Optimal Quantization and Approximation in Source Coding and Stochastic Control
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This thesis deals with non-standard optimal quantization and approximation problems in source coding and stochastic control. The first part of the thesis considers randomized quantization. Adapted from stochastic control, a general representation of randomized quantizers that is probabilistically equivalent to common models in the literature is proposed via mixtures of joint probability measures induced by deterministic quantizers. Using this general model, we prove the existence of an optimal randomized quantizer for the generalized distribution preserving quantization problem. A Shannon theoretic version of this source coding problem is also considered, in which an optimal (minimum distortion) coding of stationary and memoryless source is studied under the requirement that the quantizer's output distribution also be stationary and memoryless possibly different than source distribution. We provide a characterization of the achievable rate region where the rate region includes both the coding rate and the rate of common randomness shared between the encoder and the decoder. In the second part of the thesis, we consider the quantization problems in stochastic control from viewpoints of information transmission and computation. The first problem studies the finite-action approximation (via quantization of the action space) of deterministic stationary policies of a discrete time Markov decision process (MDP), while the second problem considers finite-state approximations (via quantization of the state space) of discrete time Markov decision process. Under certain continuity conditions on the components of the MDP, we establish that optimal policies for the finite models can approximate with arbitrary precision optimal deterministic stationary policies for the original MDP. Combining these results leads to a constructive scheme for obtaining near optimal solutions via well known algorithms developed for finite state/action MDPs. For both problems, we also obtain explicit bounds on the approximation error in terms of the number of representation points in the quantizer, under further conditions.