We propose a framework for adversarial training that relies on a sample rather than a single sample point as the fundamental unit of discrimination. Inspired by discrepancy measures and two-sample tests between probability distributions, we propose two such distributional adversaries that operate and predict on samples, and show how they can be easily implemented on top of existing models. Various experimental results show that generators trained with our distributional adversaries are much more stable and are remarkably less prone to mode collapse than traditional models trained with pointwise prediction discriminators. The application of our framework to domain adaptation also results in considerable improvement over recent state-of-the-art.
The distributional adversary bears close resemblance to two-sample tests @cite_25 , where the model takes two samples drawn from potentially distinct distributions as input and produces a discrepancy value quantifying how different two distributions are. A popular kernel-based variant is the (MMD) @cite_0 @cite_15 @cite_29 : where @math is some feature mapping, and @math is the corresponding kernel function. An identity function for @math corresponds to computing the distance of the sample means. More complex kernels result in distances of higher-order statistics between two samples.
@cite_0 @cite_15 @cite_29 @cite_25
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Given samples from distributions p and q, a two-sample test determines whether to reject the null hypothesis that p = q, based on the value of a test statistic measuring the distance between the samples. One choice of test statistic is the maximum mean discrepancy (MMD), which is a distance between embeddings of the probability distributions in a reproducing kernel Hilbert space. The kernel used in obtaining these embeddings is critical in ensuring the test has high power, and correctly distinguishes unlike distributions with high probability. A means of parameter selection for the two-sample test based on the MMD is proposed. For a given test level (an upper bound on the probability of making a Type I error), the kernel is chosen so as to maximize the test power, and minimize the probability of making a Type II error. The test statistic, test threshold, and optimization over the kernel parameters are obtained with cost linear in the sample size. These properties make the kernel selection and test procedures suited to data streams, where the observations cannot all be stored in memory. In experiments, the new kernel selection approach yields a more powerful test than earlier kernel selection heuristics. We consider training a deep neural network to generate samples from an unknown distribution given i.i.d. data. We frame learning as an optimization minimizing a two-sample test statistic—informally speaking, a good generator network produces samples that cause a two-sample test to fail to reject the null hypothesis. As our two-sample test statistic, we use an unbiased estimate of the maximum mean discrepancy, which is the centerpiece of the nonparametric kernel two-sample test proposed by [2]. We compare to the adversarial nets framework introduced by [1], in which learning is a two-player game between a generator network and an adversarial discriminator network, both trained to outwit the other. From this perspective, the MMD statistic plays the role of the discriminator. In addition to empirical comparisons, we prove bounds on the generalization error incurred by optimizing the empirical MMD. We consider training a deep neural network to generate samples from an unknown distribution given i.i.d. data. We frame learning as an optimization minimizing a two-sample test statistic---informally speaking, a good generator network produces samples that cause a two-sample test to fail to reject the null hypothesis. As our two-sample test statistic, we use an unbiased estimate of the maximum mean discrepancy, which is the centerpiece of the nonparametric kernel two-sample test proposed by (2012). We compare to the adversarial nets framework introduced by (2014), in which learning is a two-player game between a generator network and an adversarial discriminator network, both trained to outwit the other. From this perspective, the MMD statistic plays the role of the discriminator. In addition to empirical comparisons, we prove bounds on the generalization error incurred by optimizing the empirical MMD. We propose a method to optimize the representation and distinguishability of samples from two probability distributions, by maximizing the estimated power of a statistical test based on the maximum mean discrepancy (MMD). This optimized MMD is applied to the setting of unsupervised learning by generative adversarial networks (GAN), in which a model attempts to generate realistic samples, and a discriminator attempts to tell these apart from data samples. In this context, the MMD may be used in two roles: first, as a discriminator, either directly on the samples, or on features of the samples. Second, the MMD can be used to evaluate the performance of a generative model, by testing the model's samples against a reference data set. In the latter role, the optimized MMD is particularly helpful, as it gives an interpretable indication of how the model and data distributions differ, even in cases where individual model samples are not easily distinguished either by eye or by classifier.