Lyapunov analysis for FISTA under strong convexity

Luis M. Briceño-Arias

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arXiv · 2506.11785

Lyapunov analysis for FISTA under strong convexity

Luis M. Briceño-Arias

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Abstract

In this paper, we conduct a theoretical and numerical study of the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) under strong convexity assumptions. We propose an autonomous Lyapunov function that reflects the strong convexity of the objective function, whether it arises from the smooth or non-smooth component. This Lyapunov function decreases monotonically at a linear rate along the iterations of the algorithm for a fixed inertial parameter, provided that the inertial parameter is built from the total strong convexity modulus of the objective function. Our analysis shows that the resulting algorithm with optimal parameters, its Lyapunov function, and its linear convergence rate are invariant under any transfer of strong convexity between the smooth and non-smooth components. This invariance also holds for forward-backward splitting (FBS). Within this framework, we compare the performance of FBS and some FISTA variants, and find that this strategy leads FISTA to outperform all other configurations, including FBS. Moreover, we identify parameter regimes in which FBS yields better performance than FISTA when the strong convexity of the non-smooth part is not leveraged appropriately.

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BibTeX RIS

Luis M. Briceño-Arias. 2025-06-13. Lyapunov analysis for FISTA under strong convexity. https://arxiv.org/abs/2506.11785

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