Stochastic gradient methods are the workhorse (algorithms) of large-scale optimization problems in machine learning, signal processing, and other computational sciences and engineering. This paper studies Markov chain gradient descent, a variant of stochastic gradient descent where the random samples are taken on the trajectory of a Markov chain. Existing results of this method assume convex objectives and a reversible Markov chain and thus have their limitations. We establish new non-ergodic convergence under wider step sizes, for nonconvex problems, and for non-reversible finite-state Markov chains. Nonconvexity makes our method applicable to broader problem classes. Non-reversible finite-state Markov chains, on the other hand, can mix substatially faster. To obtain these results, we introduce a new technique that varies the mixing levels of the Markov chains. The reported numerical results validate our contributions.

We introduce a method which enables a recurrent dynamics model to be temporally abstract. Our approach, which we call Adaptive Skip Intervals (ASI), is based on the observation that in many sequential prediction tasks, the exact time at which events occur is irrelevant to the underlying objective. Moreover, in many situations, there exist prediction intervals which result in particularly easy-to-predict transitions. We show that there are prediction tasks for which we gain both computational efficiency and prediction accuracy by allowing the model to make predictions at a sampling rate which it can choose itself.

Entropy estimation is one of the prototypical problems in distribution property testing. To consistently estimate the Shannon entropy of a distribution on $S$ elements with independent samples, the optimal sample complexity scales sublinearly with $S$ as $\Theta(\frac{S}{\log S})$ as shown by Valiant and Valiant \cite{Valiant--Valiant2011}. Extending the theory and algorithms for entropy estimation to dependent data, this paper considers the problem of estimating the entropy rate of a stationary reversible Markov chain with $S$ states from a sample path of $n$ observations. We show that \begin{itemize} \item Provided the Markov chain mixes not too slowly, \textit{i.e.}, the relaxation time is at most $O(\frac{S}{\ln^3 S})$, consistent estimation is achievable when $n \gg \frac{S^2}{\log S}$. \item Provided the Markov chain has some slight dependency, \textit{i.e.}, the relaxation time is at least $1+\Omega(\frac{\ln^2 S}{\sqrt{S}})$, consistent estimation is impossible when $n \lesssim \frac{S^2}{\log S}$. \end{itemize} Under both assumptions, the optimal estimation accuracy is shown to be $\Theta(\frac{S^2}{n \log S})$. In comparison, the empirical entropy rate requires at least $\Omega(S^2)$ samples to be consistent, even when the Markov chain is memoryless. In addition to synthetic experiments, we also apply the estimators that achieve the optimal sample complexity to estimate the entropy rate of the English language in the Penn Treebank and the Google One Billion Words corpora, which provides a natural benchmark for language modeling and relates it directly to the widely used perplexity measure.

Despite the advances in the representational capacity of approximate distributions for variational inference, the optimization process can still limit the density that is ultimately learned. We demonstrate the drawbacks of biasing the true posterior to be unimodal, and introduce Annealed Variational Objectives (AVO) into the training of hierarchical variational methods. Inspired by Annealed Importance Sampling, the proposed method facilitates learning by incorporating energy tempering into the optimization objective. In our experiments, we demonstrate our method's robustness to deterministic warm up, and the benefits of encouraging exploration in the latent space.

Meta-learning for few-shot learning entails acquiring a prior over previous tasks and experiences, such that new tasks be learned from small amounts of data. However, a critical challenge in few-shot learning is task ambiguity: even when a powerful prior can be meta-learned from a large number of prior tasks, a small dataset for a new task can simply be too ambiguous to acquire a single model (e.g., a classifier) for that task that is accurate. In this paper, we propose a probabilistic meta-learning algorithm that can sample models for a new task from a model distribution. Our approach extends model-agnostic meta-learning, which adapts to new tasks via gradient descent, to incorporate a parameter distribution that is trained via a variational lower bound. At meta-test time, our algorithm adapts via a simple procedure that injects noise into gradient descent, and at meta-training time, the model is trained such that this stochastic adaptation procedure produces samples from the approximate model posterior. Our experimental results show that our method can sample plausible classifiers and regressors in ambiguous few-shot learning problems. We also show how reasoning about ambiguity can also be used for downstream active learning problems.

