Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of First-Order Algorithms with Momentum from the Perspective of an Inexact Gradient Descent Method

Pham Duy Khanh, Boris Mordukhovich, Dat Ba Tran

TL;DR

This work introduces the inexact gradient descent method with momentum (IGDm) as a universal framework for first-order optimization with momentum under inexact gradients. By leveraging a Lyapunov function $H_\alpha$ and the Polyak–Łojasiewicz–Kurdyka (PLK) condition, the authors establish global and local convergence results for IGDm and derive constructive convergence rates that depend on the PLK exponent $q$. They demonstrate that well-known momentum methods (EGm, SAMm, IPPm) are special cases of IGDm, carrying over the same global and local convergence guarantees under simple inexactness and step-size restrictions. Numerical experiments in derivative-free optimization corroborate the momentum advantage, showing improved performance over basic IGD in both convex and nonconvex settings. The framework thus provides a unified, rigorously analyzed approach to momentum in first-order methods with broad applicability to nonconvex and nonsmooth problems.

Abstract

This paper introduces a novel inexact gradient descent method with momentum (IGDm) considered as a general framework for various first-order methods with momentum. This includes, in particular, the inexact proximal point method (IPPm), extragradient method (EGm), and sharpness-aware minimization (SAMm). Asymptotic convergence properies of IGDm are established under both global and local assumptions on objective functions with providing constructive convergence rates depending on the Polyak-Łojasiewicz-Kurdyka (PLK) conditions for the objective function. Global convergence of EGm and SAMm for general smooth functions and of IPPM for weakly convex functions is derived in this way. Moreover, local convergence properties of EGm and SAMm for locally smooth functions as well as of IPPm for prox-regular functions are established. Numerical experiments for derivative-free optimization problems are conducted to confirm the efficiency of the momentum effects of the developed methods under inexactness of gradient computations

Convergence of First-Order Algorithms with Momentum from the Perspective of an Inexact Gradient Descent Method

TL;DR

This work introduces the inexact gradient descent method with momentum (IGDm) as a universal framework for first-order optimization with momentum under inexact gradients. By leveraging a Lyapunov function and the Polyak–Łojasiewicz–Kurdyka (PLK) condition, the authors establish global and local convergence results for IGDm and derive constructive convergence rates that depend on the PLK exponent . They demonstrate that well-known momentum methods (EGm, SAMm, IPPm) are special cases of IGDm, carrying over the same global and local convergence guarantees under simple inexactness and step-size restrictions. Numerical experiments in derivative-free optimization corroborate the momentum advantage, showing improved performance over basic IGD in both convex and nonconvex settings. The framework thus provides a unified, rigorously analyzed approach to momentum in first-order methods with broad applicability to nonconvex and nonsmooth problems.

Abstract

This paper introduces a novel inexact gradient descent method with momentum (IGDm) considered as a general framework for various first-order methods with momentum. This includes, in particular, the inexact proximal point method (IPPm), extragradient method (EGm), and sharpness-aware minimization (SAMm). Asymptotic convergence properies of IGDm are established under both global and local assumptions on objective functions with providing constructive convergence rates depending on the Polyak-Łojasiewicz-Kurdyka (PLK) conditions for the objective function. Global convergence of EGm and SAMm for general smooth functions and of IPPM for weakly convex functions is derived in this way. Moreover, local convergence properties of EGm and SAMm for locally smooth functions as well as of IPPm for prox-regular functions are established. Numerical experiments for derivative-free optimization problems are conducted to confirm the efficiency of the momentum effects of the developed methods under inexactness of gradient computations
Paper Structure (11 sections, 23 theorems, 96 equations, 2 figures, 1 table)

This paper contains 11 sections, 23 theorems, 96 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Auxiliary Results
  3. Inexact Gradient Descent Method with Momentum
  4. Algorithm Design and Some Properties
  5. Convergence Analysis
  6. Convergence of First-Order Methods with Momentum
  7. Extragradient Method with Momentum
  8. Sharpness-Aware Minimization with Momentum
  9. Inexact Proximal Point Method with Momentum
  10. Numerical Experiments
  11. Concluding Remarks and Future Research

Key Result

Lemma 2.1

Given $f:{\rm I\!R}^n\rightarrow{\rm I\!R}$ and $x,y\in\Omega$, assume that $f$ is differentiable on the line segment $[x,y]$ and its derivative is Lipschitz continuous on this segment with constant $L>0$. Then we have

Figures (2)

  • Figure 1: Inexact Gradient Descent Algorithm with Momentum
  • Figure 2: Derivative-free least-square convex and nonconvex problems

Theorems & Definitions (30)

  • Lemma 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 20 more