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Register a new userTighter Variational Representations of f-Divergences via Restriction to Probability Measures
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We show that the variational representations for f-divergences currently used in the literature can be tightened. This has implications to a number of methods recently proposed based on this representation. As an example application we use our tighter representation to derive a general f-divergence estimator based on two i.i.d. samples and derive the dual program for this estimator that performs well empirically. We also point out a connection between our estimator and MMD.
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We develop a rigorous and general framework for constructing information-theoretic divergences that subsume both $f$-divergences and integral probability metrics (IPMs), such as the $1$-Wasserstein distance. We prove under which assumptions these divergences, hereafter referred to as $(f,Gamma)$-divergences, provide a notion of `distance between probability measures and show that they can be expressed as a two-stage mass-redistribution/mass-transport process. The $(f,Gamma)$-divergences inherit features from IPMs, such as the ability to compare distributions which are not absolutely continuous, as well as from $f$-divergences, namely the strict concavity of their variational representations and the ability to control heavy-tailed distributions for particular choices of $f$. When combined, these features establish a divergence with improved properties for estimation, statistical learning, and uncertainty quantification applications. Using statistical learning as an example, we demonstrate their advantage in training generative adversarial networks (GANs) for heavy-tailed, not-absolutely continuous sample distributions. We also show improved performance and stability over gradient-penalized Wasserstein GAN in image generation.
Variational representations of divergences and distances between high-dimensional probability distributions offer significant theoretical insights and practical advantages in numerous research areas. Recently, they have gained popularity in machine learning as a tractable and scalable approach for training probabilistic models and for statistically differentiating between data distributions. Their advantages include: 1) They can be estimated from data as statistical averages. 2) Such representations can leverage the ability of neural networks to efficiently approximate optimal solutions in function spaces. However, a systematic and practical approach to improving tightness of such variational formulas, and accordingly accelerate statistical learning and estimation from data, is lacking. Here we develop such a methodology for building new, tighter variational representations of divergences. Our approach relies on improved objective functionals constructed via an auxiliary optimization problem. Furthermore, the calculation of the functional Hessian of objective functionals unveils local curvature differences around the common optimal variational solution; this quantifies and orders the tightness gains between different variational representations. Finally, numerical simulations utilizing neural-network optimization demonstrate that tighter representations can result in significantly faster learning and more accurate estimation of divergences in both synthetic and real datasets (of more than 1000 dimensions), often accelerated by nearly an order of magnitude.
Generative Adversarial Networks (GANs) are modern methods to learn the underlying distribution of a data set. GANs have been widely used in sample synthesis, de-noising, domain transfer, etc. GANs, however, are designed in a model-free fashion where no additional information about the underlying distribution is available. In many applications, however, practitioners have access to the underlying independence graph of the variables, either as a Bayesian network or a Markov Random Field (MRF). We ask: how can one use this additional information in designing model-based GANs? In this paper, we provide theoretical foundations to answer this question by studying subadditivity properties of probability divergences, which establish upper bounds on the distance between two high-dimensional distributions by the sum of distances between their marginals over (local) neighborhoods of the graphical structure of the Bayes-net or the MRF. We prove that several popular probability divergences satisfy some notion of subadditivity under mild conditions. These results lead to a principled design of a model-based GAN that uses a set of simple discriminators on the neighborhoods of the Bayes-net/MRF, rather than a giant discriminator on the entire network, providing significant statistical and computational benefits. Our experiments on synthetic and real-world datasets demonstrate the benefits of our principled design of model-based GANs.
The standard approach to supervised classification involves the minimization of a log-loss as an upper bound to the classification error. While this is a tight bound early on in the optimization, it overemphasizes the influence of incorrectly classified examples far from the decision boundary. Updating the upper bound during the optimization leads to improved classification rates while transforming the learning into a sequence of minimization problems. In addition, in the context where the classifier is part of a larger system, this modification makes it possible to link the performance of the classifier to that of the whole system, allowing the seamless introduction of external constraints.
We investigate the power of censoring techniques, first developed for learning {em fair representations}, to address domain generalization. We examine {em adversarial} censoring techniques for learning invariant representations from multiple studies (or domains), where each study is drawn according to a distribution on domains. The mapping is used at test time to classify instances from a new domain. In many contexts, such as medical forecasting, domain generalization from studies in populous areas (where data are plentiful), to geographically remote populations (for which no training data exist) provides fairness of a different flavor, not anticipated in previous work on algorithmic fairness. We study an adversarial loss function for $k$ domains and precisely characterize its limiting behavior as $k$ grows, formalizing and proving the intuition, backed by experiments, that observing data from a larger number of domains helps. The limiting results are accompanied by non-asymptotic learning-theoretic bounds. Furthermore, we obtain sufficient conditions for good worst-case prediction performance of our algorithm on previously unseen domains. Finally, we decompose our mappings into two components and provide a complete characterization of invariance in terms of this decomposition. To our knowledge, our results provide the first formal guarantees of these kinds for adversarial invariant domain generalization.
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