We uncover relations between robust MDPs and risk-sensitive MDPs. The objective of a robust MDP is to minimize a function, such as the expectation of cumulative cost, for the worst case when the parameters have uncertainties. The objective of a risk-sensitive MDP is to minimize a risk measure of the cumulative cost when the parameters are known. We show that a risk-sensitive MDP of minimizing the expected exponential utility is equivalent to a robust MDP of minimizing the worst-case expectation with a penalty for the deviation of the uncertain parameters from their nominal values, which is measured with the Kullback-Leibler divergence. We also show that a risk-sensitive MDP of minimizing an iterated risk measure that is composed of certain coherent risk measures is equivalent to a robust MDP of minimizing the worst-case expectation when the possible deviations of uncertain parameters from their nominal values are characterized with a concave function.

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.

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. 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.

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 propose a Bayesian decision making framework for control of Markov Decision Processes (MDPs) with unknown dynamics and large, possibly continuous, state, action, and parameter spaces in data-poor environments. Most of the existing adaptive controllers for MDPs with unknown dynamics are based on the reinforcement learning framework and rely on large data sets acquired by sustained direct interaction with the system or via a simulator. This is not feasible in many applications, due to ethical, economic, and physical constraints. The proposed framework addresses the data poverty issue by decomposing the problem into an offline planning stage that does not rely on sustained direct interaction with the system or simulator and an online execution stage. In the offline process, parallel Gaussian process temporal difference (GPTD) learning techniques are employed for near-optimal Bayesian approximation of the expected discounted reward over a sample drawn from the prior distribution of unknown parameters.

Monte-Carlo Tree Search (MCTS) has been successfully applied to very large POMDPs, a standard model for stochastic sequential decision-making problems. However, many real-world problems inherently have multiple goals, where multi-objective formulations are more natural. The constrained POMDP (CPOMDP) is such a model that maximizes the reward while constraining the cost, extending the standard POMDP model. To date, solution methods for CPOMDPs assume an explicit model of the environment, and thus are hardly applicable to large-scale real-world problems. In this paper, we present CC-POMCP (Cost-Constrained POMCP), an online MCTS algorithm for large CPOMDPs that leverages the optimization of LP-induced parameters and only requires a black-box simulator of the environment.

At their core, many unsupervised learning models provide a compact representation of homogeneous density mixtures, but their similarities and differences are not always clearly understood. In this work, we formally establish the relationships among latent tree graphical models (including special cases such as hidden Markov models and tensorial mixture models), hierarchical tensor formats and sum-product networks. Based on this connection, we then give a unified treatment of exponential separation in \emph{exact} representation size between deep mixture architectures and shallow ones. In contrast, for \emph{approximate} representation, we show that the conditional gradient algorithm can approximate any homogeneous mixture within $\epsilon$ accuracy by combining $O(1/\epsilon 2)$ shallow'' architectures, where the hidden constant may decrease (exponentially) with respect to the depth. Our experiments on both synthetic and real datasets confirm the benefits of depth in density estimation.

Many state-of-the-art algorithms for solving Partially Observable Markov Decision Processes (POMDPs) rely on turning the problem into a "fully observable" problem--a belief MDP--and exploiting the piece-wise linearity and convexity (PWLC) of the optimal value function in this new state space (the belief simplex). This approach has been extended to solving ρ-POMDPs--i.e., for information-oriented criteria--when the reward ρ is convex in . General ρ-POMDPs can also be turned into "fully observable" problems, but with no means to exploit the PWLC property. In this paper, we focus on POMDPs and ρ-POMDPs with λ ρ -Lipschitz reward function, and demonstrate that, for finite horizons, the optimal value function is Lipschitz-continuous. Then, value function approximators are proposed for both upper- and lower-bounding the optimal value function, which are shown to provide uniformly improvable bounds.

This paper examines the question: What kinds of distributions can be efficiently represented by Restricted Boltzmann Machines (RBMs)? We characterize the RBM's unnormalized log-likelihood function as a type of neural network (called an RBM network), and through a series of simulation results relate these networks to types that are better understood. We show the surprising result that RBM networks can efficiently compute any function that depends on the number of 1's in the input, such as parity. We also provide the first known example of a particular type of distribution which provably cannot be efficiently represented by an RBM (or equivalently, cannot be efficiently computed by an RBM network), assuming a realistic exponential upper bound on the size of the weights. By formally demonstrating that a relatively simple distribution cannot be represented efficiently by an RBM our results provide a new rigorous justification for the use of potentially more expressive generative models, such as deeper ones.

Implicit feedback is widely used in collaborative filtering methods for recommendation. It is well known that implicit feedback contains a large number of values that are \emph{missing not at random} (MNAR); and the missing data is a mixture of negative and unknown feedback, making it difficult to learn user's negative preferences. Recent studies modeled \emph{exposure}, a latent missingness variable which indicates whether an item is missing to a user, to give each missing entry a confidence of being negative feedback. However, these studies use static models and ignore the information in temporal dependencies among items, which seems to be a essential underlying factor to subsequent missingness. To model and exploit the dynamics of missingness, we propose a latent variable named \emph{user intent}'' to govern the temporal changes of item missingness, and a hidden Markov model to represent such a process.