Optimized first-order methods for smooth convex minimization

We introduce new optimized first-order methods for smooth unconstrained convex minimization. Drori and Teboulle (Math Program 145(1–2):451–482, 2014. doi:10.1007/s10107-013-0653-0) recently described a numerical method for computing the N-iteration optimal step coefficients in a class of first-order algorithms that includes gradient methods, heavy-ball methods (Polyak in USSR Comput Math Math Phys 4(5):1–17, 1964. doi:10.1016/0041-5553(64)90137-5), and Nesterov’s fast gradient methods (Nesterov in Sov Math Dokl 27(2):372–376, 1983; Math Program 103(1):127–152, 2005. doi:10.1007/s10107-004-0552-5). However, the numerical method in Drori and Teboulle (2014) is computationally expensive for large N, and the corresponding numerically optimized first-order algorithm in Drori and Teboulle (2014) requires impractical memory and computation for large-scale optimization problems. In this paper, we propose optimized first-order algorithms that achieve a convergence bound that is two times smaller than for Nesterov’s fast gradient methods; our bound is found analytically and refines the numerical bound in Drori and Teboulle (2014). Furthermore, the proposed optimized first-order methods have efficient forms that are remarkably similar to Nesterov’s fast gradient methods.