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We consider a combined restarting and adaptive backtracking strategy for thepopular Fast Iterative Shrinking-Thresholding Algorithm frequently employed foraccelerating the convergence speed of large-scale structured convexoptimization problems. Several variants of FISTA enjoy a provable linearconvergence rate for the function values $F(x_n)$ of the form $\mathcal{O}(e^{-K\sqrt{\mu/L}~n})$ under the prior knowledge of problem conditioning, i.e.of the ratio between the (\L ojasiewicz) parameter $\mu$ determining the growthof the objective function and the Lipschitz constant $L$ of its smoothcomponent. These parameters are nonetheless hard to estimate in many practicalcases. Recent works address the problem by estimating either parameter viasuitable adaptive strategies. In our work both parameters can be estimated atthe same time by means of an algorithmic restarting scheme where, at eachrestart, a non-monotone estimation of $L$ is performed. For this scheme,theoretical convergence results are proved, showing that a $\mathcal{O}(e^{-K\sqrt{\mu/L}n})$ convergence speed can still be achieved along withquantitative estimates of the conditioning. The resulting Free-FISTA algorithmis therefore parameter-free. Several numerical results are reported to confirmthe practical interest of its use in many exemplar problems.

Parameter-Free FISTA by Adaptive Restart and Backtracking