Saddle point optimization with approximate minimization oracle

A major approach to saddle point optimization minx maxy f(x, y) is a gradient based approach as is popularized by generative adversarial networks (GANs). In contrast, we analyze an alternative approach relying only on an oracle that solves a minimization problem approximately. Our approach locates approximate solutions x′ and y′ to minx′ f(x′, y) and maxy′ f(x, y′) at a given point (x, y) and updates (x, y) toward these approximate solutions (x′, y′) with a learning rate η. On locally strong convex-concave smooth functions, we derive conditions on η to exhibit linear convergence to a local saddle point, which reveals a possible shortcoming of recently developed robust adversarial reinforcement learning algorithms. We develop a heuristic approach to adapt η derivative-free and implement zero-order and first-order minimization algorithms. Numerical experiments are conducted to show the tightness of the theoretical results as well as the usefulness of the η adaptation mechanism.