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版本更新 · v3arXiv:2608.24007

Revenge of Monosemanticity: Neuron Specialization as a New Form of Feature Learning in MLPs

Amirhesam Abedsoltan, Enric Boix-Adsera, Fivos Kalogiannis, Mikhail Belkin

原文摘要

Understanding how neural networks learn and organize features is central to understanding their behavior. Much existing theory of feature learning has focused on the emergence of a global low-dimensional representation. We show that this picture is incomplete. In regression problems with clustered data, we demonstrate that multilayer perceptrons (MLPs) develop specialized neurons: individual neurons become strongly aligned with specific features relevant to a particular region of the input space. A substantial fraction of these neurons exhibit monosemanticity: they activate predominantly on inputs from the corresponding cluster. Thus, rather than learning a single global low-dimensional representation, MLPs learn a collection of local low-dimensional representations. Furthermore, we show that this ability to specialize gives MLPs a provable data-efficiency advantage over feature-learning methods based on a global low-dimensional representation.

cs.LG · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2609.38403

Advantage of Sample Complexity in Quantum PAC Learning Requires Inverse Access to State-Preparation Unitaries

Natsuto Isogai, Satoshi Yoshida, Mio Murao

原文摘要

Whether quantum computation can reduce the amount of data required for conventional machine learning is a central problem in quantum machine learning. In probably approximately correct (PAC) learning, a learner receives training examples sampled from an unknown probability distribution. Quantum PAC learning extends this framework by allowing access to quantum examples, which are quantum states that coherently encode this distribution, or to unitaries that prepare quantum examples, with or without inverse access. When only copies of quantum examples are available, the optimal worst-case sample complexity asymptotically matches that of classical learning. In contrast, access to both a preparation unitary and its inverse can improve the accuracy dependence of query complexity. However, it has remained open whether access to the preparation unitary alone suffices for this improvement. In this work, we resolve this question by determining the optimal query complexity with forward-only access, taking the worst case over compatible preparation unitaries and their dimensions. In both realizable and agnostic PAC learning, this complexity matches, up to constant factors, the optimal sample complexity achievable with classical examples or copies of quantum examples. To prove these results, we construct, for any algorithm with forward-only access, a simulator that approximates the algorithm's Haar-averaged output state using as many copies of the prepared state as the algorithm makes queries. These results show that inverse access is essential for the known improvement in accuracy dependence in this worst-case setting. Beyond PAC learning, our reduction provides a general tool for transferring lower bounds for state copies to query lower bounds for forward-only access to the corresponding preparation unitaries.

quant-ph · cs.LG · stat.ML

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arXiv ↗PDF ↗
版本更新 · v4arXiv:2506.01393

Improved Regret Bounds for Gaussian Process Upper Confidence Bound in Bayesian Optimization

Shogo Iwazaki

原文摘要

This paper addresses the Bayesian optimization problem (also referred to as the Bayesian setting of the Gaussian process bandit), where the learner seeks to minimize the regret under a function drawn from a known Gaussian process (GP). Under a Matérn kernel with a certain degree of smoothness, we show that the Gaussian process upper confidence bound (GP-UCB) algorithm achieves $\tilde{O}(\sqrt{T})$ cumulative regret with high probability. Furthermore, our analysis yields $O(\sqrt{T \ln^2 T})$ regret under a squared exponential kernel. These results fill the gap between the existing regret upper bound for GP-UCB and the best-known bound provided by Scarlett (2018). The key idea in our proof is to capture the concentration behavior of the input sequence realized by GP-UCB, enabling a more refined analysis of the GP's information gain.

cs.LG · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2510.25591

Orlicz-Sobolev with Musielak: An Efficient Regularization Approach for Graph-based IPM