What policy should be employed in a Markov decision process with uncertain parameters? Robust optimization answer to this question is to use rectangular uncertainty sets, which independently reflect available knowledge about each state, and then obtains a decision policy that maximizes expected reward for the worst-case decision process parameters from these uncertainty sets. While this rectangularity is convenient computationally and leads to tractable solutions, it often produces policies that are too conservative in practice, and does not facilitate knowledge transfer between portions of the state space or across related decision processes. In this work, we propose non-rectangular uncertainty sets that bound marginal moments of state-action features defined over entire trajectories through a decision process. This enables generalization to different portions of the state space while retaining appropriate uncertainty of the decision process. We develop algorithms for solving the resulting robust decision problems, which reduce to finding an optimal policy for a mixture of decision processes, and demonstrate the benefits of our approach experimentally.

The count-min sketch is a time- and memory-efficient randomized data structure that provides a point estimate of the number of times an item has appeared in a data stream. The count-min sketch and related hash-based data structures are ubiquitous in systems that must track frequencies of data such as URLs, IP addresses, and language n-grams. We present a Bayesian view on the count-min sketch, using the same data structure, but providing a posterior distribution over the frequencies that characterizes the uncertainty arising from the hash-based approximation. In particular, we take a nonparametric approach and consider tokens generated from a Dirichlet process (DP) random measure, which allows for an unbounded number of unique tokens. Using properties of the DP, we show that it is possible to straightforwardly compute posterior marginals of the unknown true counts and that the modes of these marginals recover the count-min sketch estimator, inheriting the associated probabilistic guarantees. Using simulated data with known ground truth, we investigate the properties of these estimators. Lastly, we also study a modified problem in which the observation stream consists of collections of tokens (i.e., documents) arising from a random measure drawn from a stable beta process, which allows for power law scaling behavior in the number of unique tokens.

We present Sequential Attend, Infer, Repeat (SQAIR), an interpretable deep generative model for image sequences. It can reliably discover and track objects through the sequence; it can also conditionally generate future frames, thereby simulating expected motion of objects. This is achieved by explicitly encoding object numbers, locations and appearances in the latent variables of the model. SQAIR retains all strengths of its predecessor, Attend, Infer, Repeat (AIR, Eslami et. al. 2016), including unsupervised learning, made possible by inductive biases present in the model structure. We use a moving multi-\textsc{mnist} dataset to show limitations of AIR in detecting overlapping or partially occluded objects, and show how \textsc{sqair} overcomes them by leveraging temporal consistency of objects. Finally, we also apply SQAIR to real-world pedestrian CCTV data, where it learns to reliably detect, track and generate walking pedestrians with no supervision.

The Probably Approximately Correct (PAC) Bayes framework (McAllester, 1999) can incorporate knowledge about the learning algorithm and (data) distribution through the use of distribution-dependent priors, yielding tighter generalization bounds on data-dependent posteriors. Using this flexibility, however, is difficult, especially when the data distribution is presumed to be unknown. We show how an e-differentially private data-dependent prior yields a valid PAC-Bayes bound, and then show how non-private mechanisms for choosing priors can also yield generalization bounds. As an application of this result, we show that a Gaussian prior mean chosen via stochastic gradient Langevin dynamics (SGLD; Welling and Teh, 2011) leads to a valid PAC-Bayes bound given control of the 2-Wasserstein distance to an e-differentially private stationary distribution. We study our datadependent bounds empirically, and show that they can be nonvacuous even when other distribution-dependent bounds are vacuous.

Event datasets include events that occur irregularly over the timeline and are prevalent in numerous domains. We introduce proximal graphical event models (PGEM) as a representation of such datasets. PGEMs belong to a broader family of models that characterize relationships between various types of events, where the rate of occurrence of an event type depends only on whether or not its parents have occurred in the most recent history. The main advantage over the state of the art models is that they are entirely data driven and do not require additional inputs from the user, which can require knowledge of the domain such as choice of basis functions or hyperparameters in graphical event models. We theoretically justify our learning of optimal windows for parental history and the choices of parental sets, and the algorithm are sound and complete in terms of parent structure learning. We present additional efficient heuristics for learning PGEMs from data, demonstrating their effectiveness on synthetic and real datasets.