Gaussian Process Upper Confidence Bound Achieves Nearly-Optimal Regret in Noise-Free Gaussian Process Bandits

Shogo Iwazaki

Keywords: Gaussian process bandits, regret analysis, near optimality

Abstract:

We study the noise-free Gaussian Process (GP) bandit problem, in which a learner seeks to minimize regret through noise-free observations of a black-box objective function that lies in a known reproducing kernel Hilbert space (RKHS).
The Gaussian Process Upper Confidence Bound (GP-UCB) algorithm is a well-known approach for GP bandits, where query points are adaptively selected based on the GP-based upper confidence bound score.
While several existing works have reported the practical success of GP-UCB, its theoretical performance remains suboptimal. However, GP-UCB often empirically outperforms other nearly-optimal noise-free algorithms that use non-adaptive sampling schemes.
This paper resolves the gap between theoretical and empirical performance by establishing a nearly-optimal regret upper bound for noise-free GP-UCB. Specifically, our analysis provides the first constant cumulative regret bounds in the noise-free setting for both the squared exponential kernel and the Matérn kernel with some degree of smoothness.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 16390