Statistical Learning under Nonstationary Mixing Processes

TL;DR

This paper extends statistical learning theory to nonstationary, $eta$-mixing processes, proposing a method that guarantees sublinear growth of cumulative excess risk under certain conditions.

Contribution

It introduces a learning approach for nonstationary $eta$-mixing processes with sublinear risk growth, combining nonstationarity and mixing relaxations.

Findings

Cumulative excess risk grows sublinearly with sample size.

The method applies to bounded VC subgraph classes.

Provides explicit rate of risk growth.

Abstract

We study a special case of the problem of statistical learning without the i.i.d. assumption. Specifically, we suppose a learning method is presented with a sequence of data points, and required to make a prediction (e.g., a classification) for each one, and can then observe the loss incurred by this prediction. We go beyond traditional analyses, which have focused on stationary mixing processes or nonstationary product processes, by combining these two relaxations to allow nonstationary mixing processes. We are particularly interested in the case of $\beta$-mixing processes, with the sum of changes in marginal distributions growing sublinearly in the number of samples. Under these conditions, we propose a learning method, and establish that for bounded VC subgraph classes, the cumulative excess risk grows sublinearly in the number of predictions, at a quantified rate.