Accelerating Nesterov's Method for Strongly Convex Functions with Lipschitz Gradient

Abstract

We modify Nesterov's constant step gradient method for strongly convex functions with Lipschitz continuous gradient described in Nesterov's book. Nesterov shows that $f(x_k) - f^* \leq L \prod_{i=1}^k (1 - \alpha_k) \| x_0 - x^* \|_2^2$ with $\alpha_k = \sqrt{\rho}$ for all $k$, where $L$ is the Lipschitz gradient constant and $\rho$ is the reciprocal condition number of $f(x)$. Hence the convergence rate is $1-\sqrt{\rho}$. In this work, we try to accelerate Nesterov's method by adaptively searching for an $\alpha_k > \sqrt{\rho}$ at each iteration. The proposed method evaluates the gradient function at most twice per iteration and has some extra Level 1 BLAS operations. Theoretically, in the worst case, it takes the same number of iterations as Nesterov's method does but doubles the gradient calls. However, in practice, the proposed method effectively accelerates the speed of convergence for many problems including a smoothed basis pursuit denoising problem.