Abstract
In this paper, the method of gaps, a technique for deriving closed-form expressions in terms of information measures for the generalization error of supervised learning algorithms, is introduced. This method relies on the notion of gaps, which characterize the variation of the expected empirical risk (when either the model or dataset is kept fixed) with respect to changes in the probability measure on the varying parameter. This distinction results in two classes of gaps: algorithm-driven gaps (fixed dataset) and data-driven gaps (fixed model). The method relies on two central observations: (i) the generalization error is the expectation of an algorithm-driven gap or a data-driven gap. In the first case, the expectation is with respect to a measure on the datasets; in the second case, it is with respect to a measure on the models. (ii) Both algorithm-driven gaps and data-driven gaps exhibit closed-form expressions in terms of relative entropies. In particular, algorithm-driven gaps involve a Gibbs probability measure on the set of models, which represents a supervised Gibbs algorithm. Alternatively, data-driven gaps involve a worst-case data-generating (WCDG) probability measure on the set of data points, which is also a Gibbs probability measure. Interestingly, such Gibbs measures, which are exogenous to the analysis of generalization, place the supervised Gibbs algorithm and the WCDG probability measure as natural references for the analysis of supervised learning algorithms. New exact expressions and all existing exact expressions for the generalization error of supervised learning algorithms can be obtained with the proposed method. Such new expressions are intended as structural and conceptual characterizations, not computational shortcuts. Finally, these expressions unveil strong connections among generalization, hypothesis testing, information measures, and Pythagorean identities.
Explore similar work
Feb 11, 2025stat.ML
Many learning algorithms can be represented as Markov processes, and understanding their generalization error is a central topic in learning theory. For specific continuous-time noisy algorithms, a prominent analysis technique relies on information-theoretic tools and the so-called ``entropy flow'' method. This technique is compatible with a broad range of assumptions and leverages the convergence properties of learning dynamics to produce meaningful generalization bounds, which can also be informative or extend to discrete-time settings. Despite their success, existing entropy flow formulations are limited to specific noise and algorithm structures (\eg, Langevin dynamics). In this work, we exploit new technical tools to extend its applicability to all learning algorithms whose iterative dynamics is governed by a time-homogeneous Markov process. Our approach builds on a principled continuous-time approximation of Markov algorithms and introduces a new, exact entropy flow formula for such processes. Within this unified framework, we establish novel connections to a well-studied family of modified logarithmic Sobolev inequalities, which we use to connect the generalization error to the ergodic properties of Markov processes. Finally, we provide a detailed analysis of all the terms appearing in our theory and demonstrate its effectiveness by deriving new generalization bounds for several concrete algorithms.
Benjamin Dupuis, Maxime Haddouche, George Deligiannidis +1
Apr 23, 2025cs.LG
We derive a differential equation that governs the evolution of the generalization gap when a model is trained by gradient descent-based methods. This differential equation is driven by two key quantities, a contraction factor that brings together trajectories corresponding to slightly different datasets, and a perturbation factor that accounts for them training on different datasets. The coupled decay of contraction and perturbation guarantees a controlled accumulation of generalization gap during training. We analyze this differential equation to show that the generalization gap is given by a quadratic form that consists of an ``effective Gram matrix'' that depends upon the training trajectory and a certain residual of the predictor at initialization. Our framework is applicable to general deep networks and smooth loss functions. In numerical experiments on different neural network architectures, datasets and sample sizes, we show that this quadratic form accurately captures the actual generalization gap. We also show how to instantiate our framework in a number of examples via analytical calculations. For example, for high-dimensional linear regression, our framework matches existing calculations of generalization gap in the literature exactly in under-parameterized, over-parameterized and critical regimes.
Rubing Yang, Pratik Chaudhari
Feb 26, 2026cs.LG
Generalization bounds for deep learning models are typically vacuous, not computable or restricted to specific model classes. In this paper, we tackle these issues by providing new disagreement-based certificates for the gap between the true risk of any two predictors. We then bound the true risk of the predictor of interest via a surrogate model that enjoys tight generalization guarantees, and by evaluating our disagreement bound on an unlabeled dataset.We empirically demonstrate the tightness of the obtained certificates and showcase the versatility of the approach by training surrogate models leveraging three different frameworks: sample compression, model compression and PAC-Bayes theory. Importantly, such guarantees are achieved without modifying the target model, nor adapting the training procedure to the generalization framework.
Mathieu Bazinet, Valentina Zantedeschi, Pascal Germain