Random Hypervolume Scalarizations for Provable Multi-Objective Black Box Optimization

Jul 12, 2020

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Single-objective black box optimization (also known as zeroth-order optimization) is the process of minimizing a scalar objective f(x), given evaluations at adaptively chosen inputs x. In this paper, we consider multi-objective optimization, where f(x) outputs a vector of possibly competing objectives and the goal is to converge to the Pareto frontier. Quantitatively, we wish to maximize the standard hypervolume indicator metric, which measures the dominated hypervolume of the entire set of chosen inputs. In this paper, we introduce a novel scalarization function, which we term the hypervolume scalarization, and show that drawing random scalarizations from an appropriately chosen distribution can be used to efficiently approximate the hypervolume indicator metric. We utilize this connection to show that Bayesian optimization with our scalarization via common acquisition functions, such as Thompson Sampling or Upper Confidence Bound, provably converges to the whole Pareto frontier by deriving tight hypervolume regret bounds on the order of O(√(T)). Furthermore, we highlight the general utility of our scalarization framework by showing that any provably convergent single-objective optimization process can be converted to a multi-objective optimization process with provable convergence guarantees.

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The International Conference on Machine Learning (ICML) is the premier gathering of professionals dedicated to the advancement of the branch of artificial intelligence known as machine learning. ICML is globally renowned for presenting and publishing cutting-edge research on all aspects of machine learning used in closely related areas like artificial intelligence, statistics and data science, as well as important application areas such as machine vision, computational biology, speech recognition, and robotics. ICML is one of the fastest growing artificial intelligence conferences in the world. Participants at ICML span a wide range of backgrounds, from academic and industrial researchers, to entrepreneurs and engineers, to graduate students and postdocs.

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