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 $\mathcal{O}(\epsilon^{-1})$ convergence rate but also demonstrates efficiency in a synthetic problem and two deep learning tasks.

翻译：摘要：双层优化在机器学习中具有广泛应用，其结构呈复杂层次性。基于梯度的方法已成为处理大规模双层问题的常用技术。然而，超梯度计算涉及海塞逆向量积，这限制了计算效率并被视为性能瓶颈。为规避求逆过程，我们借助Lanczos过程构建了一系列低维近似Krylov子空间。所构建的子空间能够以更低计算代价动态增量式逼近海塞逆向量积，从而获得超梯度的可靠估计。进一步，我们提出了一种可证明的子空间框架用于双层优化，其核心步骤是求解一个小型三对角线性系统。据我们所知，这是子空间技术首次被引入双层优化领域。这一成功尝试不仅实现了$\mathcal{O}(\epsilon^{-1})$的收敛速率，还在合成问题与两项深度学习任务中验证了其高效性。