First-Order Algorithms Converge Faster than $O(1/k)$ on Convex Problems

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Ching-Pei Lee, Stephen Wright ;
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:3754-3762, 2019.

Abstract

It is well known that both gradient descent and stochastic coordinate descent achieve a global convergence rate of $O(1/k)$ in the objective value, when applied to a scheme for minimizing a Lipschitz-continuously differentiable, unconstrained convex function. In this work, we improve this rate to $o(1/k)$. We extend the result to proximal gradient and proximal coordinate descent on regularized problems to show similar $o(1/k)$ convergence rates. The result is tight in the sense that a rate of $O(1/k^{1+\epsilon})$ is not generally attainable for any $\epsilon>0$, for any of these methods.

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