On the Convergence of Inexact Gradient Descent with Controlled Synchronization Steps

TL;DR

This paper introduces a flexible gradient-like algorithm for distributed optimization that tolerates bounded inexactness in gradient computations, uses a dynamic synchronization criterion, and guarantees convergence without requiring gradient bounds.

Contribution

It proposes a novel inexact gradient descent method with a dynamic synchronization mechanism, reducing communication overhead while ensuring convergence under mild conditions.

Findings

Algorithm converges under mild conditions.

Dynamic synchronization reduces communication costs.

Handles wide range of inexactness in gradients.

Abstract

We develop a gradient-like algorithm to minimize a sum of peer objective functions based on coordination through a peer interconnection network. The coordination admits two stages: the first is to constitute a gradient, possibly with errors, for updating locally replicated decision variables at each peer and the second is used for error-free averaging for synchronizing local replicas. Unlike many related algorithms, the errors permitted in our algorithm can cover a wide range of inexactnesses, as long as they are bounded. Moreover, we do not impose any gradient boundedness conditions for the objective functions. Furthermore, the second stage is not conducted in a periodic manner, like many related algorithms. Instead, a locally verifiable criterion is devised to dynamically trigger the peer-to-peer coordination at the second stage, so that expensive communication overhead for error-free averaging can significantly be reduced. Finally, the convergence of the algorithm is established under mild conditions.