We consider the problem of controlling a linear system with Gaussian noise and quadratic cost (LQG), using a memoryless controller that has limited capacity of the channel connecting its sensor to its actuator. We formulate this setting as a sequential rate-distortion (SRD) problem, where we minimize the rate of information required for the controller’s operation, under a constraint on its external cost. We present the optimality principle, and study the interesting and useful phenomenology of the optimal controller, such as the principled reduction of its order.
We consider the case where the controller is retentive (memory-utilizing). We can view the memory reader as one more sensor, and the memory writer as one more actuator. We can then formulate the problem of control under communication limitations, again as a sequential rate-distortion (SRD) problem of minimizing the rate of information required for the controller’s operation, under a constraint on its external cost. We show that this problem can be reduced to the memoryless case, studied in Part I. We then further investigate the form of the resulting optimal solution, and demonstrate its interesting phenomenology.
We define Passive POMDPs, where actions do not affect the world state, but still incur costs. We present a variational principle for the problem of maintaining in memory the information state that is most useful for minimizing the cost, leading to a trade-off between memory and sensing, similar to multi-terminal source coding. We analyze the problem as an equivalent joint-state MDP, and introduce an efficient and simple algorithm for finding an optimum.
We formulate the problem of optimizing an agent under both extrinsic and intrinsic constraints on its operation in a dynamical system and develop the main tools for solving it. We identify the challenging convergence properties of the optimization algorithm, such as the bifurcation structure of the update operator near phase transitions. We study the special case of linear-Gaussian dynamics and quadratic cost (LQG), where the optimal solution has a particularly simple and solvable form. We also explore the learning task, where the model of the world dynamics is unknown and sample-based updates are used instead.
We present the minimum-information principle for selective attention in reactive agents. We motivate this approach by reducing the general problem of optimal control in POMDPs, to reactive control with complex observations. We introduce a forward-backward algorithm for finding optimal selective-attention policies, and illustrate it with several examples. Finally, we analyze and explore the newly discovered phenomenon of period doubling bifurcations in this optimization process.