Loss-Sensitive Generative Adversarial Networks on Lipschitz Densities
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*New Theory Result* We analyze the generalizability of the LS-GAN, showing that the loss function and generator trained over finite examples can converge to those learned from the real distributions with a moderate number of training examples. In this paper, we present a novel Loss-Sensitive GAN (LS-GAN) that learns a loss function to separate generated samples from their real examples. An important property of the LS-GAN is it allows the generator to focus on improving poor data points that are far apart from real examples rather than wasting efforts on those samples that have already been well generated, and thus can improve the overall quality of generated samples. The theoretical analysis also shows that the LS-GAN can generate samples following the true data density. In particular, we present a regularity condition on the underlying data density, which allows us to use a class of Lipschitz losses and generators to model the LS-GAN. It relaxes the assumption that the classic GAN should have infinite modeling capacity to obtain the similar theoretical guarantee. Furthermore, we show the generalization ability of the LS-GAN by bounding the difference between the model performances over the empirical and real distributions, as well as deriving a tractable sample complexity to train the LS-GAN model in terms of its generalization ability. We also derive a non-parametric solution that characterizes the upper and lower bounds of the losses learned by the LS-GAN, both of which are cone-shaped and have non-vanishing gradient almost everywhere.
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