Bézier Flow: A Surface-Wise Gradient Descent Method for Multi-Objective Optimization

Abstract

This paper proposes a framework to construct a multi-objective optimization algorithm from a single-objective optimization algorithm by using the Bézier simplex model. Additionally, we extend the stability of optimization algorithms in the sense of Probably Approximately Correct (PAC) learning and define the PAC stability. We prove that it leads to an upper bound on the generalization error with high probability. Furthermore, we show that multi-objective optimization algorithms derived from a gradient descent-based single-objective optimization algorithm are PAC stable. We conducted numerical experiments with synthetic and real multi-objective optimization problem instances and demonstrated that our method achieved lower generalization errors than the existing multi-objective optimization algorithms.