On the Application of Danskin's Theorem to Derivative-Free Minimax Optimization
For practitioners solving black-box minimax problems, this work provides a gradient-free alternative that matches coevolutionary methods and scales better to high dimensions.
The paper applies Evolution Strategies as a stochastic gradient estimator for black-box minimax optimization, leveraging Danskin's theorem. The method achieves performance comparable to coevolutionary approaches, with advantages in high-dimensional problems and demonstrated efficacy on a real-world application.
Motivated by Danskin's theorem, gradient-based methods have been applied with empirical success to solve minimax problems that involve non-convex outer minimization and non-concave inner maximization. On the other hand, recent work has demonstrated that Evolution Strategies (ES) algorithms are stochastic gradient approximators that seek robust solutions. In this paper, we address black-box (gradient-free) minimax problems that have long been tackled in a coevolutionary setup. To this end and guaranteed by Danskin's theorem, we employ ES as a stochastic estimator for the descent direction. The proposed approach is validated on a collection of black-box minimax problems. Based on our experiments, our method's performance is comparable with its coevolutionary counterparts and favorable for high-dimensional problems. Its efficacy is demonstrated on a real-world application.