# Accelerated Distributed Nesterov Gradient Descent

## TL;DR

This paper introduces an accelerated distributed gradient descent method that improves convergence rates for convex and strongly convex functions in networked optimization, outperforming existing algorithms.

## Contribution

It develops the Acc-DNGD algorithm with provably faster convergence rates for distributed convex optimization, including linear convergence for strongly convex functions.

## Key findings

- Achieves $O(1/t^{1.4- ext{epsilon}})$ convergence for convex functions.
- Improves to $O(1/t^2)$ for composition of linear map and strongly convex functions.
- Attains linear convergence rate for strongly convex functions with condition number dependence.

## Abstract

This paper considers the distributed optimization problem over a network, where the objective is to optimize a global function formed by a sum of local functions, using only local computation and communication. We develop an Accelerated Distributed Nesterov Gradient Descent (Acc-DNGD) method. When the objective function is convex and $L$-smooth, we show that it achieves a $O(\frac{1}{t^{1.4-\epsilon}})$ convergence rate for all $\epsilon\in(0,1.4)$. We also show the convergence rate can be improved to $O(\frac{1}{t^2})$ if the objective function is a composition of a linear map and a strongly-convex and smooth function. When the objective function is $\mu$-strongly convex and $L$-smooth, we show that it achieves a linear convergence rate of $O([ 1 - C (\frac{\mu}{L})^{5/7} ]^t)$, where $\frac{L}{\mu}$ is the condition number of the objective, and $C>0$ is some constant that does not depend on $\frac{L}{\mu}$.