Online Bilevel Optimization: Regret Analysis of Online Alternating Gradient Methods

TL;DR

This paper extends online optimization to bilevel problems, introducing bilevel regret and an online alternating gradient method with regret bounds based on the path-length of solutions.

Contribution

It develops the first regret analysis framework for online bilevel optimization and proposes a new online alternating gradient algorithm leveraging smoothness.

Findings

Regret bounds are derived in terms of the path-length of solutions.

The proposed method effectively leverages smoothness for improved regret performance.

The framework generalizes online single-level regret bounds to bilevel problems.

Abstract

This paper introduces \textit{online bilevel optimization} in which a sequence of time-varying bilevel problems is revealed one after the other. We extend the known regret bounds for online single-level algorithms to the bilevel setting. Specifically, we provide new notions of \textit{bilevel regret}, develop an online alternating time-averaged gradient method that is capable of leveraging smoothness, and give regret bounds in terms of the path-length of the inner and outer minimizer sequences.