimation. The LAR inference method is studied through simulation and illustrated with data examples.

math.ST · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36581

Wasserstein approximation of measures via orthogonal frames: statistical and computational complexities

Mateo Díaz, Nicolás García Trillos, Daniel López-Castaño, Congwei Yang

原文摘要

Dimension-reduction methods such as PCA identify a subspace capturing most of the variation in the data, but not a distinguished basis within it. Selecting a basis in which the samples are sparsely represented is a recurring task in unsupervised learning, addressed by methods in factor analysis, dictionary learning, and subspace clustering. We pose this task as Wasserstein approximation of the data distribution by measures supported on the axes of an orthogonal frame, which, in turn, reduces to a nonsmooth optimization problem over the orthogonal group $\mathrm{O}(d)$. We show that, under mild assumptions, both empirical risk minimizers and a locally initialized stochastic gradient method recover the population-optimal frame at the near-parametric rate $\widetilde{O}_d(n^{-1/2})$, where $n$ is the sample size. The main difficulty is that, although the population objective is smooth, the empirical objective is not-this mismatch precludes standard covering arguments. We overcome this issue by replacing Lipschitz continuity of the empirical gradient with a coarsely Lipschitz bound, a tool that may be of independent interest. Numerical experiments support our theoretical findings.

math.ST · math.OC · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2301.12022

Epsilon-Identifiability of Causal Quantities

Ang Li, Scott Mueller, Xin Shu, Judea Pearl

原文摘要

Identifying the effects of causes and causes of effects is vital in virtually every scientific field. Often, however, the needed probabilities may not be fully identifiable from the available data sources. This paper shows how approximate identifiability is still possible for several probabilities of causation. We term this epsilon-identifiability and demonstrate its usefulness in cases where the behavior of certain subpopulations can be restricted within sufficiently narrow bounds. In particular, we show how unidentifiable causal effects and counterfactual probabilities can be epsilon-identified when such allowances are made. Often, these allowances are easily measured and reasonably assumed. Finally, epsilon-identifiability is applied to the unit selection problem.

cs.AI · math.ST

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36488

Second-order Sinkhorn Geometry and $\sqrt{n}$-Inference

Tristan Luca Saidi, Florian Gunsilius, Gonzalo Mena

原文摘要

The Sinkhorn divergence, a debiased form of entropic optimal transport, has been widely used in various scientific disciplines due to its computational and statistical tractability. Unlike the Wasserstein distance, however, the Sinkhorn divergence is not a metric and it does not naturally admit a formal Riemannian structure. These properties are central to many applications of optimal transport methods that utilize interpolation, barycentric averaging, parallel transport or other geometric operations. Recent work addresses these shortcomings by constructing a formal Riemannian tangent space and a metric tensor on the space of probability measures that quadratically approximates the Sinkhorn divergence. In this work, we extend this framework with two major threads of contributions. For a suitable class of finite-smoothness cost functions, we prove Hadamard differentiability of the Sinkhorn metric tensor and we derive the Levi-Civita covariant derivative in this geometry; we then use the covariant derivative to obtain the geodesic and parallel transport equations, each of which amount to an ODE on a fixed Reproducing Kernel Hilbert Space (RKHS) with smooth coefficients. Our second contribution is to linearize the solutions to the geodesic and parallel transport equations using standard tools from Banach space ODE theory. We show that under standard regularity conditions these linearizations imply $\sqrt{n}$-limiting Gaussian processes centered at population values for the plugin estimators of the exponential map, the logarithmic map, parallel transport and the induced Sinkhorn distance. Moreover, we show that the nonparametric bootstrap is consistent for all objects mentioned. To the best of our knowledge, these are the first $\sqrt{n}$--limit theorems and bootstrap consistency results for canonical geometric operations in a transport-based geometry on the space of probability measures.

stat.ML · cs.LG · math.ST

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36458

A Simple Presentation of Robust Membership Inference Attacks

Renaud Gaucher, Ephraim Linder, Adam Smith, Jonathan Ullman, Lydia Zakynthinou

原文摘要

In a membership inference attack, an attacker is given the data of a particular individual, and tries to determine if that individual was included in a particular dataset, given some kind of statistics derived from the dataset or a model trained on the dataset. Roughly, a membership inference attack is robust if it succeeds even when the statistics are highly noisy, regardless of how that noise is introduced. We give a simple presentation of robust membership inference attacks for mean estimation, covering both multivariate Gaussian distributions and binary product distributions. We also give a summary of the literature on robust membership inference, covering history, extensions, and connections to other research areas.

