# On the convergence of proximal gradient methods for convex simple bilevel optimization

arXiv: https://arxiv.org/abs/2305.03559 · HTML: https://mlsanity.com/paper/2305.03559/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2023.

## Built on by (most cited first, 2 of 2)

- [On the convergence of adaptive first order methods: proximal gradient and alternating minimization algorithms](https://mlsanity.com/paper/2311.18431/) (2023)
- [Adaptive Delayed-Update Cyclic Algorithm for Variational Inequalities](https://mlsanity.com/paper/2603.29128/) (2026)

## Abstract (arXiv)

This paper studies proximal gradient iterations for solving simple bilevel optimization problems where both the upper and the lower level cost functions are split as the sum of differentiable and (possibly nonsmooth) proximable 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.