Tam Le, Truyen Nguyen, Hideitsu Hino, Kenji Fukumizu

原文摘要

We study the Sobolev IPM problem for probability measures supported on a graph metric space, where critic function is constrained to lie within the unit ball defined by Sobolev norm. Sobolev IPM is intrinsically coupled with $L^p$ geometric structure within its definition, limiting its ability to incorporate other prior geometry beyond the $L^p$ paradigm. Conversely, the classic optimal transport is flexible, easy to adapt to various geometric structures by simply changing its ground cost. An important example is Orlicz-Wasserstein (OW) which utilizes \emph{Orlicz} geometric structure to generalize the $L^p$ within the standard $p$-order Wasserstein, and remarkably play a vital role to advance machine learning methodologies. Inspired by recent advantages of OW, in this work, we leverage a specific class of convex functions for Orlicz geometry to mitigate such limitation for Sobolev IPM, and propose the generalized Sobolev IPM (GSI). Our GSI approach encompasses Sobolev IPM as a special case while accommodating diverse geometric priors beyond $L^p$. It however brings up significant computational hurdles that compound those already notoriously inherent in Sobolev IPM. To address these challenges, we theoretically establish a novel connection between \emph{Orlicz-Sobolev} norm and \emph{Musielak} norm which facilitates a novel efficient regularization for GSI. By further exploiting the underlying graph structure, we show that the regularized GSI reduces to a simple univariate optimization problem, achieving notably computational efficiency, enabling its usage in practical applications. We empirically illustrate that the regularized GSI is several-order faster than the popular OW in computation, and its performances compare favorably to other transport baselines for comparing graph-based measures in document classification and topological data analysis.

cs.LG · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2509.24058

On The Variability of Concept Activation Vectors

Julia Wenkmann, Damien Garreau

原文摘要

One of the most pressing challenges in artificial intelligence is to make models more transparent to their users. Recently, explainable artificial intelligence has come up with numerous methods to tackle this challenge. A promising avenue is to use concept-based explanations, that is, high-level concepts instead of plain feature importance scores. Among this class of methods, Concept Activation Vectors (CAVs, Kim et al., 2018) stand out as one of the main protagonists. One interesting aspect of CAVs is that their computation requires sampling random examples from the train set. Therefore, the actual vectors obtained may vary depending on the randomness of this sampling. In this paper, we propose a fine-grained theoretical analysis of CAV construction in order to quantify their variability. Our results, confirmed by experiments on several real-life datasets of four different modalities, point to an universal result: the variance of CAVs declines roughly as $1/N$, where $N$ is the number of random examples. Based on this, we give practical recommendations for a resource-efficient application of the method.

cs.LG · stat.ML

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arXiv ↗PDF ↗
版本更新 · v6arXiv:2407.04495

Speed-accuracy relations for diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transport

Kotaro Ikeda, Tomoya Uda, Daisuke Okanohara, Sosuke Ito

原文摘要

We discuss a connection between a generative model, called the diffusion model, and nonequilibrium thermodynamics for the Fokker-Planck equation, called stochastic thermodynamics. Using techniques from stochastic thermodynamics, we derive the speed-accuracy relations for diffusion models, which are inequalities that relate the accuracy of data generation to the entropy production rate. This relation can be interpreted as the speed of the diffusion dynamics in the absence of the non-conservative force. From a stochastic thermodynamic perspective, our results provide quantitative insight into how best to generate data in diffusion models. The optimal learning protocol is introduced by the geodesic of space of the 2-Wasserstein distance in optimal transport theory. We numerically illustrate the validity of the speed-accuracy relations for diffusion models with different noise schedules and different data. We numerically discuss our results for optimal and suboptimal learning protocols. We also demonstrate the applicability of our results to data generation from the real-world image datasets.

cs.LG · stat.ML

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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 ↗
版本更新 · v2arXiv:2512.06615

Latent Nonlinear Denoising Score Matching for Enhanced Learning of Structured Distributions

Kaichen Shen, Wei Zhu

原文摘要

Score-based generative models commonly employ linear forward diffusions with Gaussian reference distributions, limiting their ability to directly incorporate structural information into the noising dynamics. We introduce latent nonlinear denoising score matching (LNDSM), which combines nonlinear, structure-informed latent diffusions with the variational autoencoder (VAE) training framework. In our construction, the forward diffusion is designed to admit a prescribed reference distribution, such as a Gaussian mixture model, that reflects qualitative features of the latent data distribution. We derive a computable approximation to a cross-entropy upper bound in the VAE objective using Euler-Maruyama discretization and the resulting Gaussian conditional transitions. To stabilize training, we remove two zero-mean control variates whose variances become large for small time steps. The resulting objective enables joint training of the encoder, decoder, and latent score network. Numerical experiments on benchmark image datasets demonstrate the effectiveness of LNDSM in learning structured distributions, outperforming conventional latent score-based generative modeling in sample quality and mode balance across several settings, while achieving improved training efficiency. These results highlight the potential of nonlinear, structure-informed latent diffusions for generative modeling.

