Batch Multiobjective Bayesian Optimization via Pareto Optimal Thompson Sampling
Abstract:
Numerical optimization plays a vital role in the design of complex engineered systems. Real world engineered systems are seldom designed based on a single objective; rather they involve multiple conflicting objectives where the optimal design is a careful compromise between the objectives. For instance, in aeronautics, supersonic aircraft are designed to maximize range while also minimizing sonic boom. In machine learning, models are fit for maximum accuracy with the simplest architecture possible. Optimization with multiple objectives, or multiobjective optimization, cannot necessarily leverage existing methods from single objective optimization. Further, when every objective is observed by querying an expensive stochastic zeroth-order oracle, conventional methods become intractable and, novel, “sample-efficient” (that is, minimal queries to the oracles) approaches are necessary.
One popular approach to solving multiobjective optimization, in the absence of derivatives and in a sample-efficient manner, is via Gaussian process (GP) surrogates and Bayesian optimization. This leads to a class of approaches called multiobjective Bayesian optimization (MOBO). MOBO involves the construction of a probabilistic surrogate model of all the objectives and constraints—typically GP models—which are then used to construct an acquisition function that quantifies the utility of a candidate input. An “inner” optimization problem is then solved to choose the optimal candidate in some sense. Therefore, a sequence of iterates can be generated in a greedy fashion, by optimizing the acquisition function repeatedly, that eventually leads to the solution. Existing work in MOBO are limited by one or more of the following limitations. (i) The inner optimization is highly nonconvex and/or stochastic making it non-trivial to obtain accurate solutions—this ultimately penalizes the sample efficiency. (ii) Inability to handle stochastic objectives and constraints. (iii) Inability to handle batch acquisitions. We propose to address all these limitations with our approach “Batch Pareto optimal Thompson sampling (qPOTS)”. We demonstrate that qPOTS outperforms the state of the art on several synthetic as well as real-world benchmarks, while being theoretically grounded.