cs.CR · cs.IT · cs.LG · math.ST

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.02141

On the Optimal Sample Complexity of Offline Multi-Armed Bandits with KL Regularization

Kaixuan Ji, Qiwei Di, Heyang Zhao, Qingyue Zhao, Quanquan Gu

原文摘要

Kullback-Leibler (KL) regularization is widely used in offline decision-making and offers several benefits, motivating recent work on the sample complexity of offline learning with respect to KL-regularized performance metrics. Nevertheless, the exact sample complexity of KL-regularized offline learning remains largely from fully characterized. In this paper, we study this question in the setting of multi-armed bandits (MABs). We provide a sharp analysis of KL-PCB (Zhao et al., 2026), showing that it achieves a sample complexity of $\tilde{O}(ηSAC^{π^*}/ε)$ under large regularization $η= \tilde{O}(ε^{-1})$, and a sample complexity of $\tildeΩ(SAC^{π^*}/ε^2)$ under small regularization $η= \tildeΩ(ε^{-1})$, where $η$ is the regularization parameter, $S$ is the number of contexts, $A$ is the number of arms, $C^{π^*}$ policy coverage coefficient at the optimal policy $π^*$, $ε$ is the desired sub-optimality, and $\tilde{O}$ and $\tildeΩ$ hide all poly-logarithmic factors. We further provide a pair of sharper sample complexity lower bounds, which matches the upper bounds over the entire range of regularization strengths. Overall, our results provide a nearly complete characterization of offline multi-armed bandits with KL regularization.

cs.LG · cs.AI · math.ST · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36205

Approximation Families and the Statistical Efficiency of Denoising Score Matching

Brian Miyatake

原文摘要

We study score matching over finite-dimensional approximation spaces generated by gradients of eigenfunctions of the infinitesimal generator of a forward diffusion. We compare polynomial approximation under one-dimensional Ornstein-Uhlenbeck flow with trigonometric-polynomial approximation under heat flow on the torus. In the Ornstein-Uhlenbeck setting, we show that for every non-Gaussian initial distribution uniformly comparable to the standard Gaussian, the asymptotic score-estimation variance grows exponentially along a subsequence of polynomial degrees. Averaging the score-matching loss under the diffusion semigroup, which we call operator-informed score matching (OISM), reduces the variance at each fixed degree but does not eliminate this exponential instability at sufficiently small times. A local minimax lower bound shows that the same high-degree dependence arises when estimating the polynomial projection of the score, rather than being an artifact of OISM. In contrast, under heat flow on the torus, OISM over trigonometric polynomials admits finite-sample control of the score error along the forward diffusion and yields the classical minimax rate for estimating sufficiently smooth densities under expected Kullback-Leibler loss.

stat.ML · cs.LG · math.ST

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2608.02487

Computational and Statistical Guarantees of the c-Rectified flow

Leda Wang, Zhehao Xu, Qiang Liu, Harrison H. Zhou

原文摘要

Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying c-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. More generally, for a Gaussian to log-concave coupling, the ordinary rectified flow can reach a fixed point after one step from independent initialization. In contrast, under suitable regularity and uniform-integrability assumptions, iterative c-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative c-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension \(d \ge 3\) and a nearly parametric rate for \(d=1,2\).

stat.ML · cs.LG · math.OC · math.PR · math.ST

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2411.09731

Temporal difference learning adapts to incomplete coverage: an oracle inequality over subgraphs

Wenlong Mou, Jian Qian

原文摘要

Temporal difference (TD) and Monte Carlo (MC) are two fundamental principles for value function estimation in reinforcement learning. TD is asymptotically efficient for value estimation, whereas MC requires only adequate target visitation. We show that these advantages are automatically combined by ordinary TD in the tabular setting. We identify the subgraph Bellman variance, arising from bootstrapping within a subgraph and rollouts outside it, as a fundamental quantity connecting the MC and efficient TD benchmarks. With independent trajectories, we prove a non-asymptotic oracle inequality: the mean squared error of full TD is bounded by this variance divided by the sample size for every sufficiently covered subgraph containing the target, up to a higher-order remainder whose occupancy dependence is confined to that subgraph. No subgraph is supplied to the algorithm. TD thus adapts without selecting a subgraph or requiring global coverage. A finite-sample minimax lower bound shows that the asymptotic variance of TD need not be attainable before rare states are adequately observed. Experimental studies examine the transition between these variance scales and its dependence on visitation structure.

