Epsilon-Greedy Thompson Sampling to Bayesian Optimization

TL;DR

This work empirically shows that e-greedy TS equipped with an appropriate e is more robust than its two extremes, matching or outperforming the better of the generic TS and the sample-average TS.

Abstract

Bayesian optimization (BO) has become a powerful tool for solving simulation-based engineering optimization problems thanks to its ability to integrate physical and mathematical understandings, consider uncertainty, and address the exploitation-exploration dilemma. Thompson sampling (TS) is a preferred solution for BO to handle the exploitation-exploration trade-off. While it prioritizes exploration by generating and minimizing random sample paths from probabilistic models -- a fundamental ingredient of BO -- TS weakly manages exploitation by gathering information about the true objective function after it obtains new observations. In this work, we improve the exploitation of TS by incorporating the $\varepsilon$-greedy policy, a well-established selection strategy in reinforcement learning. We first delineate two extremes of TS, namely the generic TS and the sample-average TS. The former promotes exploration, while the latter favors exploitation. We then adopt the $\varepsilon$-greedy policy to randomly switch between these two extremes. Small and large values of $\varepsilon$ govern exploitation and exploration, respectively. By minimizing two benchmark functions and solving an inverse problem of a steel cantilever beam, we empirically show that $\varepsilon$-greedy TS equipped with an appropriate $\varepsilon$ is more robust than its two extremes, matching or outperforming the better of the generic TS and the sample-average TS.

Figures (8)

• Figure 1: Sample paths from the GP posterior for $f(x) = x\sin(x)$ and distribution of their minimum locations. (a) GP predictions and five sample paths drawn from the GP posterior; (b) Approximate conditional distribution $p( x^\star|\mathcal{D})$ obtained from minimum locations of 50 sample paths.
• Figure 2: Approximate covariance functions using random features from different numbers of spectral point samples. (a) SE; (b) Matérn 5/2.
• Figure 3: Performance of EI, LCB, averaging TS, generic TS, and $\varepsilon$-greedy TS methods for the 2d Ackley and 6d Rosenbrock functions. Optimization histories for (a) the 2d Ackley function and (b) the 6d Rosenbrock function. Medians and interquartile ranges of final solutions from 100 runs of each BO method for (c) the 2d Ackley function and (d) the 6d Rosenbrock function.
• Figure 4: Initial and added points for 2d Ackley function with $N_\text{s} = 50$ and different values of $\varepsilon$.
• Figure 5: Approximate distributions of runtime for selecting a new solution point with different $\varepsilon$ values and $N_\text{s} = 50$.