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 time polynomial in the dimension and smoothness parameters of the loss function . 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  .
We propose a variant of the standard min-max framework for GANs to learn a distribution, where the discriminator can update its strategy in a greedy manner until it reaches a first-order stationary point. We give an algorithm to train such a GAN and show that it provably converges from any initial point to an approximate local equilibrium for this framework . Our algorithm runs in time polynomial in the smoothness parameters of the loss function and independent of the dimension, and allows for loss functions that can be nonconvex and nonconcave in the parameters of the generator and discriminator. Empirically, GANs trained using our algorithm consistently learn a greater number of modes than gradient descent-ascent (GDA), optimistic mirror descent (OMD), and unrolled GANs when applied to a synthetic Gaussian mixture dataset. Moreover, they perform significantly better on CIFAR-10 than OMD and GDA when comparing the mean and standard deviation of the Inception Score respectively .
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present a first-order algorithm for nonconvex-nonconcave
propose a variant of the standard
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optimization problems such as those that arise in training GANs
framework for GANs to learn a distribution, where the discriminator can update its strategy in a greedy manner until it reaches a first-order stationary point. We give an algorithm to train such a GAN and show that it provably converges from any initial point to an approximate local equilibrium for this framework
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provably converges
runs
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dimension and
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. 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
and independent of the dimension, and allows for loss functions that can be nonconvex and nonconcave in the parameters of the generator and discriminator. Empirically, GANs trained using our algorithm consistently learn a greater number of modes than gradient
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(GDA), optimistic mirror descent (OMD), and unrolled GANs when applied to a synthetic Gaussian mixture dataset. Moreover, they perform significantly better on CIFAR-10 than OMD and GDA when comparing the mean and standard deviation of the Inception Score respectively