A $\sqrt{2}$-accelerated FISTA for composite strongly convex problems

Abstract: In this paper, we propose a novel accelerated forward-backward splitting algorithm for minimizing convex composite functions, written as the sum of a smooth function and a (possibly) nonsmooth function. When the objective function is strongly convex, the method attains, to the best of our knowledge, the fastest known convergence rate, yielding a simultaneous linear and sublinear nonasymptotic bound. Our convergence analysis remains valid even when one of the two terms is only weakly convex (while the sum remains convex). The algorithm is derived by discretizing a continuous-time model of the Information-Theoretic Exact Method (ITEM), which is the optimal method for unconstrained strongly convex minimization.