Non-ergodic linear convergence property of the delayed gradient descent under the strongly convexity and the Polyak-{\L}ojasiewicz condition

TL;DR

This paper proves non-ergodic linear convergence of delayed gradient descent for strongly convex and PL-condition functions, extending previous results with weaker assumptions and larger learning rates, including stochastic variants.

Contribution

It establishes non-ergodic linear convergence rates for delayed gradient descent under weaker conditions and larger learning rates, also extending to stochastic gradient descent with delays.

Findings

Linear convergence for delayed gradient descent under strong convexity.

Extended convergence results under Polyak-{ extL}ojasiewicz condition.

Numerical experiments confirming theoretical results.

Abstract

In this work, we establish the linear convergence estimate for the gradient descent involving the delay $\tau\in\mathbb{N}$ when the cost function is $\mu$-strongly convex and $L$-smooth. This result improves upon the well-known estimates in Arjevani et al. \cite{ASS} and Stich-Karmireddy \cite{SK} in the sense that it is non-ergodic and is still established in spite of weaker constraint of cost function. Also, the range of learning rate $\eta$ can be extended from $\eta\leq 1/(10L\tau)$ to $\eta\leq 1/(4L\tau)$ for $\tau =1$ and $\eta\leq 3/(10L\tau)$ for $\tau \geq 2$, where $L >0$ is the Lipschitz continuity constant of the gradient of cost function. In a further research, we show the linear convergence of cost function under the Polyak-{\L}ojasiewicz\,(PL) condition, for which the available choice of learning rate is further improved as $\eta\leq 9/(10L\tau)$ for the large delay $\tau$. The framework of the proof for this result is also extended to the stochastic gradient descent with time-varying delay under the PL condition. Finally, some numerical experiments are provided in order to confirm the reliability of the analyzed results.