  We present a framework for designing cheap control architectures for embodied
agents. Our derivation is guided by the classical problem of universal
approximation, whereby we explore the possibility of exploiting the agent's
embodiment for a new and more efficient universal approximation of behaviors
generated by sensorimotor control. This embodied universal approximation is
compared with the classical non-embodied universal approximation. To exemplify
our approach, we present a detailed quantitative case study for policy models
defined in terms of conditional restricted Boltzmann machines. In contrast to
non-embodied universal approximation, which requires an exponential number of
parameters, in the embodied setting we are able to generate all possible
behaviors with a drastically smaller model, thus obtaining cheap universal
approximation. We test and corroborate the theory experimentally with a
six-legged walking machine. The experiments show that the sufficient controller
complexity predicted by our theory is tight, which means that the theory has
direct practical implications. Keywords: cheap design, embodiment, sensorimotor
loop, universal approximation, conditional restricted Boltzmann machine
