Online Nonsubmodular Minimization with Delayed Costs: From Full Information to Bandit Feedback

TL;DR

This paper develops algorithms with regret guarantees for online nonsubmodular minimization under delayed feedback, extending to bandit settings and unbounded delays, with applications in sparse estimation and Bayesian optimization.

Contribution

It introduces regret bounds for online nonsubmodular minimization with delays, extending convex relaxation techniques to this setting and analyzing both full information and bandit feedback.

Findings

Regret bounds hold even with unbounded delays.

Algorithms perform well in simulations for sparse estimation and Bayesian optimization.

Extension of convex relaxation analysis to nonsubmodular functions.

Abstract

Motivated by applications to online learning in sparse estimation and Bayesian optimization, we consider the problem of online unconstrained nonsubmodular minimization with delayed costs in both full information and bandit feedback settings. In contrast to previous works on online unconstrained submodular minimization, we focus on a class of nonsubmodular functions with special structure, and prove regret guarantees for several variants of the online and approximate online bandit gradient descent algorithms in static and delayed scenarios. We derive bounds for the agent's regret in the full information and bandit feedback setting, even if the delay between choosing a decision and receiving the incurred cost is unbounded. Key to our approach is the notion of $(\alpha, \beta)$-regret and the extension of the generic convex relaxation model from~\citet{El-2020-Optimal}, the analysis of which is of independent interest. We conduct and showcase several simulation studies to demonstrate the efficacy of our algorithms.