Optimal Scalarizations for Provable Multiobjective Optimization

Qiuyi Zhang

Keywords: multiobjective optimization, scalarization, linear bandits

TL;DR: Don't linearly combine your objectives: Hypervolume scalarizations provide provable and more optimal multiobjective optimization.

Abstract: Linear scalarization is a simple and widely-used technique that can be deployed in any multiobjective setting to combine diverse objectives into one reward function, but such heuristics are not theoretically understood. To that end, we perform a case study of the multiobjective stochastic linear bandits framework with $k$ objectives and our goal is to provably scalarize and explore a diverse set of optimal actions on the Pareto frontier, as measured by the dominated hypervolume. Even in this elementary convex setting, the choice of scalarizations and weight distribution surprisingly affects performance, and the natural use of linear scalarization with uniform weights is suboptimal due to a non-uniform Pareto curvature. Instead, we suggest the usage of the theoretically-inspired hypervolume scalarizations with non-adaptive uniform weights, showing that it comes with novel hypervolume regret bounds of $\tilde{O}( d T^{-1/2} + T^{-1/k})$, with optimal matching lower bounds of $\Omega(T^{-1/k})$. We support our theory with strong empirical performance of the hypervolume scalarization that consistently outperforms both the linear and Chebyshev scalarizations in high dimensions.

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