A Near-Optimal Total Complexity for the Inexact Accelerated Proximal Gradient Method via Quadratic Growth

TL;DR

This paper establishes a near-optimal complexity bound for the inexact accelerated proximal gradient method applied to conic polyhedral problems, showing improved theoretical guarantees and practical efficiency.

Contribution

It proves a new complexity bound for IAPG in conic polyhedral settings, leveraging quadratic growth conditions for faster convergence.

Findings

Achieves complexity of O(ln(1/ε)/√ε) for ε-accuracy

Demonstrates linear convergence of the inner proximal gradient loop

Numerical experiments confirm fast convergence in signal recovery

Abstract

We consider the optimization problem $\min_{x\in \mathbb R^n}{F(x):=f(x)+\omega(Ax)}$, where $f$ is an $L$-Lipschitz smooth function, and $\omega$ is a proper, lower semicontinuous, and convex function. We prove in this paper that when $\omega$ is a conic polyhedral function, the inexact accelerated proximal gradient method (IAPG), employed in a double-loop structure, achieves a total complexity of $\mathcal O(\ln(1/\varepsilon)/\sqrt{\varepsilon})$ measured by the total number of calls to the proximal operator of the convex conjugate $\omega^\star$ and the gradient of $f$ to achieve $\varepsilon$-optimality in function value. To the best of our knowledge, this improves upon the best-known complexity for IAPG. The key theoretical ingredient is a quadratic growth condition on the dual of the inexact proximal problem, which arises from the conic polyhedral structure of $\omega$ and implies linear convergence of the inner proximal gradient loop. To validate these findings, we conduct numerical experiments on a robust TV-$\ell_2$ signal recovery problem, demonstrating fast convergence.