cs.LG · math.ST · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35821

Recertifying decisions via e-closure: uniform improvements of conformal risk control and randomized e-BH

Tian Bai, Aaditya Ramdas

原文摘要

Statistical procedures often control expected loss at a prescribed level while leaving part of their risk budget unused. Building on the e-value representation of risk-controlled decisions and the e-closure principle for multiple testing, we formulate recertification, a general paradigm for improving procedures by exploiting computable slack in their validity certificates. A valid procedure's losses normalized by their risk budgets form a state-indexed e-value collection; conversely, decisions whose normalized losses are bounded by this collection inherit its risk guarantees. Recertification uses distributional structure to quantify slack, enlarge the certifying e-values while preserving validity, and construct less conservative decisions. We develop two applications. For conformal risk control (CRC) with a known loss family, recertification redistributes unused risk budget among the held-out points of each hypothetical augmented sample. The resulting controller is valid under every finite exchangeable law and no more conservative than CRC on any calibration sample. For convex losses on an interval of actions, repeated recertification converges geometrically to an admissible controller. For false discovery rate control with arbitrarily dependent e-values, recertification exploits unused conditional budgets in randomized e-BH. It yields a closed-form randomized procedure whose rejection set contains that of randomized e-BH for every realization of the shared randomization and whose expected number of rejections is maximal subject to the same conditional budget constraints. Recertifying with the larger e-collection of closed e-BH gives a second closed-form procedure whose rejection set contains that of the first. Experiments on CIFAR-100 hierarchical classification and multiple testing simulations demonstrate improved budget utilization, informativeness and power over the baselines.

stat.ME · math.ST · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2511.07414

Wasserstein-Cramér-Rao Theory of Unbiased Estimation

Nicolás García Trillos, Adam Quinn Jaffe, Bodhisattva Sen

原文摘要

The quantity of interest in the classical Cramér-Rao theory of unbiased estimation (e.g., the Cramér-Rao lower bound, its exact attainment for exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently-sampled data set from the same distribution. In this paper we are interested in a quantity which represents the instability of an estimator when its value is compared to the value for an infinitesimal additive perturbation of the original data set; we refer to this as the "sensitivity" of an estimator. The resulting theory of sensitivity is based on the Wasserstein geometry in the same way that the classical theory of variance is based on the Fisher-Rao (equivalently, Hellinger) geometry, and this insight allows us to determine a collection of results which are analogous to the classical case: a Wasserstein-Cramér-Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models in which there exist unbiased estimators achieving the lower bound exactly, and various results on the asymptotic achievement of the lower bound. More precisely, we show that Wasserstein projection estimators (and one-step approximations thereof) are asymptotically sensitivity-efficient under suitable assumptions, and that there exist Hodges-type estimators which exhibit super-sensitivity-efficiency. We use these results to treat many statistical examples, sometimes revealing new optimality properties for existing estimators and other times revealing entirely new estimators.

math.ST · math.OC · stat.ME · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.25821

Optimal Priors for Betting Tests with Bounded Payoffs: An Exact Expected-Time Characterization

Francesco Orabona

原文摘要

Testing by betting is a powerful framework that recasts hypothesis testing as a betting game. A popular approach spreads the initial wealth over fixed bets according to a prior distribution, using a universal portfolio algorithm. For suitable priors, this approach achieves the optimal expected rejection time as the error probability goes to zero at a fixed alternative. Yet, priors with this guarantee can behave quite differently near the null. This motivates the question: Which prior should we use in testing-by-betting algorithms? We study bounded scalar payoffs and, under technical conditions, prove a sharp formula for expected rejection time, uniform over all bounded alternatives as the error probability tends to zero. The formula shows how the prior's mass around the zero bet determines the cost of adapting to an unknown alternative. We characterize the priors that attain the best possible leading constant when the error probability and alternative vary together. We also show that no valid test can uniformly improve on this cost across ranges of alternatives near the null. In particular, the uniform prior can require arbitrarily more observations than suitable priors that concentrate around the zero bet. A general inversion argument converts our results into sharp expected times to reach a prescribed confidence-sequence width. We also give a concrete prior choice and an FFT-based algorithm with almost linear computational complexity.

