Momentum-based Accelerated Algorithm for Distributed Optimization under Sector-Bound Nonlinearity

TL;DR

This paper introduces an accelerated distributed optimization algorithm that incorporates momentum and handles sector-bound nonlinearities, improving convergence in dynamic, directed networks with non-convex functions.

Contribution

It presents a novel momentum-based consensus algorithm that manages sector-bound nonlinearities and dynamic directed networks, with proven convergence guarantees.

Findings

Improved convergence rate with momentum using the heavy-ball method.

Effective handling of sector-bound nonlinearities like quantization and clipping.

Applicability to dynamic directed networks with time-varying links.

Abstract

Distributed optimization advances centralized machine learning methods by enabling parallel and decentralized learning processes over a network of computing nodes. This work provides an accelerated consensus-based distributed algorithm for locally non-convex optimization using the gradient-tracking technique. The proposed algorithm (i) improves the convergence rate by adding momentum towards the optimal state using the heavy-ball method, while (ii) addressing general sector-bound nonlinearities over the information-sharing network. The link nonlinearity includes any sign-preserving odd sector-bound mapping, for example, log-scale data quantization or clipping in practical applications. For admissible momentum and gradient-tracking parameters, using perturbation theory and eigen-spectrum analysis, we prove convergence even in the presence of sector-bound nonlinearity and for locally non-convex cost functions. Further, in contrast to most existing weight-stochastic algorithms, we adopt weight-balanced (WB) network design. This WB design and perturbation-based analysis allow to handle dynamic directed network of agents to address possible time-varying setups due to link failures or packet drops.