First-order algorithms converge faster than $O(1/k)$ on convex problems

Ching-pei Lee, Stephen J. Wright

Published 2018-12-20Version 1

It has been known for many years 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 an $O(1/k^{1+\epsilon})$ rate is not generally attainable for any $\epsilon>0$, for any of these methods.