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In this work, we establish the linear convergence estimate for the gradientdescent involving the delay $\tau\in\mathbb{N}$ when the cost function is$\mu$-strongly convex and $L$-smooth. This result improves upon the well-knownestimates in Arjevani et al. \cite{ASS} and Stich-Karmireddy \cite{SK} in thesense that it is non-ergodic and is still established in spite of weakerconstraint of cost function. Also, the range of learning rate $\eta$ can beextended 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 Lipschitzcontinuity constant of the gradient of cost function. In a further research, weshow the linear convergence of cost function under thePolyak-{\L}ojasiewicz\,(PL) condition, for which the available choice oflearning rate is further improved as $\eta\leq 9/(10L\tau)$ for the large delay$\tau$. Finally, some numerical experiments are provided in order to confirmthe reliability of the analyzed results.

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