This work considers the sample and computational complexity of obtaining an ε-optimal policy in a discounted Markov Decision Process (MDP), given only access to a generative model. In this model, the learner accesses the underlying transition model via a sampling oracle that provides a sample of the next state, when given any state-action pair as input. This widely studied setting provides a natural abstraction which permits the study of sample-based planning over a long horizon, decoupled from the complexity of exploration. In this work, we study the effectiveness of the most natural plug-in approach to model-based planning: we build the maximum likelihood estimate of the transition model in the MDP from observations and then find an optimal policy in this empirical MDP. We ask arguably the most basic and unresolved question in model based planning: is the naïve "plug-in" approach, non-asymptotically, minimax optimal in the quality of the policy it finds, given a fixed sample size? Here, the non-asymptotic regime refers to when the sample size is sublinear in the model size. With access to a generative model, we resolve this question in the strongest possible sense: our main result shows that any high accuracy solution in the plug-in model constructed with N samples, provides an ε-optimal policy in the true underlying MDP (where ε is the minimax accuracy with N samples at every state, action pair). In comparison, all prior (non-asymptotically) minimax optimal results use model free approaches, such as the Variance Reduced Q-value iteration algorithm (Sidford et al., 2018a) , while the best known model-based results (e.g. Azar et al. (2013) ) require larger sample sizes in their dependence on the planning horizon or the state space. Notably, we show that the model-based approach allows the use of any efficient planning algorithm in the empirical MDP, which simplifies algorithm design as this approach does not tie the algorithm to the sampling procedure. The core of our analysis is a novel "absorbing MDP" construction to address the statistical dependency issues that arise in the analysis of model-based planning approaches, a construction which may be helpful more generally.
REF establishes similar results by estimating the parameters of the MDP model using maximum-likelihood estimation.
On the Optimality of Sparse Model-Based Planning for Markov Decision Processes.
ArXiv
ArXiv
Mathematics Computer Science