Expressive Power of Conditional Restricted Boltzmann Machines for Sensorimotor Control

We consider the causal structure of the sensorimotor loop (SML) and represent the agent's policies in terms of conditional restricted Boltzmann machines (CRBMs). CRBMs can model non-trivial conditional distributions on high dimensional input-output spaces with relatively few parameters. In addition, their Glauber dynamics can be computed efficiently to produce approximate samples. We discuss various problems related to the expressive power of these models in the context of the SML. In particular, we address the problems of universal approximation and approximation errors of conditional distributions. As in the probabilistic case, universal approximation of conditional distributions requires an exponential number of hidden units. Given a set of desirable policies, however, one can address the problem of approximating the elements of this set with as few hidden units as possible. We outline this method by considering deterministic policies as the desirable ones. Finally, we study the dimension of CRBM models, showing that most relevant cases each parameter contributes to the dimension of the set of representable policies.