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A major approach to saddle point optimization $\min_x\max_y f(x, y)$ is agradient based approach as is popularized by generative adversarial networks(GANs). In contrast, we analyze an alternative approach relying only on anoracle that solves a minimization problem approximately. Our approach locatesapproximate solutions $x'$ and $y'$ to $\min_{x'}f(x', y)$ and $\max_{y'}f(x,y')$ at a given point $(x, y)$ and updates $(x, y)$ toward these approximatesolutions $(x', y')$ with a learning rate $\eta$. On locally strongconvex--concave smooth functions, we derive conditions on $\eta$ to exhibitlinear convergence to a local saddle point, which reveals a possibleshortcoming of recently developed robust adversarial reinforcement learningalgorithms. We develop a heuristic approach to adapt $\eta$ derivative-free andimplement zero-order and first-order minimization algorithms. Numericalexperiments are conducted to show the tightness of the theoretical results aswell as the usefulness of the $\eta$ adaptation mechanism.

Saddle Point Optimization with Approximate Minimization Oracle