Generalization Bounds for Dependent Data using Online-to-Batch Conversion

2024-05-22

arXiv.org

AI Summary

This work addresses the problem of deriving generalization error upper bounds for batch learning algorithms under mixing stochastic processes (i.e., dependent data), without imposing any stability assumptions on the batch learner. The method introduces a novel analytical framework based on Online-to-Batch conversion: stability requirements are shifted to an associated online learner, and a new notion of online algorithm stability—defined via the first-order Wasserstein distance—is proposed for the first time. It is shown that the Exponentially Weighted Average (EWA) algorithm satisfies this stability condition. Consequently, the framework yields both expectation- and high-probability generalization bounds applicable to *any* batch learning algorithm; under i.i.d. assumptions, the bounds reduce to classical forms up to correction terms governed by the mixing decay rate. The resulting bounds are explicitly computable, substantially broadening the applicability and practical utility of learning theory under data dependence.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this work, we upper bound the generalization error of batch learning algorithms trained on samples drawn from a mixing stochastic process (i.e., a dependent data source) both in expectation and with high probability. Unlike previous results by Mohri et al. (2010) and Fu et al. (2023), our work does not require any stability assumptions on the batch learner, which allows us to derive upper bounds for any batch learning algorithm trained on dependent data. This is made possible due to our use of the Online-to-Batch ( OTB ) conversion framework, which allows us to shift the burden of stability from the batch learner to an artificially constructed online learner. We show that our bounds are equal to the bounds in the i.i.d. setting up to a term that depends on the decay rate of the underlying mixing stochastic process. Central to our analysis is a new notion of algorithmic stability for online learning algorithms based on Wasserstein distances of order one. Furthermore, we prove that the EWA algorithm, a textbook family of online learning algorithms, satisfies our new notion of stability. Following this, we instantiate our bounds using the EWA algorithm.

Problem

• Generalization error bounds
• Dependent data sources
• Online-to-Batch conversion framework

Innovation

• Online-to-Batch conversion framework
• Wasserstein distance stability
• EWA algorithm application