Non-ergodic linear convergence property of the delayed gradient descent under the strongly convexity and the Polyak–Łojasiewicz condition

In this work, we establish the linear convergence estimate for the gradient descent involving the delay [Formula: see text] when the cost function is [Formula: see text]-strongly convex and [Formula: see text]-smooth. This result improves upon the well-known estimates in [Y. Arjevani, O. Shamir and N. Srebro, A tight convergence analysis for stochastic gradient descent with delayed updates, Proc. Mach. Learn. Res. 117 (2020) 111–132; S. U. Stich and S. P. Karimireddy, The error-feedback framework: Better rates for SGD with delayed gradients and compressed updates, J. Mach. Learn. Res. 21(1) (2020) 9613–9648] 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 [Formula: see text] can be extended from [Formula: see text] to [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] 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–Łojasiewicz[Formula: see text](PL) condition, for which the available choice of learning rate is further improved as [Formula: see text] for the large delay [Formula: see text]. 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.