Superlinearly convergent asynchronous distributed network newton method

Abstract

The problem of minimizing a sum of local convex objective functions over a networked system captures many important applications and has received much attention in the distributed optimization field. Most of existing work focuses on development of fast distributed algorithms under the presence of a central clock. The only known algorithms with convergence guarantees for this problem in asynchronous setup could achieve either sublinear rate under totally asynchronous setting or linear rate under partially asynchronous setting (with bounded delay). In this work, we built upon existing literature to develop and analyze an asynchronous Newton based approach for solving a penalized version of the problem. We show that this algorithm converges almost surely with global linear rate and local superlinear rate in expectation. Numerical studies confirm superior performance against other existing asynchronous methods.

Nested Distributed Gradient Methods with Stochastic Computation Errors

“…Common approaches include first order methods [6]- [10], primal-dual algorithms [11], [12], gradient tracking methods *This work was supported by the DARPA award HR-001117S0039. C. Iakovidou and E. Wei are with the Department of Electrical and Computer Engineering, Northwestern University, Evanston, IL USA, chariako@u.northwestern.edu, ermin.wei@northwestern.edu [13]- [16], the alternating direction method of multipliers (ADMM) [17] and Newton methods [18], [19].…”

A Fast Distributed Asynchronous Newton-Based Optimization Algorithm

“…Rather, we assume the agents to be active based on a time invariant and not necessarily uniform probability distribution. We have studied the setting with equal activation probabilities in our previous work in [27], which is a special case of the setting in this paper.…”

“…In this case, there is no need to scale the agents' stepsizes with the inverse of their activation probabilities, i.e., in step 5 of Algorithm 1, the active agent use ε instead of ε p i . This case is studied in [27] and all the results there, are special cases of our convergence analysis in this paper.…”

A Primal-Dual Quasi-Newton Method for Exact Consensus Optimization

“…While this technique cannot be used directly in distributed optimization due to the non-sparsity of the Hessian inverse, there exist ways of using second order information to approximate the Newton step for distributed settings. This has been done for consensus optimization problems reformulated as both the penalty-based methods [17], [24] and dual-based methods [20], as well as the more recent primal-dual methods [22], [25]. These approximate Newton methods exhibit faster convergence relative to their corresponding first order methods.…”

“…Theorem 1 Consider PD-QN as introduced in (23)- (24). Consider arbitrary constants β > 1 and φ > 1 and ζ as a positive constant.…”