stat.ML · cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36311

Confidence Is Not a Promise: A Reference Audit of Typed Decision Models

Pedro Carvalho Brom

原文摘要

Typed decision models return a probability distribution over options supplied by the caller, and software tends to act on that number as if it were a guarantee. We measure what it licenses, comparing the open-weight Laya, in five checkpoints, with Jev 1.13, available only through an API. A probability licenses a statement about groups of decisions, not about one decision, and a gate that promises an accuracy among the cases it decides alone keeps that promise only with a lower bound checked on data like its traffic. Across 190 exchangeable evaluations, a threshold chosen by point estimate breaks the promise in 147, Learn then Test keeps it in all of them, and a Beta lower-bound rule decides more items on average than Learn then Test in 128 and fewer in none, without its guarantee. Calibrated in English, Jev keeps a promise of 95% in 8 of 51 locales. Ordinary typing noise breaks a promise calibrated on clean text, and, in exploratory runs, training with noise raises the noisy accuracy with no loss on clean text. In four further domains, a fine-tuned Laya needs from 250 to 8,000 labels to be non-inferior to Jev, and does not get there on LIAR2. Under drift, the preregistered claim for a Bayesian retraining loop is not supported: the loop avoids needless retraining where the change does not degrade the model, at a cost equivalent to recalibration alone, and re-qualifying the threshold in every window keeps the promise more often where it does. On the monthly series of M3, neither decider predicts the direction of the next value better than a constant answer. Confidence separates right from wrong answers with an AUROC from 0.80 to 0.96, but it does not select faithful explanations, and exact coalition games of single decisions show what an explanation can still say through an API that rounds to two decimals. We release the frozen answers with a datasheet, a promise check and the code.

cs.LG · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2403.12357

Unifying Sparse Estimation of Precision and Covariance Matrices

Aramayis Dallakyan, Mohsen Pourahmadi

原文摘要

Penalized maximum likelihood is a widely used approach to estimate a sparse precision or a sparse covariance matrix. A zero in the precision matrix encodes a conditional independence and a zero in the covariance matrix a marginal independence, and the two estimators have been developed separately, each with its own algorithm. We show that these two problems can be unified via a single reparametrization of the last column of either matrix. This reparametrization decouples the likelihood function for each problem to a one-dimensional minimization followed by a penalized multiple regression. Our approach applies to any penalty that is separable in the off-diagonal entries and linear on the diagonal. When the penalty is the $\ell_1$ norm, it unifies the graphical Lasso and the covariance graphical Lasso algorithms.

stat.CO · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36227

Directional Statistics on Circles and Spheres: A New Angle on Data

Siddhesh Kulkarni

原文摘要

Two compass bearings near north can have an ordinary arithmetic mean pointing south: the difficulty is that angles live on a circle, not a straight line. Using real wind observations, diffusion-MRI orientations, and geometric examples from machine learning, this article introduces circular and axial averages, directional variability, and probability models on circles and spheres. The guiding principle is to identify the geometry and symmetries of the observations before choosing a statistical method.

stat.ME · cs.AI · 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 · v2arXiv:2605.21341

Parametric Rates for Nonparametric Conditional-Moment Bilevel Optimization

Fares El Khoury, Houssam Zenati, Nathan Kallus, Michael Arbel, Aurélien Bibaut

原文摘要

We study functional bilevel optimization with conditional moments, where plug-in hypergradients can inherit first-order errors from nonparametrically learned conditional expectations. We introduce OBiGrad, an orthogonalized hypergradient estimator whose bias is a product of nuisance errors. Under our regression and complexity conditions, projected gradient descent with the direct plug-in hypergradient yields expected stationarity error $O(T^{-1/2}+n^{-1/2}+n^{-ν/b})$ for the best iterate, whereas OBiGrad achieves $O(T^{-1/2}+n^{-1/2}+n^{-2ν/b})$, where $b=ν(2+p)+γ$ combines the approximation and complexity exponents. Consequently, when $ν(2-p)\geγ$, OBiGrad attains the parametric statistical rate $O(T^{-1/2}+n^{-1/2})$, improving the sufficient sample-size bound relative to the direct plug-in bound for a fixed iteration budget. We illustrate these gains on instrumental-variable and fitted-Q-regression problems, examining stationary-point estimation, returned population stationarity, and regularization sensitivity.

