Accelerated gradient descent: A guaranteed bound for a heuristic restart strategy

Published: 26 Oct 2023, Last Modified: 13 Dec 2023NeurIPS 2023 Workshop PosterEveryoneRevisionsBibTeX
Keywords: convex function, convex optimization, gradient descent, Lipschitz gradient, Nestrov acceleration, restarts
Abstract: The $\mathcal{O}(1/k^2)$ convergence rate in function value of accelerated gradient descent is optimal, but there are many modifications that have been used to speed up convergence in practice. Among these modifications are restarts, that is, starting the algorithm anew with the current iteration being considered as the initial point. We focus on the adaptive restart techniques introduced by O'Donoghue and Candes, specifically their gradient restart strategy. While the gradient restart strategy is a heuristic in general, we prove that applying gradient restarts preserves and in fact improves the $\mathcal{O}(1/k^2)$ bound, hence establishing function value convergence, for one-dimensional functions. Applications of our results to separable and nearly separable functions are presented.
Submission Number: 88