Title: A Framework for Analyzing Stochastic Optimization Algorithms Under Dependence
Description: In this dissertation, a theoretical framework based on concentration inequalities for empirical processes is developed to better design iterative optimization algorithms and analyze their convergence properties in the presence of complex dependence between directions and step-sizes. Based on this framework, we proposed a stochastic away-step Frank-Wolfe algorithm and a stochastic pairwise-step Frank-Wolfe algorithm for solving strongly convex problems with polytope constraints and proved that both of those algorithms converge linearly to the optimal solution in expectation and almost surely. Numerical results showed that the proposed algorithms are faster and more stable than most of their competitors. This framework can be applied for designing and analyzing stochastic algorithms with adaptive step-sizes that are based on local curvature for self-concordant optimization problems. Notably, we proposed and analyzed a stochastic BFGS algorithm without line-search, and proved that it converges linearly globally and super-linearly locally using the framework mentioned above.

This is the first work that analyzes a fully stochastic BFGS algorithm, which also avoids time consuming or even impossible line-search steps. A third class of problems that the empirical processes framework can be applied to is to study the optimization of compositions of stochastic functions. A multi-level Monte Carlo based unbiased gradient generation method is introduced into stochastic optimization algorithms for minimizing function compositions. Based on this, standard stochastic optimization algorithms can be applied to these problems directly.