Generative modeling of high dimensional data like images is a notoriously difficult and ill-defined problem. In particular, how to evaluate a learned generative model is unclear.
 In this paper, we argue that *adversarial learning*, pioneered with generative adversarial networks (GANs), provides an interesting framework to implicitly define more meaningful task losses for unsupervised tasks, such as for generating "visually realistic" images. By relating GANs and structured prediction under the framework of statistical decision theory, we put into light links between recent advances in structured prediction theory and the choice of the divergence in GANs. We argue that the insights about the notions of "hard" and "easy" to learn losses can be analogously extended to adversarial divergences. We also discuss the attractive properties of parametric adversarial divergences for generative modeling, and perform experiments to show the importance of choosing a divergence that reflects the final task. For structured prediction and data generation the notion of final task is at the same time crucial and not well defined. Consider machine translation; the goal is to predict a good translation, but even humans might disagree on the correct translation of a sentence. Moreover, even if we settle on a ground truth, it is hard to define what it means for a candidate translation to be close to the ground truth. In the same way, for data generation, the task of generating pretty pictures or more generally realistic samples is not well defined. Nevertheless, both for structured prediction and data generation, we can try to define criteria which characterize good solutions such as grammatical correctness for translation or non-blurry pictures for image generation. By incorporating enough criteria into a task loss, one can hope to approximate the final task, which is otherwise hard to formalize.Supervised learning and structured prediction are well-defined problems once they are formulated as the minimization of such a task loss. The usual task loss in object classification is the generalization error associated with the classification error, or 0-1 loss. In machine translation, where the goal is to predict a sentence, a structured loss, such as the BLEU score BID37 , formally specifies how close the predicted sentence is from the ground truth. The generalization error is defined through this structured loss. In both cases, models can be objectively compared and evaluated with respect to the task loss (i.e., generalization error). On the other hand, we will show that it is not as obvious in generative modeling to define a task loss that correlates well with the final task of generating realistic samples.Traditionally in statistics, distribution learning is formulated as density estimation where the task loss is the expected negative-log-likelihood. Although log-likelihood works fine in low-dimension, it was shown to have many problems in high-dimension . Among others, because the Kullback-Leibler is too strong of a divergence, it can easily saturate whenever the distributions are too far apart, which makes it hard to optimize. Additionally, it was shown in BID47 that the KL-divergence is a bad proxy for the visual quality of samples.In this work we give insights on how adversarial divergences BID26 can be considered as task losses and how they address some problems of the KL by indirectly incorporating hard-to-define criteria. We define parametric adversarial divergences as the following : DISPLAYFORM0 where {f φ : X → R d ; φ ∈ Φ} is a class of parametrized functions, such as neural networks, called the discriminators in the Generative Adversarial Network (GAN) framework BID15 . The constraints Φ and the function ∆ : R d × R d → R determine properties of the resulting divergence. Using these notations, we adopt the view 1 that training a GAN can be seen as training a generator network q θ (parametrized by θ) to minimize the parametric adversarial divergence Div NN (p||q θ ), where the generator network defines the probability distribution q θ over x.Our contributions are the following:• We show that compared to traditional divergences, parametric adversarial divergences offer a good compromise in terms of sample complexity, computation, ability to integrate prior knowledge, flexibility and ease of optimization.• We relate structured prediction and generative adversarial networks using statistical decision theory, and argue that they both can be viewed as formalizing a final task into the minimization of a statistical task loss.• We explain why it is necessary to choose a divergence that adequately reflects our final task in generative modeling. We make a parallel with results in structured learning (also dealing with high-dimensional data), which quantify the importance of choosing a good objective in a specific setting.• We explore with some simple experiments how the properties of the discriminator transfer to the adversarial divergence. Our experiments suggest that parametric adversarial divergences are especially adapted to problems such as image generation, where it is hard to formally define a perceptual loss that correlates well with human judgment.• We illustrate the importance of having a parametric discriminator by running experiments with the true (nonparametric) Wasserstein, and showing its shortcomings on complex datasets, on which GANs are known to perform well.• We perform qualitative and quantitative experiments to compare maximum-likelihood and parametric adversarial divergences under two settings: very high-dimensional images, and learning data with specific constraints. We gave arguments in favor of using adversarial divergences rather than traditional divergences for generative modeling, the most important of which being the ability to account for the final task. After linking structured prediction and generative modeling under the framework of statistical decision theory, we interpreted recent results from structured prediction, and related them to the notions of strong and weak divergences. Moreover, viewing adversarial divergences as statistical task losses led us to believe that some adversarial divergences could be used as evaluation criteria in the future, replacing hand-crafted criteria which cannot usually be exhaustive. In some sense, we want to extrapolate a few desirable properties into a meaningful task loss. In the future we would like to investigate how to define meaningful evaluation criteria with minimal human intervention.
Parametric adversarial divergences implicitly define more meaningful task losses for generative modeling, we make parallels with structured prediction to study the properties of these divergences and their ability to encode the task of interest.
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