Scalable constrained optimization

Modern optimization is tasked with handling applications of increasingly large scale, chiefly due to the massive amounts of widely available data and the ever-growing reach of Machine Learning. Consequently, this area of research is under steady pressure to develop scalable and provably convergent methods capable of handling hefty, high-dimensional problems. The present dissertation contributes to recent efforts in this direction by proposing optimization algorithms with improved scalability.

Concretely, (1) We develop three novel Frank-Wolfe-type methods for minimizing convex stochastic objectives subject to stochastic linear inclusion constraints. The key feature of our algorithms is that they process only a subset of the constraints per iteration, thus gaining an edge over methods that require full passes through the data for large-scale problems. (2) We generalize Frank-Wolfe-type methods to a class of composite non-differentiable objectives --- a setting in which the classical Frank-Wolfe algorithm is known not to converge. We circumvent the difficulties related to non-differentiability by leveraging the problem structure and a modified linear minimization oracle of the constraint set, thus attaining convergence rates akin to the smooth case. (3) We propose an adaptive primal-dual algorithm for solving structured convex-concave saddle point problems, whose empirical convergence is improved as a result of tailoring the stepsizes to the local problem geometry. Importantly, our method achieves adaptivity "for-free" by using readily available quantities such as past gradients, and without relying on more expensive linesearch subroutines.

Our methods are theoretically sound and empirically grounded, as they are each accompanied by rigorous convergence guarantees and experiments showcasing their performance against relevant baselines. In a nutshell, this dissertation provides new algorithmic approaches and points of trade-off on the road toward scalably solving large optimization problems.