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Revisiting Gradient Normalization and Clipping for Nonconvex SGD under Heavy-Tailed Noise: Necessity, Sufficiency, and Acceleration

Tao Sun, Xinwang Liu, Kun Yuan

TL;DR

The paper studies SGD in nonconvex settings with heavy-tailed gradient noise and questions the necessity of gradient clipping. It shows that gradient normalization alone suffices for convergence under mild smoothness assumptions and that combining normalization with clipping yields superior convergence rates, removing logarithmic factors and improving constants. The authors extend these results to nonconvex variance-reduced algorithms and introduce an accelerated variant that leverages second-order smoothness to achieve faster rates, all without requiring a mini-batch size. The work provides a unified theory across normalization-only, clipping-only, and combined approaches, offering practical guidance for using normalization and clipping in noisy nonconvex optimization. It also discusses extensions to general normalization operators and highlights the potential for improved generalization due to relaxed minibatch constraints.

Abstract

Gradient clipping has long been considered essential for ensuring the convergence of Stochastic Gradient Descent (SGD) in the presence of heavy-tailed gradient noise. In this paper, we revisit this belief and explore whether gradient normalization can serve as an effective alternative or complement. We prove that, under individual smoothness assumptions, gradient normalization alone is sufficient to guarantee convergence of the nonconvex SGD. Moreover, when combined with clipping, it yields far better rates of convergence under more challenging noise distributions. We provide a unifying theory describing normalization-only, clipping-only, and combined approaches. Moving forward, we investigate existing variance-reduced algorithms, establishing that, in such a setting, normalization alone is sufficient for convergence. Finally, we present an accelerated variant that under second-order smoothness improves convergence. Our results provide theoretical insights and practical guidance for using normalization and clipping in nonconvex optimization with heavy-tailed noise.

Revisiting Gradient Normalization and Clipping for Nonconvex SGD under Heavy-Tailed Noise: Necessity, Sufficiency, and Acceleration

TL;DR

The paper studies SGD in nonconvex settings with heavy-tailed gradient noise and questions the necessity of gradient clipping. It shows that gradient normalization alone suffices for convergence under mild smoothness assumptions and that combining normalization with clipping yields superior convergence rates, removing logarithmic factors and improving constants. The authors extend these results to nonconvex variance-reduced algorithms and introduce an accelerated variant that leverages second-order smoothness to achieve faster rates, all without requiring a mini-batch size. The work provides a unified theory across normalization-only, clipping-only, and combined approaches, offering practical guidance for using normalization and clipping in noisy nonconvex optimization. It also discusses extensions to general normalization operators and highlights the potential for improved generalization due to relaxed minibatch constraints.

Abstract

Gradient clipping has long been considered essential for ensuring the convergence of Stochastic Gradient Descent (SGD) in the presence of heavy-tailed gradient noise. In this paper, we revisit this belief and explore whether gradient normalization can serve as an effective alternative or complement. We prove that, under individual smoothness assumptions, gradient normalization alone is sufficient to guarantee convergence of the nonconvex SGD. Moreover, when combined with clipping, it yields far better rates of convergence under more challenging noise distributions. We provide a unifying theory describing normalization-only, clipping-only, and combined approaches. Moving forward, we investigate existing variance-reduced algorithms, establishing that, in such a setting, normalization alone is sufficient for convergence. Finally, we present an accelerated variant that under second-order smoothness improves convergence. Our results provide theoretical insights and practical guidance for using normalization and clipping in nonconvex optimization with heavy-tailed noise.

Paper Structure

This paper contains 28 sections, 9 theorems, 147 equations, 2 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Main Approaches to Handling Heavy-Tailed Noise
  3. Fundamental Open Questions and Main Results
  4. Clipping: necessary or beneficial?
  5. Contributions
  6. More Related Works
  7. Assumptions and Technical Novelty
  8. Other Necessary Assumptions
  9. A Concise Explanation of How We Achieve Improved Results
  10. Gradient Normalization Alone Guarantees Convergence of SGD under Heavy-Tailed Noise
  11. NSGD under Heavy-tailed Noise
  12. NSGD-VR under Heavy-tailed Noise
  13. Improved Results of Gradient Normalization with Clipping under Heavy-tailed Noise
  14. Improved Convergence Rate of NSGDC
  15. Improved Convergence Rate of NSGDC-VR
  16. ...and 13 more sections

Key Result

Theorem 1

Let $(\bm{w}^t)_{t\geq0}$ be generated by Algorithm alg1. Suppose Assumptions ass2-ass1 hold. If $1-\theta=\min\{\frac{{\color{black}\max\{(L\Delta)^{\frac{1}{2}}, 1\}}}{\sigma^{\frac{4p-4}{3p-2}}T^{\frac{p}{3p-2}}},1\}$, $\gamma={\color{black}\sqrt{\frac{\Delta}{L}}}\frac{\sqrt{1-\theta}}{\sqrt{T}}

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Proposition 10