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On The Convergence of Euler Discretization of Finite-Time Convergent Gradient Flows

Siqi Zhang, Mouhacine Benosman, Orlando Romero

TL;DR

This work analyzes the forward-Euler discretization of finite-time convergent flows, namely the $q$-rescaled gradient flow ($q$-RGF) and the $q$-signed gradient flow ($q$-SGF), for gradient-dominated objectives. By leveraging hybrid-system and Lyapunov tools, it proves closeness between discretized iterations and their continuous-time counterparts and derives linear convergence guarantees in both deterministic and stochastic settings, with line-search extensions for nonuniform smoothness. Key contributions include (i) discretization closeness results, (ii) iteration-complexity bounds with explicit rates, (iii) stochastic-rate guarantees under variance and SPB-type assumptions, and (iv) empirical validation on academic benchmarks and SVHN showing faster convergence than standard methods. The findings offer practical, faster first-order methods for nonconvex optimization and deepen the understanding of discretizing non-Lipschitz finite-time flows in optimization.

Abstract

In this study, we investigate the performance of two novel first-order optimization algorithms, namely the rescaled-gradient flow (RGF) and the signed-gradient flow (SGF). These algorithms are derived from the forward Euler discretization of finite-time convergent flows, comprised of non-Lipschitz dynamical systems, which locally converge to the minima of gradient-dominated functions. We first characterize the closeness between the continuous flows and the discretizations, then we proceed to present (linear) convergence guarantees of the discrete algorithms (in the general and the stochastic case). Furthermore, in cases where problem parameters remain unknown or exhibit non-uniformity, we further integrate the line-search strategy with RGF/SGF and provide convergence analysis in this setting. We then apply the proposed algorithms to academic examples and deep neural network training, our results show that our schemes demonstrate faster convergences against standard optimization alternatives.

On The Convergence of Euler Discretization of Finite-Time Convergent Gradient Flows

TL;DR

This work analyzes the forward-Euler discretization of finite-time convergent flows, namely the -rescaled gradient flow (-RGF) and the -signed gradient flow (-SGF), for gradient-dominated objectives. By leveraging hybrid-system and Lyapunov tools, it proves closeness between discretized iterations and their continuous-time counterparts and derives linear convergence guarantees in both deterministic and stochastic settings, with line-search extensions for nonuniform smoothness. Key contributions include (i) discretization closeness results, (ii) iteration-complexity bounds with explicit rates, (iii) stochastic-rate guarantees under variance and SPB-type assumptions, and (iv) empirical validation on academic benchmarks and SVHN showing faster convergence than standard methods. The findings offer practical, faster first-order methods for nonconvex optimization and deepen the understanding of discretizing non-Lipschitz finite-time flows in optimization.

Abstract

In this study, we investigate the performance of two novel first-order optimization algorithms, namely the rescaled-gradient flow (RGF) and the signed-gradient flow (SGF). These algorithms are derived from the forward Euler discretization of finite-time convergent flows, comprised of non-Lipschitz dynamical systems, which locally converge to the minima of gradient-dominated functions. We first characterize the closeness between the continuous flows and the discretizations, then we proceed to present (linear) convergence guarantees of the discrete algorithms (in the general and the stochastic case). Furthermore, in cases where problem parameters remain unknown or exhibit non-uniformity, we further integrate the line-search strategy with RGF/SGF and provide convergence analysis in this setting. We then apply the proposed algorithms to academic examples and deep neural network training, our results show that our schemes demonstrate faster convergences against standard optimization alternatives.

Paper Structure

This paper contains 21 sections, 13 theorems, 81 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related Work
  4. Preliminaries: Optimization Algorithms as Discrete-Time Systems
  5. Analysis of Euler Discretization of RGF/SGF
  6. Continuous-Time Convergence of $q$-RGF/SGF
  7. Closeness of the Discretization
  8. Convergence Rate Analysis
  9. Convergence Rates in The Stochastic Case
  10. Stochastic $q$-RGF
  11. Stochastic $q$-SGF
  12. Extensions to Non-Uniform Smoothness and Gradient Dominance
  13. Problem Setting and Algorithm Design
  14. Convergence Analysis
  15. Numerical Experiments
  16. ...and 6 more sections

Key Result

Theorem 3.1

With Assumption assume:gradient_dominance, let $c>0$ and $q\in (p,\infty]$. Then, any maximal solution $x(\cdot)$, in the sense of Filippov, to the $q$-RGF (eq:qRGF) or $q$-SGF (eq:qSGF) will converge in finite time to $x^\star$, provided that $\|x(0)-x^\star\|_2>0$ is sufficiently small. More preci In particular, given any compact and positively invariant subset $S\subset\mathcal{D}$, both flows

Figures (3)

  • Figure 1: Results of discretization of RGF and SGF on Academic Examples \ref{['eq:numerical_academic']}
  • Figure 2: Losses for several optimization algorithms run on the GPU-VGG16-SVHN experiment: Train loss (left), test loss (right)
  • Figure 3: Training loss vs. computation time for the GPU-VGG16-SVHN experiment

Theorems & Definitions (24)

  • Example 2.1
  • Example 2.2
  • Remark 3.1
  • Theorem 3.1: ourICMLpaperanon
  • Theorem 3.2: Closeness
  • Definition 3.1: $(T,\epsilon)$-closeness
  • Lemma 3.1: Closeness of continuous and discrete solutions ST10
  • Remark 4.1
  • Theorem 4.1: Convergence Rate
  • Remark 4.2
  • ...and 14 more