On the Convergence of Proximal Gradient Methods for Convex Simple Bilevel Optimization

• Published in: Journal of optimization theory and applications Vol. 204; no. 3; p. 51
• Main Authors: Latafat, Puya, Themelis, Andreas, Villa, Silvia, Patrinos, Panagiotis
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2025; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Convergence; Convexity; Cost function; Electrical engineering; Engineering; Mathematics; Mathematics and Statistics; Methods; Operations Research/Decision Theory; Optimization; Theory of Computation; 65K05; 90C30; Convex optimization; 90C25; Locally Lipschitz gradient; 90C06; Bilevel programming; Adaptive proximal gradient methods
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-024-02564-6

This paper studies proximal gradient iterations for addressing simple bilevel optimization problems where both the upper and the lower level cost functions are split as the sum of differentiable and (possibly nonsmooth) prox-friendly functions. We develop a novel convergence recipe for iteration-varying stepsizes that relies on Barzilai-Borwein type local estimates for the differentiable terms. Leveraging the convergence recipe, under global Lipschitz gradient continuity, we establish convergence for a nonadaptive stepsize sequence, without requiring any strong convexity or linesearch. In the locally Lipschitz differentiable setting, we develop an adaptive linesearch method that introduces a systematic adaptive scheme enabling large and nonmonotonic stepsize sequences while being insensitive to the choice of hyperparameters and initialization. Numerical simulations are provided showcasing favorable convergence speed of our methods.