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

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/k1+ε) is not generally attainable for any ε>0, for any of these methods.