Convex Optimization and Extensions, with a View Toward Large-Scale Problems

Convex Optimization and Extensions, with a View Toward Large-Scale Problems Wenbo Gao Machine learning is a major source of interesting optimization problems of current interest. These problems tend to be challenging because of their enormous scale, which makes it difficult to apply traditional optimization algorithms. We explore three avenues to designing algorithms suited to handling these challenges, with a view toward large-scale ML tasks. The first is to develop better general methods for unconstrained minimization. The second is to tailor methods to the features of modern systems, namely the availability of distributed computing. The third is to use specialized algorithms to exploit specific problem structure. Chapters 2 and 3 focus on improving quasi-Newton methods, a mainstay of unconstrained optimization. In Chapter 2, we analyze an extension of quasi-Newton methods wherein we use block updates, which add curvature information to the Hessian approximation on a higher-dimensional subspace. This defines a family of methods, Block BFGS, that form a spectrum between the classical BFGS method and Newton’s method, in terms of the amount of curvature information used. We show that by adding a correction step, the Block BFGS method inherits the convergence guarantees of BFGS for deterministic problems, most notably a Q-superlinear convergence rate for strongly convex problems. To explore the tradeoff between reduced iterations and greater work per iteration of block methods, we present a set of numerical experiments. In Chapter 3, we focus on the problem of step size determination. To obviate the need for line searches, and for pre-computing fixed step sizes, we derive an analytic step size, which we call curvature-adaptive, for self-concordant functions. This adaptive step size allows us to generalize the damped Newton method of Nesterov to other iterative methods, including gradient descent and quasi-Newton methods. We provide simple proofs of convergence, including superlinear convergence for adaptive BFGS, allowing us to obtain superlinear convergence without line searches. In Chapter 4, we move from general algorithms to hardware-influenced algorithms. We consider a form of distributed stochastic gradient descent that we call Leader SGD, which is inspired by the Elastic Averaging SGD method. These methods are intended for distributed settings where communication between machines may be expensive, making it important to set their consensus mechanism. We show that LSGD avoids an issue with spurious stationary points that affects EASGD, and provide a convergence analysis of LSGD. In the stochastic strongly convex setting, LSGD converges at the rate O( 1 k ) with diminishing step sizes, matching other distributed methods. We also analyze the impact of varying communication delays, stochasticity in the selection of the leader points, and under what conditions LSGD may produce better search directions than the gradient alone. In Chapter 5, we switch again to focus on algorithms to exploit problem structure. Specifically, we consider problems where variables satisfy multiaffine constraints, which motivates us to apply the Alternating Direction Method of Multipliers (ADMM). Problems that can be formulated with such a structure include representation learning (e.g with dictionaries) and deep learning. We show that ADMM can be applied directly to multiaffine problems. By extending the theory of nonconvex ADMM, we prove that ADMM is convergent on multiaffine problems satisfying certain assumptions, and more broadly, analyze the theoretical properties of ADMM for general problems, investigating the effect of different types of structure.