A Provably Convergent and Practical Algorithm for Min-Max Optimization with Applications to GANs

We present a first-order algorithm for nonconvex-nonconcave min-max optimization problems such as those that arise in training GANs. Our algorithm provably converges in $\mathrm{poly}(d,L, b)$ steps for any loss function $f:\mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ which is $b$-bounded with ${L}$-Lipschitz gradient. To achieve convergence, we 1) give a novel approximation to the global strategy of the max-player based on first-order algorithms such as gradient ascent, and 2) empower the min-player to look ahead and simulate the max-player’s response for arbitrarily many steps, but restrict the min-player to move according to updates sampled from a stochastic gradient oracle. Our algorithm, when used to train GANs on synthetic and real-world datasets, does not cycle, results in GANs that seem to avoid mode collapse, and achieves a training time per iteration and memory requirement similar to gradient descent-ascent.