No-Regret Gaussian Process Optimization of Time-Varying Functions

TL;DR

This paper introduces a novel Gaussian Process-based method for optimizing time-varying functions, achieving no-regret with minimal additional queries by leveraging uncertainty injection and heteroscedastic regression.

Contribution

It proposes W-SparQ-GP-UCB, an online algorithm that attains no-regret in time-varying settings with few extra queries, and establishes theoretical bounds on query complexity.

Findings

Achieves no-regret with vanishing additional queries.

Provides lower bounds on the number of extra queries needed.

Links time-variation degree to regret rates.

Abstract

Sequential optimization of black-box functions from noisy evaluations has been widely studied, with Gaussian Process bandit algorithms such as GP-UCB guaranteeing no-regret in stationary settings. However, for time-varying objectives, it is known that no-regret is unattainable under pure bandit feedback unless strong and often unrealistic assumptions are imposed. In this article, we propose a novel method to optimize time-varying rewards in the frequentist setting, where the objective has bounded RKHS norm. Time variations are captured through uncertainty injection (UI), which enables heteroscedastic GP regression that adapts past observations to the current time step. As no-regret is unattainable in general in the strict bandit setting, we relax the latter allowing additional queries on previously observed points. Building on sparse inference and the effect of UI on regret, we propose W-SparQ-GP-UCB, an online algorithm that achieves no-regret with only a vanishing number of additional queries per iteration. To assess the theoretical limits of this approach, we establish a lower bound on the number of additional queries required for no-regret, proving the efficiency of our method. Finally, we provide a comprehensive analysis linking the degree of time-variation of the function to achievable regret rates, together with upper and lower bounds on the number of additional queries needed in each regime.