Enhanced Bilevel Optimization via Bregman Distance
This work addresses computational bottlenecks in bilevel optimization for machine learning applications such as hyperparameter tuning, offering incremental improvements with specific gains.
The paper tackles the high computational complexity and lack of support for nonsmooth regularization in bilevel optimization by proposing new methods using Bregman distance, achieving lower computational complexities than previous best results and outperforming existing approaches in tasks like data hyper-cleaning and hyper-representation learning.
Bilevel optimization has been recently used in many machine learning problems such as hyperparameter optimization, policy optimization, and meta learning. Although many bilevel optimization methods have been proposed, they still suffer from the high computational complexities and do not consider the more general bilevel problems with nonsmooth regularization. In the paper, thus, we propose a class of enhanced bilevel optimization methods with using Bregman distance to solve bilevel optimization problems, where the outer subproblem is nonconvex and possibly nonsmooth, and the inner subproblem is strongly convex. Specifically, we propose a bilevel optimization method based on Bregman distance (BiO-BreD) to solve deterministic bilevel problems, which achieves a lower computational complexity than the best known results. Meanwhile, we also propose a stochastic bilevel optimization method (SBiO-BreD) to solve stochastic bilevel problems based on stochastic approximated gradients and Bregman distance. Moreover, we further propose an accelerated version of SBiO-BreD method (ASBiO-BreD) using the variance-reduced technique, which can achieve a lower computational complexity than the best known computational complexities with respect to condition number $κ$ and target accuracy $ε$ for finding an $ε$-stationary point. We conduct data hyper-cleaning task and hyper-representation learning task to demonstrate that our new algorithms outperform related bilevel optimization approaches.