Minimax-Optimal Off-Policy Evaluation with Linear Function Approximation

This paper studies the statistical theory of batch data reinforcement learning with function approximation. Consider the off-policy evaluation problem, which is to estimate the cumulative value of a new target policy from logged history generated by unknown behavioral policies. We study a regression-based fitted Q iteration method, and show that it is equivalent to a model-based method that estimates a conditional mean embedding of the transition operator. We prove that this method is information-theoretically optimal and has nearly minimal estimation error. In particular, by leveraging contraction property of Markov processes and martingale concentration, we establish a finite-sample instance-dependent error upper bound and a nearly-matching minimax lower bound. The policy evaluation error depends sharply on a restricted $\chi^2$-divergence over the function class between the long-term distribution of the target policy and the distribution of past data. This restricted $\chi^2$-divergence is both instance-dependent and function-class-dependent. It characterizes the statistical limit of off-policy evaluation. Further, we provide an easily computable confidence bound for the policy evaluator, which may be useful for optimistic planning and safe policy improvement.

Learning low-dimensional state embeddings and metastable clusters from time series data

This paper studies how to find compact state embeddings from high-dimensional Markov state trajectories, where the transition kernel has a small intrinsic rank. In the spirit of diffusion map, we propose an efficient method for learning a low-dimensional state embedding and capturing the process's dynamics. This idea also leads to a kernel reshaping method for more accurate nonparametric estimation of the transition function. State embedding can be used to cluster states into metastable sets, thereby identifying the slow dynamics. Sharp statistical error bounds and misclassification rate are proved. Experiment on a simulated dynamical system shows that the state clustering method indeed reveals metastable structures. We also experiment with time series generated by layers of a Deep-Q-Network when playing an Atari game. The embedding method identifies game states to be similar if they share similar future events, even though their raw data are far different.

State Aggregation Learning from Markov Transition Data

State aggregation is a model reduction method rooted in control theory and reinforcement learning. It reduces the complexity of engineering systems by mapping the system's states into a small number of meta-states. In this paper, we study the unsupervised estimation of unknown state aggregation structures based on Markov trajectories. We formulate the state aggregation of Markov processes into a nonnegative factorization model, where left and right factor matrices correspond to aggregation and disaggregation distributions respectively. By leveraging techniques developed in the context of topic modeling, we propose an efficient polynomial-time algorithm for computing the estimated state aggregation model. Under some "anchor state" assumption, we show that one can reliably recover the state aggregation structure from sample transitions with high probability. Sharp divergence error bounds are proved for the estimated aggregation and disaggregation distributions, and experiments with Manhattan traffic data are provided.

Adaptive Low-Nonnegative-Rank Approximation for State Aggregation of Markov Chains

This paper develops a low-nonnegative-rank approximation method to identify the state aggregation structure of a finite-state Markov chain under an assumption that the state space can be mapped into a handful of meta-states. The number of meta-states is characterized by the nonnegative rank of the Markov transition matrix. Motivated by the success of the nuclear norm relaxation in low rank minimization problems, we propose an atomic regularizer as a convex surrogate for the nonnegative rank and formulate a convex optimization problem. Because the atomic regularizer itself is not computationally tractable, we instead solve a sequence of problems involving a nonnegative factorization of the Markov transition matrices by using the proximal alternating linearized minimization method. Two methods for adjusting the rank of factorization are developed so that local minima are escaped. One is to append an additional column to the factorized matrices, which can be interpreted as an approximation of a negative subgradient step. The other is to reduce redundant dimensions by means of linear combinations. Overall, the proposed algorithm very likely converges to the global solution. The efficiency and statistical properties of our approach are illustrated on synthetic data. We also apply our state aggregation algorithm on a Manhattan transportation data set and make extensive comparisons with an existing method.