stat.ML · cs.LG

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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:2610.36149

Anytime Valid Confidence Bands for Distribution Functions

Bror Hultberg, Dave Zachariah, Antônio H. Ribeiro

原文摘要

Nonparametric inference often requires characterizing an unknown function, such as a cumulative distribution function (CDF), across its entire domain rather than a specific point. We develop confidence bands for distribution functions of a data stream that are valid simultaneously over all times and all points, even when each observation may depend arbitrarily on the past. We begin by deriving such bands for the time-averaged conditional CDF and show how their widths are determined by the depth of a binary search tree. We then show that this result is an instance of a framework through which one can construct valid confidence bands uniformly over time and over an arbitrary index set for any indexed family of time-averaged distribution functions. We illustrate the framework on importance-weighted and right-censored CDFs, as well as multivariate CDFs.

stat.ML · cs.LG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2502.17061

A Mechanistic Interpretation of Prescribed-Parameter Convolutional Kernels

Jorge Marco Blanco, Rubén Cuevas Rumín

原文摘要

Prescribed-parameter convolutional transforms achieve strong time series classification performance using fixed kernels, simple pooling operations, and a linear classifier. Their success raises a mechanistic question: which properties of the representation make this construction effective? We study the proportion of positive values (PPV), a pooling operator shared by ROCKET, MiniROCKET, and MultiROCKET. We establish exact invariance to circular translations under periodic boundaries and quantify departures from invariance under finite boundaries. A complementary analysis bounds the sensitivity of a smooth PPV surrogate. We then define the population pairwise geometry induced by a distribution of convolutional parameters and show how a finite random kernel bank concentrates around that geometry. This separates the role of the parameter distribution from the number of sampled kernels. Experiments on the UCR classification collection examine effective dimensionality, distance correlation with the input, and classification accuracy using PPV-only, unit-energy ROCKET kernels. Increasing kernel count produces diminishing representational gains. Increasing kernel length generally raises effective dimensionality and distance correlation, while the best test performance occurs at intermediate lengths and training accuracy remains close to perfect. Together, these results support an interpretation in which nonlinear pooling creates a richer but structured representation that can favour linear separation, while further representational richness need not improve generalisation.

stat.ML · cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36138

Symmetry Breaking Is the Price of Finite Capacity

Maggie Chlon, Leon Chlon

原文摘要

If rotating a problem never changes its answer, one might expect its best model to be rotation-invariant; we show that this fails, predictably and by an exact amount, when capacity is limited. A symmetry groups a model's directions into blocks of equally strong directions, a symmetric model must use each block whole, and when the budget cuts a block every optimal model breaks the symmetry. For a Gaussian problem with a $k$-unit bilinear predictor, the log-loss price of symmetry is exactly $(U_k-I_k)/(2σ^2)$, the signal of the best $k$ directions minus that of the best whole blocks. Because an attention head compares tokens through a similarity of rank at most its head dimension, whatever computation precedes it, the same price governs heads fitted to a symmetric similarity by squared error, and heads trained this way on three-dimensional directions break symmetry at exactly the predicted head dimensions, with the predicted rotation sensitivity (RMSE 0.004). Under softmax attention, flatter attention restores the predicted symmetric head where tested, while with peaked attention broken heads beat every whole-shell symmetric head at 16 of 20 head dimensions. We give a procedure that separates capacity-driven breaking from other causes.

stat.ML · cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36110

Dirichlet Process Mixtures within Semiparametric Bayesian Networks: A Novel Conditional Density Estimation Approach

