Bayesian nonparametrics (BNP) models appear to be perfectly suited for the era of big data (Jordan, 2011), in which ever-expanding databases of high-dimensional data cannot be dealt with simplistically. Generative processes priors like the Dirichlet process (Ferguson, 1973) or the Indian buffet process (Griffiths and Ghahramani, 2011) allow for modeling latent variables like clusters or otherwise unobservable features in our data and adapting the complexity of the model in accordance to the complexity of the data. Even if we had some understanding of the latent structure in the data, we would not necessarily know their exact forms and implications in the model a priori. The BNP solution, which divides the data into discrete features and clusters, fosters interpretable models that would naturally lead to new hypotheses about the information in such databases (Kim et al., 2015). For example, in a general medical records dataset containing billions of observations, a cluster (or feature) composed of 0.001% of the population still includes tens of thousands of people.

Learning in models with discrete latent variables is challenging due to high variance gradient estimators. Generally, approaches have relied on control variates to reduce the variance of the REINFORCE estimator. Recent work (Jang et al. 2016, Maddison et al. 2016) has taken a different approach, introducing a continuous relaxation of discrete variables to produce low-variance, but biased, gradient estimates. In this work, we combine the two approaches through a novel control variate that produces low-variance, \emph{unbiased} gradient estimates. Then, we introduce a modification to the continuous relaxation and show that the tightness of the relaxation can be adapted online, removing it as a hyperparameter. We show state-of-the-art variance reduction on several benchmark generative modeling tasks, generally leading to faster convergence to a better final log-likelihood.

Density ratio estimation (DRE) [18, 11, 27] is an important tool in various branches of machine learning and statistics. Due to its ability of directly modelling the differences between two probability density functions, DRE finds its applications in change detection [13, 6], twosample test [32] and outlier detection [1, 26]. In recent years, a sampling framework called Generative Adversarial Network (GAN) (see e.g., [9, 19]) uses the density ratio function to compare artificial samples from a generative distribution and real samples from an unknown distribution. DRE has also been widely discussed in statistical literatures for adjusting nonparametric density estimation [5], stabilizing the estimation of heavy tailed distribution [7] and fitting multiple distributions at once [8]. However, as a density ratio function can grow unbounded, DRE can suffer from robustness and stability issues: a few corrupted points may completely mislead the estimator (see Figure 2 in Section 6 for example).

This paper is about index policies for minimizing (frequentist) regret in a stochastic multi-armed bandit model, inspired by a Bayesian view on the problem. Our main contribution is to prove that the Bayes-UCB algorithm, which relies on quantiles of posterior distributions, is asymptotically optimal when the reward distributions belong to a one-dimensional exponential family, for a large class of prior distributions. We also show that the Bayesian literature gives new insight on what kind of exploration rates could be used in frequentist, UCB-type algorithms. Indeed, approximations of the Bayesian optimal solution or the Finite Horizon Gittins indices provide a justification for the kl-UCB+ and kl-UCB-H+ algorithms, whose asymptotic optimality is also established.

We propose a novel method to directly learn a stochastic transition operator whose repeated application provides generated samples. Traditional undirected graphical models approach this problem indirectly by learning a Markov chain model whose stationary distribution obeys detailed balance with respect to a parameterized energy function. The energy function is then modified so the model and data distributions match, with no guarantee on the number of steps required for the Markov chain to converge. Moreover, the detailed balance condition is highly restrictive: energy based models corresponding to neural networks must have symmetric weights, unlike biological neural circuits. In contrast, we develop a method for directly learning arbitrarily parameterized transition operators capable of expressing non-equilibrium stationary distributions that violate detailed balance, thereby enabling us to learn more biologically plausible asymmetric neural networks and more general non-energy based dynamical systems. The proposed training objective, which we derive via principled variational methods, encourages the transition operator to "walk back" in multi-step trajectories that start at data-points, as quickly as possible back to the original data points. We present a series of experimental results illustrating the soundness of the proposed approach, Variational Walkback (VW), on the MNIST, CIFAR-10, SVHN and CelebA datasets, demonstrating superior samples compared to earlier attempts to learn a transition operator. We also show that although each rapid training trajectory is limited to a finite but variable number of steps, our transition operator continues to generate good samples well past the length of such trajectories, thereby demonstrating the match of its non-equilibrium stationary distribution to the data distribution. Source Code: http://github.com/anirudh9119/walkback_nips17

