  This paper aims at theoretically and empirically comparing two standard
optimization criteria for Reinforcement Learning: i) maximization of the mean
value and ii) minimization of the Bellman residual. For that purpose, we place
ourselves in the framework of policy search algorithms, that are usually
designed to maximize the mean value, and derive a method that minimizes the
residual $\|T_* v_\pi - v_\pi\|_{1,\nu}$ over policies. A theoretical analysis
shows how good this proxy is to policy optimization, and notably that it is
better than its value-based counterpart. We also propose experiments on
randomly generated generic Markov decision processes, specifically designed for
studying the influence of the involved concentrability coefficient. They show
that the Bellman residual is generally a bad proxy to policy optimization and
that directly maximizing the mean value is much better, despite the current
lack of deep theoretical analysis. This might seem obvious, as directly
addressing the problem of interest is usually better, but given the prevalence
of (projected) Bellman residual minimization in value-based reinforcement
learning, we believe that this question is worth to be considered.
