Adaptive Inertial Method

TL;DR

The paper introduces the Adaptive Inertial Method (AIM), a new accelerated optimization framework that adaptively adjusts inertial parameters, unifies quasi-Newton and Newton methods, and demonstrates superior empirical performance.

Contribution

It presents AIM, a novel adaptive inertial framework for first-order optimization that achieves accelerated convergence and unifies quasi-Newton and Newton methods without Hessian inversions.

Findings

Achieves a convergence rate of O(1/k) under convexity and Lipschitz conditions.

Transforms into a quasi-Newton method with accelerated rate O(1/k^2).

Demonstrates superior performance in numerical experiments.

Abstract

In this paper, we introduce the Adaptive Inertial Method (AIM), a novel framework for accelerated first-order methods through a customizable inertial term. We provide a rigorous convergence analysis establishing a global convergence rate of O(1/k) under mild conditions, requiring only convexity and local Lipschitz differentiability of the objective function. Our method enables adaptive parameter selection for the inertial term without manual tuning. Furthermore, we derive the particular form of the inertial term that transforms AIM into a new Quasi-Newton method. Notably, under specific circumstances, AIM coincides with the regularized Newton method, achieving an accelerated rate of O(1/k^2) without Hessian inversions. Through extensive numerical experiments, we demonstrate that AIM exhibits superior performance across diverse optimization problems, highlighting its practical effectiveness.