Kernelized Wasserstein Natural Gradient

Many machine learning problems can be expressed as the optimization of some cost functional over a parametric family of probability distributions. It is often beneficial to solve such optimization problems using natural gradient methods. These methods are invariant to the parametrization of the family, and thus can yield more effective optimization. Unfortunately, computing the natural gradient is challenging as it requires inverting a high dimensional matrix at each iteration. We propose a general framework to approximate the natural gradient for the Wasserstein metric, by leveraging a dual formulation of the metric restricted to a Reproducing Kernel Hilbert Space. Our approach leads to an estimator for gradient direction that can trade-off accuracy and computational cost, with theoretical guarantees. We verify its accuracy on simple examples, and show the advantage of using such an estimator in classification tasks on Cifar10 and Cifar100 empirically.

Counterfactual Distribution Regression for Structured Inference

We consider problems in which a system receives external \emph{perturbations} from time to time. For instance, the system can be a train network in which particular lines are repeatedly disrupted without warning, having an effect on passenger behavior. The goal is to predict changes in the behavior of the system at particular points of interest, such as passenger traffic around stations at the affected rails. We assume that the data available provides records of the system functioning at its "natural regime" (e.g., the train network without disruptions) and data on cases where perturbations took place. The inference problem is how information concerning perturbations, with particular covariates such as location and time, can be generalized to predict the effect of novel perturbations. We approach this problem from the point of view of a mapping from the counterfactual distribution of the system behavior without disruptions to the distribution of the disrupted system. A variant on \emph{distribution regression} is developed for this setup.

A Kernel Stein Test for Comparing Latent Variable Models

We propose a nonparametric, kernel-based test to assess the relative goodness of fit of latent variable models with intractable unnormalized densities. Our test generalises the kernel Stein discrepancy (KSD) tests of (Liu et al., 2016, Chwialkowski et al., 2016, Yang et al., 2018, Jitkrittum et al., 2018) which required exact access to unnormalized densities. Our new test relies on the simple idea of using an approximate observed-variable marginal in place of the exact, intractable one. As our main theoretical contribution, we prove that the new test, with a properly corrected threshold, has a well-controlled type-I error. In the case of models with low-dimensional latent structure and high-dimensional observations, our test significantly outperforms the relative maximum mean discrepancy test (Bounliphone et al., 2015) , which cannot exploit the latent structure.

Maximum Mean Discrepancy Gradient Flow

We construct a Wasserstein gradient flow of the maximum mean discrepancy (MMD) and study its convergence properties. The MMD is an integral probability metric defined for a reproducing kernel Hilbert space (RKHS), and serves as a metric on probability measures for a sufficiently rich RKHS. We obtain conditions for convergence of the gradient flow towards a global optimum, that can be related to particle transport when optimizing neural networks. We also propose a way to regularize this MMD flow, based on an injection of noise in the gradient. This algorithmic fix comes with theoretical and empirical evidence. The practical implementation of the flow is straightforward, since both the MMD and its gradient have simple closed-form expressions, which can be easily estimated with samples.

Kernel Instrumental Variable Regression

Instrumental variable regression is a strategy for learning causal relationships in observational data. If measurements of input X and output Y are confounded, the causal relationship can nonetheless be identified if an instrumental variable Z is available that influences X directly, but is conditionally independent of Y given X. The classic two-stage least squares algorithm (2SLS) simplifies the estimation problem by modeling all relationships as linear functions. We propose kernel instrumental variable regression (KIV), a nonparametric generalization of 2SLS, modeling relations among X, Y, and Z as nonlinear functions in reproducing kernel Hilbert spaces (RKHSs). We prove the consistency of KIV under mild assumptions, and derive conditions under which the convergence rate achieves the minimax optimal rate for unconfounded, one-stage RKHS regression. In doing so, we obtain an efficient ratio between training sample sizes used in the algorithm's first and second stages. In experiments, KIV outperforms state of the art alternatives for nonparametric instrumental variable regression.

