Generalization of Gibbs and Langevin Monte Carlo Algorithms in the Interpolation Regime

2025-10-07

AI Summary

This work addresses the theoretical analysis of Gibbs algorithm generalization in the overparameterized interpolation regime. We derive a data-dependent upper bound on test error that remains stable under Langevin Monte Carlo (LMC) approximation. Methodologically, we integrate empirical risk minimization with a statistical physics–inspired analytical framework. Our key contribution is revealing that generalization performance in the low-temperature interpolation regime can be anticipated by small training error in the high-temperature regime—thereby providing an interpretable generalization mechanism even for “impossible learning” scenarios such as random labels. Theoretical results are empirically validated on MNIST and CIFAR-10: the proposed bound is nontrivial, uniformly upper-bounds test error under both true and random labels, and significantly outperforms conventional complexity-based bounds.

Technology Category

• Machine Learning: Learning Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

The paper provides data-dependent bounds on the test error of the Gibbs algorithm in the overparameterized interpolation regime, where low training errors are also obtained for impossible data, such as random labels in classification. The bounds are stable under approximation with Langevin Monte Carlo algorithms. Experiments on the MNIST and CIFAR-10 datasets verify that the bounds yield nontrivial predictions on true labeled data and correctly upper bound the test error for random labels. Our method indicates that generalization in the low-temperature, interpolation regime is already signaled by small training errors in the more classical high temperature regime.

Problem

• Analyzing generalization bounds for Gibbs algorithms with overparameterized interpolation
• Studying test error stability under Langevin Monte Carlo approximations
• Investigating generalization signals from high to low temperature regimes

Innovation

• Provides data-dependent bounds for Gibbs algorithm generalization
• Stable approximation using Langevin Monte Carlo algorithms
• Validates bounds on MNIST and CIFAR-10 datasets