# Parameter-Free FISTA by Adaptive Restart and Backtracking

arXiv: https://arxiv.org/abs/2307.14323 · HTML: https://mlsanity.com/paper/2307.14323/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jul 2023.

## Built on by (most cited first, 4 of 4)

- [Adaptive Backtracking Line Search](https://mlsanity.com/paper/2408.13150/) (2024)
- [Variance Reduction and Low Sample Complexity in Stochastic Optimization via Proximal Point Method](https://mlsanity.com/paper/2402.08992/) (2024)
- [Why do we regularise in every iteration for imaging inverse problems?](https://mlsanity.com/paper/2411.00688/) (2024)
- [Stochastic Adaptive Gradient Descent Without Descent](https://mlsanity.com/paper/2509.14969/) (2025)

## Abstract (arXiv)

We consider a combined restarting and adaptive backtracking strategy for the popular Fast Iterative Shrinking-Thresholding Algorithm 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 F(xn) of the form O( e^-K√μ/L~n) under the prior knowledge of problem conditioning, i.e. of the ratio between the (L ojasiewicz) parameter μ determining the growth of the objective function and the Lipschitz constant L 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 non-monotone estimation of L is performed. For this scheme, theoretical convergence results are proved, showing that a O( e^-K√μ/Ln) convergence speed can still be achieved along with quantitative estimates of the conditioning. The resulting Free-FISTA algorithm is therefore parameter-free. Several numerical results are reported to confirm the practical interest of its use in many exemplar problems.