Victor Alejandre, Pedro Larrañaga, Concha Bielza

原文摘要

Semiparametric Bayesian networks (SPBNs) are flexible probabilistic graphical models for representing joint distributions of continuous variables through lower-dimensional conditional probability distributions (CPDs). Existing SPBN models use kernel density estimators (KDEs) for non-parametric CPD modeling, but their one-component-per-observation mixture representation can lead to increased inference costs and density estimation errors, which can be limiting. In this work, we define a new CPD modeling approach for non-parametric components based on Bayesian non-parametric Dirichlet Process Mixtures (DPMs) within SPBNs. DPMs provide a data-adaptive clustering of observations, yielding a more compact representation while retaining the flexibility of non-parametric density estimation, unlike KDE. This results in lower inference costs while improving density estimation performance. We implement the proposed DPM-based SPBN (DPM-SPBN) by extending the open-source PyBNesian package and evaluate it on synthetic and real-world data. For a given network structure, DPM-based CPDs significantly outperform state-of-the-art Bayesian network models for continuous variables in density estimation, achieving lower absolute log-likelihood errors on synthetic data. We further investigate the impact of the proposed CPD modeling on structure learning methods relying on CPD-based scores. DPM-based SPBNs achieve lower structure learning and density estimation errors than KDE-based SPBNs. Experiments on real-world data corroborate these findings, identifying DPM-based CPD modeling as the most effective alternative among the evaluated non-parametric approaches for continuous variables.

cs.LG · cs.AI · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.12341

Multi-Variable Conformal Prediction: Optimizing Prediction Sets using Multiple Calibration Variables

Laura Lützow, Simone Garatti, Marco C. Campi, Lars Lindemann, Matthias Althoff

原文摘要

Split conformal prediction constructs prediction sets with finite-sample coverage guarantees, but its calibration stage is limited to adjusting a single threshold parameter. Hence, the shape of the prediction sets must be fixed beforehand, typically by fitting it on a reserved part of the calibration data. We introduce multi-variable conformal prediction (MCP), a framework that jointly optimizes and calibrates multiple prediction-set parameters in a single optimization problem, eliminating calibration data split without sacrificing finite-sample coverage guarantees. Building on scenario theory, a principled framework for certifying data-driven decisions, we propose two tractable variants: RemMCP, grounded in constrained optimization with constraint removal, which generalizes split conformal prediction; and RelMCP, based on penalized optimization with constraint relaxation, which also supports non-convex problems, albeit at a higher computational cost. In experiments with ellipsoidal and multi-modal prediction sets, RemMCP and a heuristic, adaptive version of RelMCP consistently meet the target coverage, with lower variance across calibration runs than baselines with calibration data split, and in most settings with smaller or comparable prediction sets.

stat.ML · cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36097

Minimax-Optimal Small-Loss Bounds for Online Linear Optimisation on $\ell_p$ Balls

Jack J. Mayo

原文摘要

We study full-information online linear optimisation on the unit $\ell_p$ ball in $\mathbb{R}^d$, for fixed $2\le p<\infty$ and losses of dual norm at most one. Let $Δ_T$ denote the cumulative loss of the best fixed action above each round's minimum. A follow-the-regularised-leader (FTRL) algorithm attains, on every loss sequence and without knowing $T$ or $Δ_T$, regret $O_p(1+\min\{Δ_T^{1-1/p},\, d^{1/2-1/p}\sqrt{Δ_T}\})$. Matching lower bounds show that this rate is minimax for each fixed $p$, and its constant $(2+o(1))p/\log p$ is optimal as $p\to\infty$ among guarantees uniform in $d$ and $T$. The algorithm uses a bounded radial regulariser and switches at most once to its quadratic augmentation. On some paths where follow-the-leader (FTL) has regret one, its regret grows with $T$. A switching extension keeps the sharp constant and also guarantees $\mathrm{Reg}_T\le 2\,\mathrm{Reg}_T^{\mathrm{FTL}}$. For i.i.d. losses whose mean has no zero coordinates, we compute the exact $\log T$ coefficient of the expected regret, which separates FTL's variance term from the bias of the regulariser.

cs.LG · math.OC · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2509.00924

When is Fine-Tuning Scalable? Certified Recovery from Noisy Data and Linear-Cost Fine-Tuning on Structured Task Shifts

