Dependence in Constrained Bayesian Optimization

Constrained Bayesian optimization optimizes a black-box objective function subject to black-box constraints. For simplicity, most existing works assume that multiple constraints are independent. To ask, when and how does dependence between constraints help? , we remove this assumption and implement probability of feasibility with dependence (Dep-PoF) by applying multiple output Gaussian processes (MOGPs) as surrogate models and using expectation propagation to approximate the probabilities. We compare Dep-PoF and the independent version PoF. We propose two new acquisition functions incorporating Dep-PoF and test them on synthetic and practical benchmarks. Our results are largely negative: incorporating dependence between the constraints does not help much. Empirically, incorporating dependence between constraints may be useful if: (i) the solution is on the boundary of the feasible region(s) or (ii) the feasible set is very small. When these conditions are satisfied, the predictive covariance matrix from the MOGP may be poorly approximated by a diagonal matrix and the off-diagonal matrix elements may become important. Dep-PoF may apply to settings where (i) the constraints and their dependence are totally unknown and (ii) experiments are so expensive that any slightly better Bayesian optimization procedure is preferred. But, in most cases, Dep-PoF is indistinguishable from PoF.