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.

本研究讨论了有关有向随机神经网络中条件概率分布的表示能力，证明了其可以用于表示条件马尔可夫随机场和带有限制支持的条件分布，研究了通用逼近器的最小尺寸、最大模型逼近误差以及可表示的条件分布集合的维数等等。我们贡献了新的工具，改进了现有有关有限支持限制玻尔兹曼机概率模型的研究结果。

条件受限玻尔兹曼机的几何与表现力