Linearly convergent bilevel optimization with single-step inner methods
提出一种介于牛顿法和隐函数法之间的双层优化求解方法，通过预处理最优性条件交替更新内层、伴随方程和外层，证明梯度下降和前后向分裂的线性收敛，在图像去噪和去卷积中表现良好。
Abstract We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to solve the full-system optimality conditions, but to precondition them to alternate between taking steps of simple conventional methods for the inner problem, the adjoint equation, and the outer problem. While the inner objective has to be smooth, the outer objective may be nonsmooth subject to a prox-contractivity condition. We prove linear convergence of the approach for combinations of gradient descent and forward-backward splitting with exact and inexact solution of the adjoint equation. We demonstrate good performance on learning the regularization parameter for anisotropic total variation image denoising, and the convolution kernel for image deconvolution.