In this paper we consider the problem of computing an $\epsilon$-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in $O(1)$ time. Given such a DMDP with states $\states$, actions $\actions$, discount factor $\gamma\in(0,1)$, and rewards in range $[0, 1]$ we provide an algorithm which computes an $\epsilon$-optimal policy with probability $1 - \delta$ where {\it both} the run time spent and number of sample taken is upper bounded by \[ O\left[\frac{|\cS||\cA|}{(1-\gamma)^3 \epsilon^2} \log \left(\frac{|\cS||\cA|}{(1-\gamma)\delta \epsilon} \right) \log\left(\frac{1}{(1-\gamma)\epsilon}\right)\right] ~. \] For fixed values of $\epsilon$, this improves upon the previous best known bounds by a factor of $(1 - \gamma)^{-1}$ and matches the sample complexity lower bounds proved in \cite{azar2013minimax} up to logarithmic factors. We also extend our method to computing $\epsilon$-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound.

Several applications of Reinforcement Learning suffer from instability due to high variance. This is especially prevalent in high dimensional domains. Regularization is a commonly used technique in machine learning to reduce variance, at the cost of introducing some bias. Most existing regularization techniques focus on spatial (perceptual) regularization. Yet in reinforcement learning, due to the nature of the Bellman equation, there is an opportunity to also exploit temporal regularization based on smoothness in value estimates over trajectories. This paper explores a class of methods for temporal regularization. We formally characterize the bias induced by this technique using Markov chain concepts. We illustrate the various characteristics of temporal regularization via a sequence of simple discrete and continuous MDPs, and show that the technique provides improvement even in high-dimensional Atari games.

Symmetric determinantal point processes (DPP) are a class of probabilistic models that encode the random selection of items that have a repulsive behavior. They have attracted a lot of attention in machine learning, where returning diverse sets of items is sought for. Sampling and learning these symmetric DPP's is pretty well understood. In this work, we consider a new class of DPP's, which we call signed DPP's, where we break the symmetry and allow attractive behaviors. We set the ground for learning signed DPP's through a method of moments, by solving the so called principal assignment problem for a class of matrices $K$ that satisfy $K_{i,j}=\pm K_{j,i}$, $i\neq j$, in polynomial time.

Understanding and interpreting how machine learning (ML) models make decisions have been a big challenge. While recent research has proposed various technical approaches to provide some clues as to how an ML model makes individual predictions, they cannot provide users with an ability to inspect a model as a complete entity. In this work, we propose a novel technical approach that augments a Bayesian non-parametric regression mixture model with multiple elastic nets. Using the enhanced mixture model, we can extract generalizable insights for a target model through a global approximation. To demonstrate the utility of our approach, we evaluate it on different ML models in the context of image recognition. The empirical results indicate that our proposed approach not only outperforms the state-of-the-art techniques in explaining individual decisions but also provides users with an ability to discover the vulnerabilities of the target ML models.

The problem of estimating an unknown discrete distribution from its samples is a fundamental tenet of statistical learning. Over the past decade, it attracted significant research effort and has been solved for a variety of divergence measures. Surprisingly, an equally important problem, estimating an unknown Markov chain from its samples, is still far from understood. We consider two problems related to the min-max risk (expected loss) of estimating an unknown k-state Markov chain from its n sequential samples: predicting the conditional distribution of the next sample with respect to the KL-divergence, and estimating the transition matrix with respect to a natural loss induced by KL or a more general f-divergence measure. For the first measure, we determine the min-max prediction risk to within a linear factor in the alphabet size, showing it is \Omega(k\log\log n/n) and O(k^2\log\log n/n). For the second, if the transition probabilities can be arbitrarily small, then only trivial uniform risk upper bounds can be derived. We therefore consider transition probabilities that are bounded away from zero, and resolve the problem for essentially all sufficiently smooth f-divergences, including KL-, L_2-, Chi-squared, Hellinger, and Alpha-divergences.

We present the Multi-value Rule Set (MRS) for interpretable classification with feature efficient presentations. Compared to rule sets built from single-value rules, MRS adopts a more generalized form of association rules that allows multiple values in a condition. Rules of this form are more concise than classical singlevalue rules in capturing and describing patterns in data. Our formulation also pursues a higher efficiency of feature utilization, which reduces possible cost in data collection and storage. We propose a Bayesian framework for formulating an MRS model and develop an efficient inference method for learning a maximum a posteriori, incorporating theoretically grounded bounds to iteratively reduce the search space and improve the search efficiency. Experiments on synthetic and realworld data demonstrate that MRS models have significantly smaller complexity and fewer features than baseline models while being competitive in predictive accuracy. Human evaluations show that MRS is easier to understand and use compared to other rule-based models.

Infants are experts at playing, with an amazing ability to generate novel structured behaviors in unstructured environments that lack clear extrinsic reward signals. We seek to mathematically formalize these abilities using a neural network that implements curiosity-driven intrinsic motivation. Using a simple but ecologically naturalistic simulated environment in which an agent can move and interact with objects it sees, we propose a "world-model" network that learns to predict the dynamic consequences of the agent's actions. Simultaneously, we train a separate explicit "self-model" that allows the agent to track the error map of its world-model. It then uses the self-model to adversarially challenge the developing world-model. We demonstrate that this policy causes the agent to explore novel and informative interactions with its environment, leading to the generation of a spectrum of complex behaviors, including ego-motion prediction, object attention, and object gathering. Moreover, the world-model that the agent learns supports improved performance on object dynamics prediction, detection, localization and recognition tasks. Taken together, our results are initial steps toward creating flexible autonomous agents that self-supervise in realistic physical environments.

In this paper, we study the problems of principal Generalized Eigenvector computation and Canonical Correlation Analysis in the stochastic setting. We propose a simple and efficient algorithm, Gen-Oja, for these problems. We prove the global convergence of our algorithm, borrowing ideas from the theory of fastmixing 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.

Deep latent variable models (DLVMs) combine the approximation abilities of deep neural networks and the statistical foundations of generative models. Variational methods are commonly used for inference; however, the exact likelihood of these models has been largely overlooked. The purpose of this work is to study the general properties of this quantity and to show how they can be leveraged in practice. We focus on important inferential problems that rely on the likelihood: estimation and missing data imputation. First, we investigate maximum likelihood estimation for DLVMs: in particular, we show that most unconstrained models used for continuous data have an unbounded likelihood function. This problematic behaviour is demonstrated to be a source of mode collapse. We also show how to ensure the existence of maximum likelihood estimates, and draw useful connections with nonparametric mixture models. Finally, we describe an algorithm for missing data imputation using the exact conditional likelihood of a DLVM. On several data sets, our algorithm consistently and significantly outperforms the usual imputation scheme used for DLVMs.

Gaussian Processes (GPs) are a generic modelling tool for supervised learning. While they have been successfully applied on large datasets, their use in safety-critical applications is hindered by the lack of good performance guarantees. To this end, we propose a method to learn GPs and their sparse approximations by directly optimizing a PAC-Bayesian bound on their generalization performance, instead of maximizing the marginal likelihood. Besides its theoretical appeal, we find in our evaluation that our learning method is robust and yields significantly better generalization guarantees than other common GP approaches on several regression benchmark datasets.