Regularization in reinforcement learning

This thesis studies the reinforcement learning and planning problems that are modeled by a discounted Markov Decision Process (MDP) with a large state space and finite action space. We follow the value-based approach in which a function approximator is used to estimate the optimal value function. The choice of function approximator, however, is nontrivial, as it depends on both the number of data samples and the MDP itself. The goal of this work is to introduce flexible and statistically-efficient algorithms that find close to optimal policies for these problems without much prior information about them. The recurring theme of this thesis is the application of the regularization technique to design value function estimators that choose their estimates from rich function spaces. We introduce regularization-based Approximate Value/Policy Iteration algorithms, analyze their statistical properties, and provide upper bounds on the performance loss of the resulted policy compared to the optimal one. The error bounds show the dependence of the performance loss on the number of samples, the capacity of the function space to which the estimated value function belongs, and some intrinsic properties of the MDP itself. Remarkably, the dependence on the number of samples in the task of policy evaluation is minimax optimal. We also address the problem of automatic parameter-tuning of reinforcement learning/planning algorithms and introduce a complexity regularization-based model selection algorithm. We prove that the algorithm enjoys an oracle-like property and it may be used to achieve adaptivity: the performance is almost as good as the performance of the unknown best parameters. Our two other contributions are used to analyze the aforementioned algorithms. First, we analyze the rate of convergence of the estimation error in regularized least-squares regression when the data is exponentially beta-mixing. We prove that up to a logarithmic factor, the convergence rate is the same as the optimal minimax rate available for the i.i.d. case. Second, we attend to the question of how the errors at each iteration of the approximate policy/value iteration influence the quality of the resulting policy. We provide results that highlight some new aspects of these algorithms.

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