DeepAI

# Proximal Subgradient Norm Minimization of ISTA and FISTA

For first-order smooth optimization, the research on the acceleration phenomenon has a long-time history. Until recently, the mechanism leading to acceleration was not successfully uncovered by the gradient correction term and its equivalent implicit-velocity form. Furthermore, based on the high-resolution differential equation framework with the corresponding emerging techniques, phase-space representation and Lyapunov function, the squared gradient norm of Nesterov's accelerated gradient descent () method at an inverse cubic rate is discovered. However, this result cannot be directly generalized to composite optimization widely used in practice, e.g., the linear inverse problem with sparse representation. In this paper, we meticulously observe a pivotal inequality used in composite optimization about the step size s and the Lipschitz constant L and find that it can be improved tighter. We apply the tighter inequality discovered in the well-constructed Lyapunov function and then obtain the proximal subgradient norm minimization by the phase-space representation, regardless of gradient-correction or implicit-velocity. Furthermore, we demonstrate that the squared proximal subgradient norm for the class of iterative shrinkage-thresholding algorithms (ISTA) converges at an inverse square rate, and the squared proximal subgradient norm for the class of faster iterative shrinkage-thresholding algorithms (FISTA) is accelerated to convergence at an inverse cubic rate.

• 48 publications
• 13 publications
• 7 publications
09/19/2022

### Gradient Norm Minimization of Nesterov Acceleration: o(1/k^3)

In the history of first-order algorithms, Nesterov's accelerated gradien...
12/13/2022

### Linear Convergence of ISTA and FISTA

In this paper, we revisit the class of iterative shrinkage-thresholding ...
12/12/2022

### Revisiting the acceleration phenomenon via high-resolution differential equations

Nesterov's accelerated gradient descent (NAG) is one of the milestones i...
12/29/2016

### Geometric descent method for convex composite minimization

In this paper, we extend the geometric descent method recently proposed ...
05/11/2022

### A globally convergent fast iterative shrinkage-thresholding algorithm with a new momentum factor for single and multi-objective convex optimization

Convex-composite optimization, which minimizes an objective function rep...
02/09/2015

### Projected Nesterov's Proximal-Gradient Algorithm for Sparse Signal Reconstruction with a Convex Constraint

We develop a projected Nesterov's proximal-gradient (PNPG) approach for ...
08/13/2013

### Composite Self-Concordant Minimization

We propose a variable metric framework for minimizing the sum of a self-...