Apprenticeship Learning via Frank-Wolfe

T om Zahavy, Alon Cohen, Haim Kaplan and Yishay Mansour Google Research, Tel Aviv Abstract We consider the applications of the Frank-Wolfe (FW) algorithm for Apprenticeship Learning (AL). In this setting, we are given a Markov Decision Process (MDP) without an explicit reward function. Instead, we observe an expert that acts according to some policy, and the goal is to find a policy whose feature expectations are closest to those of the expert policy. We formulate this problem as finding the projection of the feature expectations of the expert on the feature expectations polytope - the convex hull of the feature expectations of all the deterministic policies in the MDP . We show that this formulation is equivalent to the AL objective and that solving this problem using the FW algorithm is equivalent well-known Projection method of Abbeel and Ng (2004). This insight allows us to analyze AL with tools from convex optimization literature and derive tighter convergence bounds on AL. Specifically, we show that a variation of the FW method that is based on taking "away steps" achieves a linear rate of convergence when applied to AL and that a stochastic version of the FW algorithm can be used to avoid precise estimation of feature expectations. We also experimentally show that this version outperforms the FW baseline. To the best of our knowledge, this is the first work that shows linear convergence rates for AL. 1 Introduction We consider sequential decision making in the Markov decision process (MDP) formalism. Given an MDP, the optimal policy and its value function are characterized by the Bellman equations and can be computed via value or policy iteration. This makes the MDP model useful in problems where we can specify the MDP model (states, actions, reward, transitions) appropriately. However, in many real-world problems, it is often hard to define a reward function, such that the optimal policy with respect to this reward produces the desired behavior.