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Convergence of SGD with momentum in the nonconvex case: A time window-based analysis

Junwen Qiu, Bohao Ma, Andre Milzarek

TL;DR

This work develops a time window-based framework to analyze SGDM in nonconvex optimization, addressing the challenge that standard descent properties may fail under stochastic momentum. By introducing auxiliary iterates $\boldsymbol{z}^k$ and a merit function $\mathcal{M}$, the authors obtain an approximate descent and establish global convergence: $f(\boldsymbol{x}^k)\to f^*$ and $\|\nabla f(\boldsymbol{x}^k)\|\to0$ almost surely. Under the Łojasiewicz inequality, they further prove iterate convergence to a stationary point without requiring a priori bounded iterates, and derive convergence rates that depend on the Łojasiewicz exponent $\theta$ and the step-size schedule, including polynomial and exponential families. The results extend stochastic momentum theory to nonconvex settings, offering practical insights into the convergence and rate behavior of SGDM in noisy, nonconvex landscapes common in large-scale learning systems.

Abstract

The stochastic gradient descent method with momentum (SGDM) is a common approach for solving large-scale and stochastic optimization problems. Despite its popularity, the convergence behavior of SGDM remains less understood in nonconvex scenarios. This is primarily due to the absence of a sufficient descent property and challenges in simultaneously controlling the momentum and stochastic errors in an almost sure sense. To address these challenges, we investigate the behavior of SGDM over specific time windows, rather than examining the descent of consecutive iterates as in traditional studies. This time window-based approach simplifies the convergence analysis and enables us to establish the iterate convergence result for SGDM under the Łojasiewicz property. We further provide local convergence rates which depend on the underlying Łojasiewicz exponent and the utilized step size schemes.

Convergence of SGD with momentum in the nonconvex case: A time window-based analysis

TL;DR

This work develops a time window-based framework to analyze SGDM in nonconvex optimization, addressing the challenge that standard descent properties may fail under stochastic momentum. By introducing auxiliary iterates and a merit function , the authors obtain an approximate descent and establish global convergence: and almost surely. Under the Łojasiewicz inequality, they further prove iterate convergence to a stationary point without requiring a priori bounded iterates, and derive convergence rates that depend on the Łojasiewicz exponent and the step-size schedule, including polynomial and exponential families. The results extend stochastic momentum theory to nonconvex settings, offering practical insights into the convergence and rate behavior of SGDM in noisy, nonconvex landscapes common in large-scale learning systems.

Abstract

The stochastic gradient descent method with momentum (SGDM) is a common approach for solving large-scale and stochastic optimization problems. Despite its popularity, the convergence behavior of SGDM remains less understood in nonconvex scenarios. This is primarily due to the absence of a sufficient descent property and challenges in simultaneously controlling the momentum and stochastic errors in an almost sure sense. To address these challenges, we investigate the behavior of SGDM over specific time windows, rather than examining the descent of consecutive iterates as in traditional studies. This time window-based approach simplifies the convergence analysis and enables us to establish the iterate convergence result for SGDM under the Łojasiewicz property. We further provide local convergence rates which depend on the underlying Łojasiewicz exponent and the utilized step size schemes.
Paper Structure (21 sections, 12 theorems, 130 equations, 1 figure, 2 tables)

This paper contains 21 sections, 12 theorems, 130 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Modeling the Stochastic Process and Assumptions
  5. Time Window Techniques
  6. Limitations of Classical Approaches
  7. The Time Window
  8. Iterate Bounds and Descent-type Property
  9. Convergence Analysis
  10. Global Convergence
  11. Iterate Convergence under the Łojasiewicz Inequality
  12. Proof of Iterate Convergence Theorem
  13. Convergence Rates
  14. Main Result and Discussions
  15. Derivations of Rates under Polynomial Step Sizes
  16. ...and 6 more sections

Key Result

Lemma 3.1

Assume that $\{\alpha_k\}_k$ satisfies $\lim_{k\to\infty} \alpha_k =0$ and $\sum_{k=1}^\infty \alpha_k = \infty$. Then, for any given time window ${\sf T}>0$ and $\delta\in[0,1)$, there exists $K_\delta \in \mathbb{N}$ such that $\delta {\sf T} \leq \Delta_{\gamma_k,\gamma_{k+1}} \leq {\sf T}$ for a

Figures (1)

  • Figure 1: Time window and indices. Illustration for $\alpha_k = \frac{1}{k}$ and ${\sf T} = 1$.

Theorems & Definitions (19)

  • Lemma 3.1
  • Lemma 3.2: Error estimate
  • Lemma 3.3: Iterate bounds
  • Lemma 3.4: Approximate descent
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Remark 4.1
  • Lemma 4.2
  • proof
  • ...and 9 more