Jack M. Buckingham

Jack M. Buckingham, Ivo Couckuyt, Juergen Branke

Bayesian optimization is a sample-efficient method for solving expensive, black-box optimization problems. Stochastic programming concerns optimization under uncertainty where, typically, average performance is the quantity of interest. In the first stage of a two-stage problem, here-and-now decisions must be made in the face of uncertainty, while in the second stage, wait-and-see decisions are made after the uncertainty has been resolved. Many methods in stochastic programming assume that the objective is cheap to evaluate and linear or convex. We apply Bayesian optimization to solve non-convex, two-stage stochastic programs which are black-box and expensive to evaluate as, for example, is often the case with simulation objectives. We formulate a knowledge-gradient-based acquisition function to jointly optimize the first- and second-stage variables, establish a guarantee of asymptotic consistency, and provide a computationally efficient approximation. We demonstrate comparable empirical results to an alternative we formulate with fewer approximations, which alternates its focus between the two variable types, and superior empirical results over the state of the art and the standard, naïve, two-step benchmark.

Jack M. Buckingham, Sebastian Rojas Gonzalez, Juergen Branke

Multi-objective Bayesian optimization aims to find the Pareto front of trade-offs between a set of expensive objectives while collecting as few samples as possible. In some cases, it is possible to evaluate the objectives separately, and a different latency or evaluation cost can be associated with each objective. This decoupling of the objectives presents an opportunity to learn the Pareto front faster by avoiding unnecessary, expensive evaluations. We propose a scalarization based knowledge gradient acquisition function which accounts for the different evaluation costs of the objectives. We prove asymptotic consistency of the estimator of the optimum for an arbitrary, D-dimensional, real compact search space and show empirically that the algorithm performs comparably with the state of the art and significantly outperforms versions which always evaluate both objectives.

Jack M. Buckingham, Ivo Couckuyt, Juergen Branke

Bayesian optimization (BO) is a popular, sample-efficient technique for expensive, black-box optimization. One such problem arising in manufacturing is that of maximizing the reliability, or equivalently minimizing the probability of a failure, of a design which is subject to random perturbations - a problem that can involve extremely rare failures ($P_\mathrm{fail} = 10^{-6}-10^{-8}$). In this work, we propose two BO methods based on Thompson sampling and knowledge gradient, the latter approximating the one-step Bayes-optimal policy for minimizing the logarithm of the failure probability. Both methods incorporate importance sampling to target extremely small failure probabilities. Empirical results show the proposed methods outperform existing methods in both extreme and non-extreme regimes.