Nested Distributed Gradient Methods with Stochastic Computation Errors

Abstract

In this work, we consider the problem of a network of agents collectively minimizing a sum of convex functions. The agents in our setting can only access their local objective functions and exchange information with their immediate neighbors. Motivated by applications where computation is imperfect, including, but not limited to, empirical risk minimization (ERM) and online learning, we assume that only noisy estimates of the local gradients are available. To tackle this problem, we adapt a class of Nested Distributed Gradient methods (NEAR-DGD) to the stochastic gradient setting. These methods have minimal storage requirements, are communication aware and perform well in settings where gradient computation is costly, while communication is relatively inexpensive. We investigate the convergence properties of our method under standard assumptions for stochastic gradients, i.e. unbiasedness and bounded variance. Our analysis indicates that our method converges to a neighborhood of the optimal solution with a linear rate for local strongly convex functions and appropriate constant steplengths. We also show that distributed optimization with stochastic gradients achieves a noise reduction effect similar to mini-batching, which scales favorably with network size. Finally, we present numerical results to demonstrate the effectiveness of our method.

S-NEAR-DGD: A Flexible Distributed Stochastic Gradient Method for Inexact Communication

“…Our method is based on a class of flexible algorithms (NEAR-DGD) [57] that permit the trade-off of computation and communication to best accommodate the application setting. In this work, we generalize our previous results analyzing NEAR-DGD in the presence of either deterministically quantized communication [58] or stochastic gradient errors [59], and unify them under a common, fully stochastic framework. We provide theoretical results to demonstrate that S-NEAR-DGD converges to a neighborhood of the optimal solution with geometric rate, and that if an error-correction mechanism is incorporated to consensus, then the total communication error induced by inexact communication is independent of the number of consensus rounds peformed by our algorithm.…”

“…We considered quantized communication using deterministic (D) algorithms (e.g. rounding to the nearest integer with no uncertainty) in [58], while a variant of NEAR-DGD that utilizes stochastic gradient approximations only was presented in [59]. This work unifies and generalizes these methods.…”

On the Convergence of Nested Decentralized Gradient Methods With Multiple Consensus and Gradient Steps

“…1. distributed first-order primal algorithms [15][16][17][18][19][20][21][22][23][24]: arXiv:2006.01665v1 [math.OC] 31 May 2020 methods that use only gradient information and operate in primal space (i.e., directly on problem (1.3)); 2. nested [1,[25][26][27][28]: methods that decompose the communication and computation steps and perform them sequentially;…”