  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.
