Parameter-Free FISTA by Adaptive Restart and Backtracking
Jean–François Aujol, Luca Calatroni, Charles Dossal, Hippolyte Labarrière, Aude Rondepierre · SIAM Journal on Optimization · 2024
Abstract. We consider a combined restarting and adaptive backtracking strategy for the popular fast iterative shrinking-thresholding algorithm (FISTA) frequently employed for accelerating the convergence speed of large-scale structured convex optimization problems. Several variants of FISTA enjoy a provable linear convergence rate for the function values [Formula: see text] of the form [Formula: see text] under the prior knowledge of problem conditioning, i.e., of the ratio between the (Łojasiewicz) parameter [Formula: see text] determining the growth of the objective function and the Lipschitz constant [Formula: see text] of its smooth component. These parameters are nonetheless hard to estimate in many practical cases. Recent works address the problem by estimating either parameter via suitable adaptive strategies. In our work both parameters can be estimated at the same time by means of an algorithmic restarting scheme where, at each restart, a nonmonotone estimation of [Formula: see text] is performed. For this scheme, theoretical convergence results are proved, showing that an [Formula: see text] convergence speed can still be achieved along with quantitative estimates of the conditioning. The resulting free-FISTA is therefore parameter-free. Several numerical results are reported to confirm the practical interest of its use in many example problems.