Adaptive Gradient Normalization and Independent Sampling for (Stochastic) Generalized-Smooth Optimization (2410.14054v2)

Abstract: Recent studies have shown that many nonconvex machine learning problems satisfy a generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms are not fully adapted to such generalized-smooth nonconvex geometry and encounter significant technical limitations on their convergence analysis. In this work, we first analyze the convergence of adaptively normalized gradient descent under function geometries characterized by generalized-smoothness and generalized P{\L} condition, revealing the advantage of adaptive gradient normalization. Our results provide theoretical insights into adaptive normalization across various scenarios.For stochastic generalized-smooth nonconvex optimization, we propose \textbf{I}ndependent-\textbf{A}daptively \textbf{N}ormalized \textbf{S}tochastic \textbf{G}radient \textbf{D}escent, which leverages adaptive gradient normalization, independent sampling, and gradient clipping to achieve an $\mathcal{O}(\epsilon^{-4})$ sample complexity under relaxed noise assumptions. Experiments on large-scale nonconvex generalized-smooth problems demonstrate the fast convergence of our algorithm.

• The paper introduces I-NSGD, which independently normalizes stochastic gradients to reduce bias and enhance convergence.
• The paper analyzes a normalized gradient descent method under a generalized Polyak-Łojasiewicz condition to achieve linear convergence in deterministic settings.
• Numerical experiments on tasks like nonconvex phase retrieval validate the effectiveness of I-NSGD over traditional SGD methods.

Independently-Normalized SGD for Generalized-Smooth Nonconvex Optimization

The paper introduces a novel approach to nonconvex optimization, specifically addressing the limitations of existing algorithms designed for generalized-smooth nonconvex optimization. The authors propose new methodologies for both deterministic and stochastic settings to enhance convergence rates under relaxed assumptions.

Overview

The concept of generalized-smoothness is central to the paper, characterized by the dependency of the smoothness parameter on the gradient norm, extending beyond traditional nonconvex optimization. This condition applies to various complex machine learning problems such as distributionally-robust optimization and meta-learning, indicating broad applicability.

The existing literature often relies on stochastic gradient descent (SGD) and its variants. However, these algorithms generally suffer from practical limitations due to ill-conditioned smoothness parameters when gradients are large. Addressing this, the authors propose a normalized gradient descent strategy for deterministic settings and an independently-normalized stochastic gradient descent (I-NSGD) algorithm for stochastic scenarios.

Deterministic Setting

In the deterministic framework, the authors analyze the convergence of a normalized gradient descent algorithm under a generalized Polyak-{\L}ojasiewicz (P{\L}) condition. This analysis provides insights into optimizing algorithm parameters like learning rate and normalization scale to align with the geometry of the function. The convergence rates are shown to be sensitive to the parameters of the P{\L} condition, offering linear convergence under specific conditions.

Stochastic Generalized-Smooth Nonconvex Optimization

For the stochastic setting, the introduction of I-NSGD marks a significant advancement. Traditional SGD methods suffer from biases due to dependencies in the gradient estimation process. I-NSGD mitigates this by independently normalizing stochastic gradients, leading to improved stability and reduced bias. This approach achieves an $\mathcal{O}(\epsilon^{-4})$ sample complexity without relying on excessively large batch sizes or strong assumptions typical of earlier methods.

Implications and Numerical Results

The authors present numerical experiments on nonconvex phase retrieval and distributionally-robust optimization. The I-NSGD algorithm demonstrates superior convergence compared to existing methods, particularly in tackling generalized-smooth problems. These results align with the theoretical improvements predicted, confirming the algorithm's practical benefits.

Future Directions

The paper hints at potential future developments involving acceleration techniques such as momentum or variance reduction, integrated with independent sampling to further refine sample complexity and convergence efficiency in generalized-smooth nonconvex optimization.

Conclusion

This research contributes significantly to the optimization of generalized-smooth nonconvex problems by addressing critical limitations of existing algorithms. Through innovative strategies such as I-NSGD, the authors push forward the understanding and practical capability of gradient-based optimizations in complex settings, paving the way for further investigations and optimizations in similar domains.