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. In the online stage, the action with the maximum expected return with respect to the posterior distribution of the parameters is selected. This is achieved by an approximation of the posterior distribution using a Markov Chain Monte Carlo (MCMC) algorithm, followed by constructing multiple Gaussian processes over the parameter space for efficient prediction of the means of the expected return at the MCMC sample. The effectiveness of the proposed framework is demonstrated using a simple dynamical system model with continuous state and action spaces, as well as a more complex model for a metastatic melanoma gene regulatory network observed through noisy synthetic gene expression data.

In many real-world reinforcement learning (RL) problems, besides optimizing the main objective function, an agent must concurrently avoid violating a number of constraints. In particular, besides optimizing performance, it is crucial to guarantee thesafety of an agent during training as well as deployment (e.g., a robot should avoid taking actions - exploratory or not - which irrevocably harm its hardware). Toincorporate safety in RL, we derive algorithms under the framework of constrained Markov decision processes (CMDPs), an extension of the standard Markov decision processes (MDPs) augmented with constraints on expected cumulative costs.Our approach hinges on a novel Lyapunov method. We define and present a method for constructing Lyapunov functions, which provide an effective wayto guarantee the global safety of a behavior policy during training via a set of local linear constraints. Leveraging these theoretical underpinnings, we show how to use the Lyapunov approach to systematically transform dynamic programming (DP) and RL algorithms into their safe counterparts. To illustrate their effectiveness, we evaluate these algorithms in several CMDP planning and decision-making tasks on a safety benchmark domain. Our results show that our proposed method significantly outperforms existing baselines in balancing constraint satisfactionand performance.

Expectation-Maximization (EM) is a popular tool for learning latent variable models, but the vanilla batch EM does not scale to large data sets because the whole data set is needed at every E-step. Stochastic Expectation Maximization (sEM) reduces the cost of E-step by stochastic approximation. However, sEM has a slower asymptotic convergence rate than batch EM, and requires a decreasing sequence of step sizes, which is difficult to tune. In this paper, we propose a variance reduced stochastic EM (sEM-vr) algorithm inspired by variance reduced stochastic gradient descent algorithms. We show that sEM-vr has the same exponential asymptotic convergence rate as batch EM. Moreover, sEM-vr only requires a constant step size to achieve this rate, which alleviates the burden of parameter tuning. We compare sEM-vr with batch EM, sEM and other algorithms on Gaussian mixture models and probabilistic latent semantic analysis, and sEM-vr converges significantly faster than these baselines.

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. In the experiments, we demonstrate that CC-POMCP converges to the optimal stochastic action selection in CPOMDP and pushes the state-of-the-art by being able to scale to very large problems.

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.

In this paper, we study the problems of principle Generalized Eigenvector computation and Canonical Correlation Analysis in the stochastic setting. We propose a simple and efficient algorithm for these problems. We prove the global convergence of our algorithm, borrowing ideas from the theory of fast-mixing Markov chains and two-Time-Scale Stochastic Approximation, showing that it achieves the optimal rate of convergence. In the process, we develop tools for understanding stochastic processes with Markovian noise which might be of independent interest.

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 allows proposing two algorithms derived from HSVI which are empirically evaluated on various benchmark problems.

The notion of ``policy regret'' in online learning is supposed to capture the reactions of the adversary to the actions taken by the learner, which more traditional notions such as external regret do not take into account. We revisit this notion of policy regret, and first show that there are online learning settings in which policy regret and external regret are incompatible: any sequence of play which does well with respect to one must do poorly with respect to the other. We then focus on the game theoretic setting, when the adversary is a self-interested agent. In this setting we show that the external regret and policy regret are not in conflict, and in fact that a wide class of algorithms can ensure both as long as the adversary is also using such an algorithm. We also define a new notion of equilibrium which we call a ``policy equilibrium'', and show that no-policy regret algorithms will have play which converges to such an equilibrium. Relating this back to external regret, we show that coarse correlated equilibria (which no-external regret players will converge to) are a strict subset of policy equilibria. So in game-theoretic settings every sequence of play with no external regret also has no policy regret, but the converse is not true.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space the time-discretization error can dominate when we are near the boundary of the space. We demonstrate that because of this, current SGMCMC methods for the simplex struggle with sparse simplex spaces; when many of the components are close to zero. Unfortunately, many popular large-scale Bayesian models, such as network or topic models, require inference on sparse simplex spaces. To avoid the biases caused by this discretization error, we propose the stochastic Cox-Ingersoll-Ross process (SCIR), which removes all discretization error and we prove that samples from the SCIR process are asymptotically unbiased. We discuss how this idea can be extended to target other constrained spaces. Use of the SCIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.

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. The resulting framework captures the dynamic item missingness and incorporate it into matrix factorization (MF) for recommendation. We also explore two types of constraints to achieve a more compact and interpretable representation of \emph{user intents}. Experiments on real-world datasets demonstrate the superiority of our method against state-of-the-art recommender systems.