Bilevel Learning via Inexact Stochastic Gradient Descent

TL;DR

This paper develops a theoretical framework for inexact stochastic bilevel optimization, proving convergence rates under decaying accuracy and step size schedules, with practical experiments in image processing confirming the results.

Contribution

It advances the theory of inexact stochastic bilevel optimization by establishing convergence guarantees with decaying accuracy and step sizes, bridging the gap between theory and large-scale applications.

Findings

Decreasing step sizes improve stability in practice.

Accuracy scheduling is more critical than step size strategy.

Adaptive preconditioning enhances performance.

Abstract

Bilevel optimization is a central tool in machine learning for high-dimensional hyperparameter tuning. Its applications are vast; for instance, in imaging it can be used for learning data-adaptive regularizers and optimizing forward operators in variational regularization. These problems are large in many ways: a lot of data is usually available to train a large number of parameters, calling for stochastic gradient-based algorithms. However, exact gradients with respect to parameters (so-called hypergradients) are not available, and their precision is usually linearly related to computational cost. Hence, algorithms must solve the problem efficiently without unnecessary precision. The design of such methods is still not fully understood, especially regarding how accuracy requirements and step size schedules affect theoretical guarantees and practical performance. Existing approaches introduce stochasticity at both the upper level (e.g., in sampling or mini-batch estimates) and the lower level (e.g., in solving the inner problem) to improve generalization, but they typically fix the number of lower-level iterations, which conflicts with asymptotic convergence assumptions. In this work, we advance the theory of inexact stochastic bilevel optimization. We prove convergence and establish rates under decaying accuracy and step size schedules, showing that with optimal configurations convergence occurs at an $\mathcal{O}(k^{-1/4})$ rate in expectation. Experiments on image denoising and inpainting with convex ridge regularizers and input-convex networks confirm our analysis: decreasing step sizes improve stability, accuracy scheduling is more critical than step size strategy, and adaptive preconditioning (e.g., Adam) further boosts performance. These results bridge theory and practice, providing convergence guarantees and practical guidance for large-scale imaging problems.