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Optimal Transport Relaxations with Application to Wasserstein GANs
We propose a family of relaxations of the optimal transport problem which
regularize the problem by introducing an additional minimization step over a
small region around one of the underlying transporting measures. The type of
regularization that we obtain is related to smoothing techniques studied in the
optimization literature. When using our approach to estimate optimal transport
costs based on empirical measures, we obtain statistical learning bounds which
are useful to guide the amount of regularization, while maintaining good
generalization properties. To illustrate the computational advantages of our
regularization approach, we apply our method to training Wasserstein GANs. We
obtain running time improvements, relative to current benchmarks, with no
deterioration in testing performance (via FID). The running time improvement
occurs because our new optimality-based threshold criterion reduces the number
of expensive iterates of the generating networks, while increasing the number
of actor-critic iterations