A Separation Principle for Control in the Age of Deep Learning

We review the problem of defining and inferring a state for a control system based on complex, high-dimensional, highly uncertain measurement streams, such as videos. Such a state, or representation, should contain all and only the information needed for control and discount nuisance variability in the data. It should also have finite complexity, ideally modulated depending on available resources. This representation is what we want to store in memory in lieu of the data, as it separates the control task from the measurement process. For the trivial case with no dynamics, a representation can be inferred by minimizing the information bottleneck Lagrangian in a function class realized by deep neural networks. The resulting representation has much higher dimension than the data (already in the millions) but is smaller in the sense of information content, retaining only what is needed for the task. This process also yields representations that are invariant to nuisance factors and have maximally independent components. We extend these ideas to the dynamic case, where the representation is the posterior density of the task variable given the measurements up to the current time, which is in general much simpler than the prediction density maintained by the classical Bayesian filter. Again, this can be finitely parameterized using a deep neural network, and some applications are already beginning to emerge. No explicit assumption of Markovianity is needed; instead, complexity trades off approximation of an optimal representation, including the degree of Markovianity.