Lookahead Bayesian Optimization with Inequality Constraints
We consider the task of optimizing an objective function subject to inequality constraints when both the objective and the constraints are expensive to evaluate. Bayesian optimization (BO) is a popular way to tackle optimization problems with expensive objective function evaluations, but has mostly been applied to unconstrained problems. Several BO approaches have been proposed to address expensive constraints but are limited to greedy strategies maximizing immediate reward. To address this limitation, we propose a lookahead approach that selects the next evaluation in order to maximize the long-term feasible reduction of the objective function. We present numerical experiments demonstrating the performance improvements of such a lookahead approach compared to several greedy BO algorithms, including constrained expected improvement (EIC) and predictive entropy search with constraint (PESC).
This paper seems a continuation of last year: Bayesian optimization with a finite budget... where the authors have added new elements to deal with inequality constraints. The method uses a approximation of a lookahead strategy by dynamic programming. For the constrained case, the authors propose an heuristic that combines the EIc criterion for all the steps except for the last one were the mean function is used. The authors claim that the mean function has an exploitative behaviour, although it has been previously shown that it might be misleading [A].
      A considerably amount of the text, including Figure 1, can be mostly found in [16]. Although it is nice to have an self-contained paper as much as possible, that space could be used to explain better the selection of the acquisition heuristic and present alternatives. For example, the experiments should show the result a single-step posterior mean acquisition function, which will correspond to h=0 as a baseline for each of the functions/experiments.
      Concerning the experiments, there are some details that should be improved or addressed in the paper:
      - Why P2 does not include h=3?
      - How is it possible that EIc for P2 goes upwards around n=25?
      - The plots only shows median values without error bars. I can understand that for that, for such number of repetitions, the error bars might be small, but that should be adressed. Furthermore, a common problem of lookahead methods is that, while most of the time outperform greedy methods, they can also fail catastrophically for selecting the wrong path. This would results in some lookahead trials resulting in poor results. Using the average instead of the median would show also the robustness of the method.
      - The number of iterations is fairly small for the standard in Bayesian optimization. In fact, it can bee seen that most methods have not reach convergence at that point. This is specially problematic in P1 and P4 where the plots seems to cross exactly at the end of experiments,
      - It is unclear why the performance decreases after h=2. If possible, the authors should provide an intuition behind that results. Furthermore, the fact that for most of the experiments h=1 is the optimal strategy seems to indicate that the whole dynamic programing and heuristics might be excessive for CBO, or that more complex and higher-dimensional problems are need to illustrate the benefits of the strategy.
      
      [A] Jones, Donald R. "A taxonomy of global optimization methods based on response surfaces." Journal of global optimization 21.4 (2001): 345-383.