Anastasis Kratsios, Tin Sum Cheng, Bum Jun Kim, Daniel Roy

原文摘要

Transformers often transfer across related tasks with little fine-tuning; we study this question in noisy nonparametric regression for continuous maps $f,g:[0,1]^d\to\mathbb{R}^D$, representing the source and shifted tasks, using a single-query, single-head transformer $\mathcal{T}(\cdot)=V\operatorname{Softmax}(-α(\cdot,K))$ with $\ell^{\infty}$-based alignment scores $α$. Rather than assume that gradient-based training reaches an empirical risk minimizer, we give a model-specific two-stage procedure \textbf{certifying} source-task recovery and out-of-distribution (OOD) adaptation. With high probability, the resulting attention head recovers $f$ to arbitrary accuracy $\varepsilon>0$ in every fixed $L^p([0,1]^d)$ norm, $1\le p<\infty$, from noisy observations. For Lipschitz targets, its rate is minimax-optimal up to logarithmic factors without assuming oracle access to an asymptotic ERM. If $D=1$ and $f$ is Pfaffian than the values matrix factors into $AC$ with $A$ scaling at-most linearly $\mathcal{O}(\varepsilon^{-1})$ in the reciprocal approximation error $\varepsilon>0$. Importantly, our training procedure encodes all our identified (potentially) shared geometric structure from $f$ only into the values matrix factor $C$ and the keys matrix $K$ allowing for rapid access, and all $f$-specific structure only into $A$. Our main result concerns the OOD phase; where we identify geometric structures, which if shared by the source $f$ and OOD $g$ task; then only only the values matrix factor $A$ needs to be fine-tuned to guarantee an $L^p$-recovery of $g$. Critically, fine-tuning only $A$ is a convex problem which involving an \textit{at-most linear} number of parameters $\mathcal{O}(\varepsilon^{-1})$; dramatically cutting down on the worst-case scaling of $\mathcal{O}(\varepsilon^{-\mathbf{d}})$ unavoidable in the source-training phase.

stat.ML · cs.LG · cs.NE · math.NA · math.PR

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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 ↗
新论文 · v1arXiv:2610.36018

Anchored Stein Variational Gradient Descent Algorithm for Non-smooth Sampling

Thanh Dang, Matthew X. Burns, Jiaming Liang

原文摘要

We introduce a Stein variational framework for sampling from a target density $p\propto e^{-U}$ when the target potential $U$ is possibly non-smooth. The key idea is to introduce a smoother reference potential $U_0$ and the anchor weight $a=e^{U-U_0}$. The resulting anchored Stein variational gradient descent (SVGD) algorithm is able to target $p\propto e^{-U}$ without computing the gradient of $U$ and include standard SVGD as a special case. In particular, we allow the target potential $U$ to be locally Lipschitz while the reference potential $U_0$ is continuously differentiable, along with other suitable assumptions. Under this low regularity regime, we establish a Kullback-Leibler first variation formula for anchored perturbations, identify the corresponding steepest descent direction in the reproducing kernel Hilbert space and prove a nonasymptotic convergence bound for the anchored SVGD algorithm in terms of an anchored kernelized Stein discrepancy. Finally, we also provide guidance on constructing a suitable reference potential $U_0$ based on $U$.

stat.ML · cs.LG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2605.21060

Divide et Calibra: Multiclass Local Calibration via Vector Quantization

Cesare Barbera, Lorenzo Perini, Giovanni De Toni, Andrea Passerini, Andrea Pugnana

原文摘要

Accurate and well-calibrated Machine Learning (ML) models are mandatory in high-stakes settings, yet effective multiclass calibration remains challenging: global approaches assume calibration errors are homogeneous across the latent space, while local methods often rely on representation-space dimensionality reduction, discarding latent directions relevant to miscalibration. To address these issues, we propose a compositional approach to multiclass calibration, where region-specific calibration maps are constructed from shared factors. We instantiate this idea via Vector Quantization (VQ), which induces a structured partition of the representation space, and an indexed parameterization of Dirichlet concentrations that enables parameter sharing across regions. Our approach learns heterogeneous calibration maps that generalize well even to sparse regions of the latent space. Experiments on benchmark datasets show significant improvements in local calibration while maintaining competitive global calibration and predictive performance.

