The Generalization Ability of Online Algorithms for Dependent Data

TL;DR

This paper analyzes how online learning algorithms perform on dependent data, providing high-probability error bounds and convergence rates under mixing conditions, with implications for stochastic optimization.

Contribution

It establishes generalization error bounds for stable online algorithms on dependent data using martingale techniques, extending prior results to mixing processes.

Findings

Error bounds hold for convex loss functions.

Sharp convergence rates for strongly convex losses.

Applications to linear regression, logistic regression, and boosting.

Abstract

We study the generalization performance of online learning algorithms trained on samples coming from a dependent source of data. We show that the generalization error of any stable online algorithm concentrates around its regret--an easily computable statistic of the online performance of the algorithm--when the underlying ergodic process is $\beta$- or $\phi$-mixing. We show high probability error bounds assuming the loss function is convex, and we also establish sharp convergence rates and deviation bounds for strongly convex losses and several linear prediction problems such as linear and logistic regression, least-squares SVM, and boosting on dependent data. In addition, our results have straightforward applications to stochastic optimization with dependent data, and our analysis requires only martingale convergence arguments; we need not rely on more powerful statistical tools such as empirical process theory.