No-Regret Bayesian Optimization with Unknown Equality and Inequality Constraints using Exact Penalty Functions

Abstract

Bayesian optimization (BO) methods have been successfully applied to many challenging black-box optimization problems involving expensive-to-evaluate functions. Although BO is often applied to problems with only simple box constraints, it has recently been extended to the constrained black-box optimization setting in which testing feasibility is just as expensive as evaluating performance. Existing literature on the topic has focused on empirical performance of different constrained BO methods, meaning convergence guarantees to the global solution have yet to be established. In this paper, we propose a new constrained BO strategy that uses the notion of exact penalty functions to achieve asymptotic convergence to the global optimum under certain conditions (i.e., we prove it is a no penalty-regret algorithm). We present rates on the convergence of cumulative penalty-regret in terms of the maximal information gain of the objective and constraint functions. Moreover, we show how the proposed algorithm can directly handle black-box equality constraints, which has been a key limitation of alternative approaches. Finally, we demonstrate that a practical implementation of our method is able to outperform state-of-the-art constrained BO methods on problems with and without equality constraints.