math.ST · cs.IT · stat.ME

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35656

Provably Efficient Q-learning for Robust Reinforcement Learning

Yuchen Jiao, Gen Li

原文摘要

Standard reinforcement learning (RL) algorithms based on Markov Decision Processes (MDPs) are vulnerable to sim-to-real gaps. To address this issue, distributionally robust RL algorithms based on robust MDPs (RMDPs) have been proposed, for ensuring that agents can perform reliably to unpredictable conditions in real-world environments. Despite of the rich literature, \revision{an efficient model-free method is still lacking} for robust RL, particularly for Markovian data, which is more relevant in practical applications. In this work, we take a step towards settling this problem by proposing a sample efficient (asynchronous Q-learning) algorithm for robust RL. In particular, consider a $γ$-discounted infinite-horizon RMDP with state space $\mathcal{S}$ and action space $\mathcal{A}$, where the uncertainty set is specified by the total variation (TV) distance with uncertainty level $σ$. Assuming access to a single trajectory of Markovian samples induced by a behavior policy and the nominal transition kernel, the sample complexity of the proposed Q-learning algorithm --- to achieve an entrywise $\varepsilon$-approximation of the optimal Q-function --- is bounded by $$ \frac{1}{μ_{\mathsf{min}}\varepsilon^2(1-γ)^3\max\{1-γ,σ\}} + \frac{t_{\mathsf{mix}}}{μ_{\mathsf{min}}(1-γ)} $$ up to logarithmic factors. Here, $t_{\mathsf{mix}}$ and $μ_{\mathsf{min}}$ denote the mixing time and the minimum state-action occupancy probability of the sample trajectory, respectively. This result is nearly minimax optimal, with an additional factor of $\frac{1}{1-γ}$. Moreover, it achieves a significant improvement over existing bounds.

stat.ML · cs.AI · cs.LG · math.ST

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35490

Minimax Rates for Conditional Generative Learning under Wasserstein Loss

Sungwon Park, Kyoung-Sook Moon, Hongjoong Kim

原文摘要

We study minimax rates for conditional generative learning under integrated Wasserstein loss, with Hölder continuous conditional laws and an unrestricted covariate distribution. We evaluate unrestricted generators through their conditional output distributions. For expected Wasserstein loss, the minimax rate agrees with the rate under uniform covariates and follows from balancing localization bias against unconditional distribution estimation error. Higher moments of the loss can instead be dominated by unobserved covariate regions, which force an additional risk contribution of order $n^{-1}$, where $n$ is the sample size. An adaptive localization argument gives matching upper bounds. The results cover unrestricted response laws and possibly vanishing Besov densities for $W_1$, and extend to higher Wasserstein orders under uniform response density bounds and integrability conditions. A multiscale lower bound closes the logarithmic gap in unconditional $W_1$ minimax rates in dimension two for both unrestricted distributions and Besov density classes. We also establish minimax rates for conditional Brenier maps under density, integrability, and uniform curvature conditions.

math.ST · cs.LG · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35330

On the asymptotic behaviour of $α$-divergence generalized posteriors

Ioar Casado-Telletxea, Omar Rivasplata

原文摘要

We study the asymptotic behaviour of the solutions to the unconstrained generalized variational inference problem regularized by an $α$-divergence. Under mild conditions, the minimizer is explicit up to a scalar normalizing constant, and we show that it admits a rescaling under which it converges in $L^1$ to a fixed non-Gaussian law, at a concentration scale around the empirical loss minimizer that departs from the typical $n^{-1/2}$ in the classical Bernstein--von Mises theorem. For $α\in(0,1)$ and $d<2/(1-α)$ the limit is a Student-$t$ distribution and the posterior width shrinks faster than the sampling error of the empirical minimizer. Conversely, for $α>1$ the limiting distribution is a compactly supported Pearson type-II distribution, and the posterior width shrinks more slowly than the sampling error. The KL divergence ($α=1$) is the unique member of the $α$-divergence family whose posterior width matches the sampling scale and whose limit is Gaussian. We discuss the implications of these qualitatively different regimes for the frequentist coverage of credible regions, and a toy example on intercept-only logistic regression illustrates the theoretical claims.

