Accelerated-gradient-based generalized Levenberg–Marquardt method with oracle complexity bound and local quadratic convergence
提出一种新的广义Levenberg–Marquardt方法，用于最小化凸函数与光滑复合函数的和，该方法同时具有迭代复杂度界、预言复杂度界和局部二次收敛性，实验表明在神经网络分类和非负矩阵分解中表现良好。
Abstract Minimizing the sum of a convex function and a composite function appears in various fields. The generalized Levenberg–Marquardt (LM) method, also known as the prox-linear method, has been developed for such optimization problems. The method iteratively solves strongly convex subproblems with a damping term. This study proposes a new generalized LM method for solving the problem with a smooth composite function. The method enjoys three theoretical guarantees: iteration complexity bound, oracle complexity bound, and local convergence under a Hölderian growth condition. The local convergence results include local quadratic convergence under the quadratic growth condition; this is the first to extend the classical result for least-squares problems to a general smooth composite function. In addition, this is the first LM method with both an oracle complexity bound and local quadratic convergence under standard assumptions. These results are achieved by carefully controlling the damping parameter and solving the subproblems by the accelerated proximal gradient method equipped with a particular termination condition. Experimental results show that the proposed method performs well in practice for several instances, including classification with a neural network and nonnegative matrix factorization.