cs.LG · cs.AI · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35748

Model Selection for Off-Policy Evaluation via Pairwise Bellman Validation

Lars van der Laan, Vasilis Syrgkanis, Nathan Kallus

原文摘要

Model selection and hyperparameter tuning pose a central challenge in off-policy evaluation (OPE) because errors of fitted Q-functions and occupancy-ratio estimates cannot be directly evaluated from held-out transitions. Bellman-residual and minimax validation methods often require auxiliary regressions or optimization over separate critic classes, adding computational and tuning costs. We introduce pairwise Bellman validation (PBV), a tuning-free minimax procedure for selecting among finite classes of fitted Q-functions and occupancy-ratio estimators using held-out transitions. PBV constructs critics from pairwise candidate differences, requiring only finitely many empirical moment calculations without auxiliary validation models or separate critic optimization. We establish finite-sample oracle inequalities relative to the best available candidate, even under candidate-class misspecification. Our key insight is that Q-function errors and log ratios between candidate and true occupancy ratios provide oracle critics whose Bellman and adjoint Bellman moments control estimation error. Pairwise candidate differences approximate these unknown critics, while penalties account for comparator approximation error. For Q-function selection, we first develop unweighted PBV, which evaluates pairwise Bellman moments under the offline distribution and admits oracle guarantees under a pairwise coercivity condition relating the offline distribution to the target-policy dynamics. We then develop target-aligned PBV (TA-PBV), which uses marginalized importance weighting to provide error control without pairwise coercivity. For occupancy-ratio selection, we develop adjoint PBV (A-PBV), which evaluates adjoint Bellman moments against log-ratio critics. On Hopper and Deep OPE, unweighted PBV performs competitively, while target-occupancy weighting can improve performance in settings with controlled overlap.

stat.ML · cs.LG · econ.EM

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35740

LUCID: Local Understanding of Covariate Importance with Conditional Density Estimation

Yixiao Wang, Srikar Katta, Jon Donnelly, Emanuele Borgonovo, Cynthia Rudin

原文摘要

Computing local-level variable importance (VI) is critical for understanding machine learning models. Local permutation-based VI metrics offer a simple yet powerful approach, measuring how much a model's output changes when perturbing its inputs. However, existing perturbation strategies often ignore feature dependence or account for it only through restricted approximations. Perturbing a feature without fully respecting this dependence can generate unrealistic counterfactuals, thereby evaluating the model in regions unsupported by the observed data distribution. We address this gap by introducing a new model-agnostic local VI metric and estimator, Local Understanding of Covariate Importance with conditional Density estimation (LUCID). LUCID perturbs features of interest using plausible values drawn from the full conditional distribution given the remaining observed features. We establish theoretical connections between LUCID and existing VI measures, and provide an estimator with convergence guarantees and uncertainty quantification. Finally, we demonstrate LUCID's stable performance and practical utility across simulated and real-world examples.

cs.LG · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35733

Multi-Objective Reinforcement Learning Using Routed Ensembles and Trajectory Attribution

Zhenshuo Zhang, Ruoxuan Xiong, Hongyang R. Zhang

原文摘要

Multi-objective reinforcement learning is a fundamental problem for modern AI, with applications spanning post-training, reasoning models, and robotic control. A central challenge is to learn policies that balance competing objectives, such as helpfulness versus honesty in safety and alignment, or speed versus energy in robotics. The optimal policy can vary sharply with objective-weighting combinations, and the extent of this variation depends crucially on the degree of conflict among objectives, making per-preference policy learning sample-inefficient. In this paper, we introduce objective-conflict coefficients and derive near-optimal regret bounds for online settings using this measure, thereby quantifying the impact of misalignment on learning efficiency. Then, we design routed ensemble policy learning, an algorithm that maintains a small set of basis policies and dynamically routes trajectories to the policy with the highest potential reward. Finally, we consider trajectory selection as another approach to mitigate objective conflict, and design a kernel surrogate trajectory attribution algorithm that estimates how feedback signals from a set of trajectories influence policy learning on other trajectories. The kernel can capture nonlinear interactions such as synergy and antagonism between trajectories. We evaluate our approach across diverse environments and benchmarks: routed ensembles improve average performance by 4.3% over baselines for online environments from Meta-World and MO-Gymnasium, and kernel surrogate attribution improves correlation with leave-one-out analysis over existing baselines by 50%. When applied to trajectory selection, kernel-based attribution yields a 27% improvement in reward over baseline attribution methods.

cs.LG · cs.AI · cs.MA · cs.RO · stat.ML

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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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