On Local Optimizers of Acquisition Functions in Bayesian Optimization
This addresses a practical bottleneck in Bayesian optimization for researchers and practitioners, but it is incremental as it builds on existing methods without introducing new paradigms.
The paper tackles the problem of using local optimizers instead of global optimizers for acquisition functions in Bayesian optimization, analyzing their performance in terms of instantaneous regrets and showing that starting from multiple initial conditions can mitigate issues, with numerical experiments confirming the theoretical findings.
Bayesian optimization is a sample-efficient method for finding a global optimum of an expensive-to-evaluate black-box function. A global solution is found by accumulating a pair of query point and its function value, repeating these two procedures: (i) modeling a surrogate function; (ii) maximizing an acquisition function to determine where next to query. Convergence guarantees are only valid when the global optimizer of the acquisition function is found at each round and selected as the next query point. In practice, however, local optimizers of an acquisition function are also used, since searching for the global optimizer is often a non-trivial or time-consuming task. In this paper we consider three popular acquisition functions, PI, EI, and GP-UCB induced by Gaussian process regression. Then we present a performance analysis on the behavior of local optimizers of those acquisition functions, in terms of {\em instantaneous regrets} over global optimizers. We also introduce an analysis, allowing a local optimization method to start from multiple different initial conditions. Numerical experiments confirm the validity of our theoretical analysis.