We present a novel technique for learning the mass matrices in samplers obtained from discretized dynamics that preserve some energy function. Existing adaptive samplers use Riemannian preconditioning techniques, where the mass matrices are functions of the parameters being sampled. This leads to significant complexities in the energy reformulations and resultant dynamics, often leading to implicit systems of equations and requiring inversion of high-dimensional matrices in the leapfrog steps. Our approach provides a simpler alternative, by using existing dynamics in the sampling step of a Monte Carlo EM framework, and learning the mass matrices in the M step with a novel online technique. We also propose a way to adaptively set the number of samples gathered in the E step, using sampling error estimates from the leapfrog dynamics. Along with a novel stochastic sampler based on Nos\'{e}-Poincar\'{e} dynamics, we use this framework with standard Hamiltonian Monte Carlo (HMC) as well as newer stochastic algorithms such as SGHMC and SGNHT, and show strong performance on synthetic and real high-dimensional sampling scenarios; we achieve sampling accuracies comparable to Riemannian samplers while being significantly faster.

A key task in Bayesian statistics is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). However, without any assumptions, sampling (even approximately) can be #P-hard, and few works have provided "beyond worst-case" guarantees for such settings. For log-concave distributions, classical results going back to Bakry and \'Emery (1985) show that natural continuous-time Markov chains called Langevin diffusions mix in polynomial time. The most salient feature of log-concavity violated in practice is uni-modality: commonly, the distributions we wish to sample from are multi-modal. In the presence of multiple deep and well-separated modes, Langevin diffusion suffers from torpid mixing. We address this problem by combining Langevin diffusion with simulated tempering. The result is a Markov chain that mixes more rapidly by transitioning between different temperatures of the distribution. We analyze this Markov chain for the canonical multi-modal distribution: a mixture of gaussians (of equal variance). The algorithm based on our Markov chain provably samples from distributions that are close to mixtures of gaussians, given access to the gradient of the log-pdf. For the analysis, we use a spectral decomposition theorem for graphs (Gharan and Trevisan, 2014) and a Markov chain decomposition technique (Madras and Randall, 2002).

We present MILABOT: a deep reinforcement learning chatbot developed by the Montreal Institute for Learning Algorithms (MILA) for the Amazon Alexa Prize competition. MILABOT is capable of conversing with humans on popular small talk topics through both speech and text. The system consists of an ensemble of natural language generation and retrieval models, including template-based models, bag-of-words models, sequence-to-sequence neural network and latent variable neural network models. By applying reinforcement learning to crowdsourced data and real-world user interactions, the system has been trained to select an appropriate response from the models in its ensemble. The system has been evaluated through A/B testing with real-world users, where it performed significantly better than many competing systems. Due to its machine learning architecture, the system is likely to improve with additional data.

We propose a statistical model for natural language that begins by considering language as a monoid, then representing it in complex matrices with a compatible translation invariant probability measure. We interpret the probability measure as arising via the Born rule from a translation invariant matrix product state.

Implicit probabilistic models are a flexible class of models defined by a simulation process for data. They form the basis for theories which encompass our understanding of the physical world. Despite this fundamental nature, the use of implicit models remains limited due to challenges in specifying complex latent structure in them, and in performing inferences in such models with large data sets. In this paper, we first introduce hierarchical implicit models (HIMs). HIMs combine the idea of implicit densities with hierarchical Bayesian modeling, thereby defining models via simulators of data with rich hidden structure. Next, we develop likelihood-free variational inference (LFVI), a scalable variational inference algorithm for HIMs. Key to LFVI is specifying a variational family that is also implicit. This matches the model's flexibility and allows for accurate approximation of the posterior. We demonstrate diverse applications: a large-scale physical simulator for predator-prey populations in ecology; a Bayesian generative adversarial network for discrete data; and a deep implicit model for text generation.