Papers › LancBiO: dynamic Lanczos-aided bilevel optimization via Krylov subspace

LancBiO: dynamic Lanczos-aided bilevel optimization via Krylov subspace

4 Apr 2024arXiv:2404.03331archive 2025-07-28

Yan Yang, Bin Gao, Ya-xiang Yuan

Bilevel optimization, with broad applications in machine learning, has an intricate hierarchical structure. Gradient-based methods have emerged as a common approach to large-scale bilevel problems. However, the computation of the hyper-gradient, which involves a Hessian inverse vector product, confines the efficiency and is regarded as a bottleneck. To circumvent the inverse, we construct a sequence of low-dimensional approximate Krylov subspaces with the aid of the Lanczos process. As a result, the constructed subspace is able to dynamically and incrementally approximate the Hessian inverse vector product with less effort and thus leads to a favorable estimate of the hyper-gradient. Moreover, we propose a provable subspace-based framework for bilevel problems where one central step is to solve a small-size tridiagonal linear system. To the best of our knowledge, this is the first time that subspace techniques are incorporated into bilevel optimization. This successful trial not only enjoys 𝒪(ϵ⁻¹) convergence rate but also demonstrates efficiency in a synthetic problem and two deep learning tasks.

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