math.ST · stat.ML

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arXiv ↗PDF ↗
版本更新 · v5arXiv:2407.11353

Sharp Generalization for Nonparametric Regression in Interpolation Space by Over-Parameterized Neural Networks Trained with Preconditioned Gradient Descent and Early Stopping

Yingzhen Yang, Ping Li

原文摘要

We study nonparametric regression using an over-parameterized two-layer neural network trained with algorithmic guarantees in this paper. We consider the setting where the training features are drawn uniformly from the unit sphere in $\RR^d$, and the target function lies in an interpolation space commonly studied in statistical learning theory. We demonstrate that training the neural network with a novel Preconditioned Gradient Descent (PGD) algorithm, equipped with early stopping, achieves a sharp regression rate of $\cO(n^{-\frac{2αs'}{2αs'+1}})$, where $α$ parametrizes the polynomial eigenvalue decay of the regular Neural Tangent Kernel (NTK), $λ_j\asymp j^{-2α}$, and $2α=d/(d-1)$ under the spherical uniform design considered here. The target function is assumed to belong to the interpolation space $\bth{\cH_K}^{s'}$ with $s'\ge3$. This rate is even sharper than the currently known nearly-optimal rate of $\cO(n^{-\frac{2αs'}{2αs'+1}})\log^2(1/δ)$~\citep{Li2024-edr-general-domain}, where $n$ is the size of the training data and $δ\in (0,1)$ is a small probability. Our analysis is based on two key technical contributions. First, we present a principled decomposition of the network output at each PGD step into a function in the reproducing kernel Hilbert space (RKHS) of a newly induced integral kernel, and a residual function with small $L^{\infty}$-norm. Second, leveraging this decomposition, we apply local Rademacher complexity theory to tightly control the complexity of the function class comprising all the neural network functions obtained in the PGD iterates. Our results show how PGD changes the effective kernel in the reference training dynamics, inducing an integral kernel with lower complexity than the regular NTK arising from vanilla GD.

stat.ML · cs.LG · math.ST

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2501.02624

Simultaneous analysis of approximate leave-one-out cross-validation and mean-field inference

Pierre C Bellec

原文摘要

Approximate Leave-One-Out Cross-Validation (ALO-CV) is a method that has been proposed to estimate the generalization error of a regularized estimator in the high-dimensional regime where dimension and sample size are of the same order, the so-called ``proportional regime''. A new analysis is developed to derive the consistency of ALO-CV for non-differentiable regularizers under Gaussian covariates and strong convexity. Using a conditioning argument, the difference between the ALO-CV weights and their counterparts in mean-field inference is shown to be small. Combined with upper bounds between the mean-field inference estimate and the leave-one-out quantity, this provides a proof that ALO-CV approximates the leave-one-out quantity up to negligible error terms. Linear models with square loss, robust linear regression and single-index models are explicitly treated.

math.ST · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.34484

Intrinsic Gradient Barriers in Pushforward Deep Generative Models

Edric Tam, Qingyang Liu, Barbara Engelhardt, Ray Bai

原文摘要

Broad classes of deep generative models (DGMs) operate by learning to map a simple source distribution, such as a standard Gaussian, to a complex high-dimensional target distribution. Despite their success, DGMs often empirically struggle to accurately represent distributions with heavy tails or well-separated modes. In this work, we show that these difficulties reflect intrinsic properties of the underlying transport problem rather than particular neural network architectures or training procedures. We establish quantitative lower bounds on the gradients and Jacobians of pushforward mappings that transport log-concave source distributions to such targets. For heavy-tailed target distributions, the required gradient magnitude grows as the target develops increasingly large moments. For multimodal target distributions, the low-probability regions between well-separated modes similarly force large gradients in the transport map. We discuss how such large derivatives relate to trade-offs between expressivity and numerical stability. Our results hold under substantially weaker regularity conditions than previous analyses and remain stable under perturbations of the target distribution. Finally, we show that these obstructions can be mitigated when the goal is approximation of a given density rather than generative modeling. Specifically, we propose simple classes of bijective reparametrizations that can alleviate the difficult geometric structure and reduce the gradient burden placed on the learned generative map. Numerical experiments support our theoretical analysis.