Exponential Family Estimation via Adversarial Dynamics Embedding

We present an efficient algorithm for maximum likelihood estimation (MLE) of the general exponential family, even in cases when the energy function is represented by a deep neural network. We consider the primal-dual view of the MLE for the kinectics augmented model, which naturally introduces an adversarial dual sampler. The sampler will be represented by a novel neural network architectures, dynamics embeddings, mimicking the dynamical-based samplers, e.g., Hamiltonian Monte-Carlo and its variants. The dynamics embedding parametrization inherits the flexibility from HMC, and provides tractable entropy estimation of the augmented model. Meanwhile, it couples the adversarial dual samplers with the primal model, reducing memory and sample complexity. We further show that several existing estimators, including contrastive divergence (Hinton, 2002), score matching (Hyv\"arinen, 2005), pseudo-likelihood (Besag, 1975), noise-contrastive estimation (Gutmann and Hyv\"arinen, 2010), non-local contrastive objectives (Vickrey et al., 2010), and minimum probability flow (Sohl-Dickstein et al., 2011), can be recast as the special cases of the proposed method with different prefixed dual samplers. Finally, we empirically demonstrate the superiority of the proposed estimator against existing state-of-the-art methods on synthetic and real-world benchmarks.

Learning deep kernels for exponential family densities

The kernel exponential family is a rich class of distributions,which can be fit efficiently and with statistical guarantees by score matching. Being required to choose a priori a simple kernel such as the Gaussian, however, limits its practical applicability. We provide a scheme for learning a kernel parameterized by a deep network, which can find complex location-dependent local features of the data geometry. This gives a very rich class of density models, capable of fitting complex structures on moderate-dimensional problems. Compared to deep density models fit via maximum likelihood, our approach provides a complementary set of strengths and tradeoffs: in empirical studies, the former can yield higher likelihoods, whereas the latter gives better estimates of the gradient of the log density, the score, which describes the distribution's shape.

Kernel Exponential Family Estimation via Doubly Dual Embedding

We investigate penalized maximum log-likelihood estimation for exponential family distributions whose natural parameter resides in a reproducing kernel Hilbert space. Key to our approach is a novel technique, doubly dual embedding, that avoids computation of the partition function. This technique also allows the development of a flexible sampling strategy that amortizes the cost of Monte-Carlo sampling in the inference stage. The resulting estimator can be easily generalized to kernel conditional exponential families. We furthermore establish a connection between infinite-dimensional exponential family estimation and MMD-GANs, revealing a new perspective for understanding GANs. Compared to current score matching based estimators, the proposed method improves both memory and time efficiency while enjoying stronger statistical properties, such as fully capturing smoothness in its statistical convergence rate while the score matching estimator appears to saturate. Finally, we show that the proposed estimator can empirically outperform state-of-the-art methods in both kernel exponential family estimation and its conditional extension.

On gradient regularizers for MMD GANs

We propose a principled method for gradient-based regularization of the critic of GAN-like models trained by adversarially optimizing the kernel of a Maximum Mean Discrepancy (MMD). We show that controlling the gradient of the critic is vital to having a sensible loss function, and devise a method to enforce exact, analytical gradient constraints at no additional cost compared to existing approximate techniques based on additive regularizers. The new loss function is provably continuous, and experiments show that it stabilizes and accelerates training, giving image generation models that outperform state-of-the art methods on $160 \times 160$ CelebA and $64 \times 64$ unconditional ImageNet.

Informative Features for Model Comparison

Given two candidate models, and a set of target observations, we address the problem of measuring the relative goodness of fit of the two models. We propose two new statistical tests which are nonparametric, computationally efficient (runtime complexity is linear in the sample size), and interpretable. As a unique advantage, our tests can produce a set of examples (informative features) indicating the regions in the data domain where one model fits significantly better than the other. In a real-world problem of comparing GAN models, the test power of our new test matches that of the state-of-the-art test of relative goodness of fit, while being one order of magnitude faster.