Convergence Rate Analysis of Accelerated Forward-Backward Algorithm with Generalized Nesterov Momentum Scheme

TL;DR

This paper introduces a generalized Nesterov momentum scheme for the accelerated forward-backward algorithm, achieving improved convergence rates and demonstrating better performance in support vector machine optimization.

Contribution

It proposes a flexible momentum scheme with a power parameter, extending AFBA's convergence analysis and showing empirical improvements over existing methods.

Findings

Achieves $o(1/k^{2 heta})$ convergence rate with the new scheme.

Demonstrates superior performance in SVM training tasks.

Provides a versatile parameter selection framework for different scenarios.

Abstract

Nesterov's accelerated forward-backward algorithm (AFBA) is an efficient algorithm for solving a class of two-term convex optimization models consisting of a differentiable function with a Lipschitz continuous gradient plus a nondifferentiable function with a closed form of its proximity operator. It has been shown that the iterative sequence generated by AFBA with a modified Nesterov's momentum scheme converges to a minimizer of the objective function with an $o\left(\frac{1}{k^2}\right)$ convergence rate in terms of the function value (FV-convergence rate) and an $o\left(\frac{1}{k}\right)$ convergence rate in terms of the distance between consecutive iterates (DCI-convergence rate). In this paper, we propose a more general momentum scheme with an introduced power parameter $\omega\in(0,1]$ and show that AFBA with the proposed momentum scheme converges to a minimizer of the objective function with an $o\left(\frac{1}{k^{2\omega}}\right)$ FV-convergence rate and an $o\left(\frac{1}{k^{\omega}}\right)$ DCI-convergence rate. The generality of the proposed momentum scheme provides us a variety of parameter selections for different scenarios, which makes the resulting algorithm more flexible to achieve better performance. We then employ AFBA with the proposed momentum scheme to solve the smoothed hinge loss $\ell_1$-support vector machine model. Numerical results demonstrate that the proposed generalized momentum scheme outperforms two existing momentum schemes.