stat.ML · cs.LG · math.ST

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2503.03206Google Research

An Analytical Theory of Spectral Bias in the Learning Dynamics of Diffusion Models

Binxu Wang, Cengiz Pehlevan

原文摘要

We develop an analytical framework for understanding how the generated distribution evolves during diffusion model training. Leveraging a Gaussian-equivalence principle, we solve the full-batch gradient-flow dynamics of linear and convolutional denoisers and integrate the resulting probability-flow ODE, yielding analytic expressions for the generated distribution. The theory exposes a universal inverse-variance spectral law: the time for an eigen- or Fourier mode to match its target variance scales as $τ\proptoλ^{-1}$, so high-variance (coarse) structure is mastered orders of magnitude sooner than low-variance (fine) detail. Extending the analysis to deep linear networks and circulant full-width convolutions shows that weight sharing merely multiplies learning rates -- accelerating but not eliminating the bias -- whereas local convolution introduces a qualitatively different bias. Experiments on Gaussian and natural-image datasets confirm the spectral law persists in deep MLP-based UNet. Convolutional U-Nets, however, display rapid near-simultaneous emergence of many modes, implicating local convolution in reshaping learning dynamics. These results underscore how data covariance governs the order and speed with which diffusion models learn, and they call for deeper investigation of the unique inductive biases introduced by local convolution.

cs.LG · cs.CV · math.ST · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2510.02305Google Research

Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive

Tyler Farghly, Peter Potaptchik, Samuel Howard, George Deligiannidis, Jakiw Pidstrigach

原文摘要

Diffusion models have achieved state-of-the-art performance, demonstrating remarkable generalisation capabilities across diverse domains. However, the mechanisms underpinning these strong capabilities remain only partially understood. A leading conjecture, based on the manifold hypothesis, attributes this success to their ability to adapt to low-dimensional geometric structure within the data. This work provides evidence for this conjecture, focusing on how such phenomena could result from the formulation of the learning problem through score matching. We inspect the role of implicit regularisation by investigating the effect of smoothing minimisers of the empirical score matching objective. Our theoretical and empirical results confirm that smoothing the score function -- or equivalently, smoothing in the log-density domain -- produces smoothing tangential to the data manifold. In addition, we show that the manifold along which the diffusion model generalises can be controlled by choosing an appropriate smoothing.

cs.LG · cs.AI · math.ST · stat.ML

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arXiv ↗PDF ↗
版本更新 · v4arXiv:2110.15073Google Research

MMD Aggregated Two-Sample Test

Antonin Schrab, Ilmun Kim, Mélisande Albert, Béatrice Laurent, Benjamin Guedj, Arthur Gretton

原文摘要

We propose two novel nonparametric two-sample kernel tests based on the Maximum Mean Discrepancy (MMD). First, for a fixed kernel, we construct an MMD test using either permutations or a wild bootstrap, two popular numerical procedures to determine the test threshold. We prove that this test controls the probability of type I error non-asymptotically. Hence, it can be used reliably even in settings with small sample sizes as it remains well-calibrated, which differs from previous MMD tests which only guarantee correct test level asymptotically. When the difference in densities lies in a Sobolev ball, we prove minimax optimality of our MMD test with a specific kernel depending on the smoothness parameter of the Sobolev ball. In practice, this parameter is unknown and, hence, the optimal MMD test with this particular kernel cannot be used. To overcome this issue, we construct an aggregated test, called MMDAgg, which is adaptive to the smoothness parameter. The test power is maximised over the collection of kernels used, without requiring held-out data for kernel selection (which results in a loss of test power), or arbitrary kernel choices such as the median heuristic. We prove that MMDAgg still controls the level non-asymptotically, and achieves the minimax rate over Sobolev balls, up to an iterated logarithmic term. Our guarantees are not restricted to a specific type of kernel, but hold for any product of one-dimensional translation invariant characteristic kernels. We provide a user-friendly parameter-free implementation of MMDAgg using an adaptive collection of bandwidths. We demonstrate that MMDAgg significantly outperforms alternative state-of-the-art MMD-based two-sample tests on synthetic data satisfying the Sobolev smoothness assumption, and that, on real-world image data, MMDAgg closely matches the power of tests leveraging the use of models such as neural networks.

stat.ML · cs.LG · math.ST · stat.ME

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arXiv ↗PDF ↗