Why is Normalization Preferred? A Worst-Case Complexity Theory for Stochastically Preconditioned SGD under Heavy-Tailed Noise
arXiv:2602.13413v1 Announce Type: new Abstract: We develop a worst-case complexity theory for stochastically preconditioned stochastic gradient descent (SPSGD) and its accelerated variants under heavy-tailed noise, a setting that encompasses widely used adaptive methods such as Adam, RMSProp, and Shampoo. We assume the stochastic gradient noise has a finite $p$-th moment for some $p \in (1,2]$, and measure convergence after $T$ iterations. While clipping and normalization are parallel tools for stabilizing training of SGD under heavy-tailed noise, there is a fundamental separation in their worst-case properties in stochastically preconditioned settings. We demonstrate that normalization guarantees convergence to a first-order stationary point at rate $\mathcal{O}(T^{-\frac{p-1}{3p-2}})$ when problem parameters are known, and $\mathcal{O}(T^{-\frac{p-1}{2p}})$ when problem parameters are unknown, matching the optimal rates for normalized SGD, respectively. In contrast, we prove that cl
arXiv:2602.13413v1 Announce Type: new Abstract: We develop a worst-case complexity theory for stochastically preconditioned stochastic gradient descent (SPSGD) and its accelerated variants under heavy-tailed noise, a setting that encompasses widely used adaptive methods such as Adam, RMSProp, and Shampoo. We assume the stochastic gradient noise has a finite $p$-th moment for some $p \in (1,2]$, and measure convergence after $T$ iterations. While clipping and normalization are parallel tools for stabilizing training of SGD under heavy-tailed noise, there is a fundamental separation in their worst-case properties in stochastically preconditioned settings. We demonstrate that normalization guarantees convergence to a first-order stationary point at rate $\mathcal{O}(T^{-\frac{p-1}{3p-2}})$ when problem parameters are known, and $\mathcal{O}(T^{-\frac{p-1}{2p}})$ when problem parameters are unknown, matching the optimal rates for normalized SGD, respectively. In contrast, we prove that clipping may fail to converge in the worst case due to the statistical dependence between the stochastic preconditioner and the gradient estimates. To enable the analysis, we develop a novel vector-valued Burkholder-type inequality that may be of independent interest. These results provide a theoretical explanation for the empirical preference for normalization over clipping in large-scale model training.
Executive Summary
The article presents a worst-case complexity theory for stochastically preconditioned stochastic gradient descent (SPSGD) and its accelerated variants under heavy-tailed noise, which is relevant to adaptive methods like Adam, RMSProp, and Shampoo. The study compares clipping and normalization techniques, demonstrating that normalization guarantees convergence to a first-order stationary point at optimal rates, while clipping may fail to converge due to statistical dependence between the stochastic preconditioner and gradient estimates. The authors introduce a novel vector-valued Burkholder-type inequality to facilitate the analysis, providing a theoretical basis for the empirical preference for normalization in large-scale model training.
Key Points
- ▸ SPSGD and its variants under heavy-tailed noise are analyzed for worst-case complexity.
- ▸ Normalization guarantees convergence at optimal rates, while clipping may fail.
- ▸ A novel vector-valued Burkholder-type inequality is developed for the analysis.
- ▸ Theoretical results explain the empirical preference for normalization in training.
Merits
Theoretical Rigor
The article provides a rigorous theoretical framework for understanding the behavior of SPSGD under heavy-tailed noise, which is crucial for advancing the field of optimization.
Practical Relevance
The findings have direct implications for the practical implementation of optimization algorithms in machine learning, particularly in large-scale model training.
Novel Contribution
The introduction of a novel vector-valued Burkholder-type inequality adds significant value to the mathematical toolkit for analyzing stochastic optimization methods.
Demerits
Assumptions and Generalizability
The assumptions made about the stochastic gradient noise and the specific settings analyzed may limit the generalizability of the results to other scenarios.
Complexity of Analysis
The complexity of the analysis may make it challenging for practitioners to directly apply the findings without a deep understanding of the underlying mathematical concepts.
Expert Commentary
The article presents a significant advancement in the theoretical understanding of stochastically preconditioned stochastic gradient descent under heavy-tailed noise. The rigorous analysis and the introduction of a novel mathematical tool, the vector-valued Burkholder-type inequality, are particularly noteworthy. The findings provide a clear theoretical basis for the empirical preference for normalization over clipping in large-scale model training, which is a critical insight for practitioners. However, the assumptions made in the study may limit its applicability to certain scenarios, and the complexity of the analysis may pose a challenge for direct implementation. Overall, this work contributes valuable knowledge to the field of optimization and has practical implications for machine learning and statistical analysis.
Recommendations
- ✓ Further research should explore the generalizability of these findings to other optimization algorithms and noise distributions.
- ✓ Efforts should be made to develop more accessible explanations and tools based on these theoretical insights to facilitate their adoption by practitioners.
