Determining Convergence for Expected Improvement-Based Bayesian Optimization

Bayesian optimization routines may have theoretical convergence results, but determining whether a run has converged in practice can be a subjective task. This paper provides a framework inspired by statistical process control for monitoring an optimization run for convergence. The maximum Expected Improvement (EI) tends to decrease during an optimization run, but decreasing EI is not sufficient for convergence. We consider both a decrease in EI as well as local stability of the variance in order to assess for convergence. The EI process is made more numerically stable through an expected log-normal approximation. An Exponentially Weighted Moving Average control chart is adapted for automated convergence analysis, which allows assessment of stability of both the EI and its variance. The success of the methodology is demonstrated on several examples.