Improved Dynamic Regret for Non-degenerate Functions

TL;DR

This paper improves the theoretical bounds of dynamic regret in online learning by introducing the squared path-length and leveraging multiple gradient queries, extending results to semi-strongly convex and self-concordant functions.

Contribution

It introduces the squared path-length as a new regularity measure and demonstrates that multiple gradient queries can further tighten dynamic regret bounds for various convexity conditions.

Findings

Dynamic regret can be bounded by the minimum of path-length and squared path-length.

Multiple gradient queries improve the regret bounds for strongly convex functions.

Theoretical extension to semi-strongly convex and self-concordant functions.

Abstract

Recently, there has been a growing research interest in the analysis of dynamic regret, which measures the performance of an online learner against a sequence of local minimizers. By exploiting the strong convexity, previous studies have shown that the dynamic regret can be upper bounded by the path-length of the comparator sequence. In this paper, we illustrate that the dynamic regret can be further improved by allowing the learner to query the gradient of the function multiple times, and meanwhile the strong convexity can be weakened to other non-degenerate conditions. Specifically, we introduce the squared path-length, which could be much smaller than the path-length, as a new regularity of the comparator sequence. When multiple gradients are accessible to the learner, we first demonstrate that the dynamic regret of strongly convex functions can be upper bounded by the minimum of the path-length and the squared path-length. We then extend our theoretical guarantee to functions that are semi-strongly convex or self-concordant. To the best of our knowledge, this is the first time that semi-strong convexity and self-concordance are utilized to tighten the dynamic regret.