Mamajiwala, Mariya; (2023) Stochastic schemes motivated by energetics and Riemannian geometry: optimization, Markov chain Monte Carlo, optimal control and data assimilation. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Statistical algorithms not only involve drawing realizations from a given distribution or estimating the parameters of the related density, but a wider class of problems such as optimal control, data assimilation and non-convex optimization. Unlike a deterministic search algorithm, e.g. one based on quasi-Newton updates, stochastic search schemes can make use of concepts from both deterministic dynamics and stochastic theory of noise to strike a balance between exploitation and exploration in the design space. In the quest for more efficacious schemes, researchers have drawn on ideas from contemporary physics and differential geometry in arriving at suitably constrained dynamical systems that guide the search, and the work in this dissertation is similarly inspired. To start with, a survey of the state-of-the-art is presented in Chapter 1 to motivate and put in perspective the work in the chapters to follow. Chapter 2 dwells on a Riemannian geometric approach to non-convex optimization, wherein the flow that minimizes a given objective function with progressing iterations is constrained to live on a manifold defined using a metric derivable by treating the objective function as energy. Specifically, the underlying dynamical system is designed as a geometrically adapted Langevin stochastic differential equation (SDE). The same adaptation, albeit with a Riemannian metric given by the Fisher information matrix obtainable from the available likelihood, is used in Chapter 3 to arrive at an MCMC method. In Chapter 4, a time-recursive scheme for stochastic optimal control is proposed using SDEs integrated strictly forward in time, thus bypassing the computationally inexpedient forward-backward route to solve the Hamilton-Jacobi-Bellman (HJB) equation. We address the combined state-parameter estimation problem via stochastic filtering in Chapter 5, with a new proposal for the parameter dynamics for higher accuracy and faster convergence. The thesis is concluded in Chapter 6 with a summary and scope for future research.

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