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Convergence Rate Analysis for Monotone Accelerated Proximal Gradient Method

Zepeng Wang, Juan Peypouquet

Abstract

We analyze the convergence rate of the monotone accelerated proximal gradient method, which can be used to solve structured convex composite optimization problems. A linear convergence rate is established when the smooth part of the objective function is strongly convex, without knowledge of the strong convexity parameter. This is the fastest convergence rate known for this algorithm. As a byproduct, we also establish the boundedness of the iterates in the convex setting, and prove that the limit points of the iterates are all minimizers of the objective function.

Convergence Rate Analysis for Monotone Accelerated Proximal Gradient Method

Abstract

We analyze the convergence rate of the monotone accelerated proximal gradient method, which can be used to solve structured convex composite optimization problems. A linear convergence rate is established when the smooth part of the objective function is strongly convex, without knowledge of the strong convexity parameter. This is the fastest convergence rate known for this algorithm. As a byproduct, we also establish the boundedness of the iterates in the convex setting, and prove that the limit points of the iterates are all minimizers of the objective function.

Paper Structure

This paper contains 7 sections, 8 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Energy Estimations
  3. Preliminaries
  4. Energy Decrease
  5. Convergence Analysis I: the Convex Case
  6. Convergence Analysis II: the Strongly Convex Case
  7. Conclusions

Key Result

proposition 1

Let Hypothesis Hypo: F hold and $s\in \left( 0, \frac{1}{L} \right]$. Let $(x_k)_{k\ge 1}$ and $(y_k)_{k\ge 1}$ be generated according to Algo: M-APM, and consider the sequence $(E_k)_{k\ge 0}$ defined by E: E_k. Then,

Theorems & Definitions (17)

  • remark 1
  • proposition 1
  • proof
  • remark 2
  • theorem 1
  • proof
  • remark 3
  • lemma 1
  • proof
  • lemma 2
  • ...and 7 more