Block stochastic gradient iteration for convex and nonconvex optimization

The stochastic gradient (SG) method can minimize an objective function composed of a large number of differentiable functions, or solve a stochastic optimization problem, to a moderate accuracy. The block coordinate descent/update (BCD) method, on the other hand, handles problems with multiple blocks of variables by updating them one at a time; when the blocks of variables are easier to update individually than together, BCD has a lower per-iteration cost. This paper introduces a method that combines the features of SG and BCD for problems with many components in the objective and with multiple (blocks of) variables. Specifically, a block stochastic gradient (BSG) method is proposed for solving both convex and nonconvex programs. At each iteration, BSG approximates the gradient of the differentiable part of the objective by randomly sampling a small set of data or sampling a few functions from the sum term in the objective, and then, using those samples, it updates all the blocks of variables in either a deterministic or a randomly shuffled order. Its convergence for both convex and nonconvex cases are established in different senses. In the convex case, the proposed method has the same order of convergence rate as the SG method. In the nonconvex case, its convergence is established in terms of the expected violation of a first-order optimality condition. The proposed method was numerically tested on problems including stochastic least squares and logistic regression, which are convex, as well as low-rank tensor recovery and bilinear logistic regression, which are nonconvex

On the Convergence of Alternating Direction Lagrangian Methods for Nonconvex Structured Optimization Problems

Nonconvex and structured optimization problems arise in many engineering applications that demand scalable and distributed solution methods. The study of the convergence properties of these methods is in general difficult due to the nonconvexity of the problem. In this paper, two distributed solution methods that combine the fast convergence properties of augmented Lagrangian-based methods with the separability properties of alternating optimization are investigated. The first method is adapted from the classic quadratic penalty function method and is called the Alternating Direction Penalty Method (ADPM). Unlike the original quadratic penalty function method, in which single-step optimizations are adopted, ADPM uses an alternating optimization, which in turn makes it scalable. The second method is the well-known Alternating Direction Method of Multipliers (ADMM). It is shown that ADPM for nonconvex problems asymptotically converges to a primal feasible point under mild conditions and an additional condition ensuring that it asymptotically reaches the standard first order necessary conditions for local optimality are introduced. In the case of the ADMM, novel sufficient conditions under which the algorithm asymptotically reaches the standard first order necessary conditions are established. Based on this, complete convergence of ADMM for a class of low dimensional problems are characterized. Finally, the results are illustrated by applying ADPM and ADMM to a nonconvex localization problem in wireless sensor networks.Comment: 13 pages, 6 figure

Asynchronous and Parallel Distributed Pose Graph Optimization

We present Asynchronous Stochastic Parallel Pose Graph Optimization (ASAPP), the first asynchronous algorithm for distributed pose graph optimization (PGO) in multi-robot simultaneous localization and mapping. By enabling robots to optimize their local trajectory estimates without synchronization, ASAPP offers resiliency against communication delays and alleviates the need to wait for stragglers in the network. Furthermore, ASAPP can be applied on the rank-restricted relaxations of PGO, a crucial class of non-convex Riemannian optimization problems that underlies recent breakthroughs on globally optimal PGO. Under bounded delay, we establish the global first-order convergence of ASAPP using a sufficiently small stepsize. The derived stepsize depends on the worst-case delay and inherent problem sparsity, and furthermore matches known result for synchronous algorithms when there is no delay. Numerical evaluations on simulated and real-world datasets demonstrate favorable performance compared to state-of-the-art synchronous approach, and show ASAPP's resilience against a wide range of delays in practice.Comment: full paper with appendice