TensorGRaD: Tensor Gradient Robust Decomposition for Memory-Efficient Neural Operator Training (2501.02379v2)

Abstract: Scientific problems require resolving multi-scale phenomena across different resolutions and learning solution operators in infinite-dimensional function spaces. Neural operators provide a powerful framework for this, using tensor-parameterized layers to capture complex, multi-dimensional relationships. However, scaling neural operators to high-resolution problems leads to significant computational demands, making the training of industrial-scale models prohibitive. In this work, we introduce \textbf{TensorGRaD}, a novel method that directly addresses the memory challenges associated with optimizing large tensor-structured weights. Our approach, based on a \texit{robust tensor decomposition}, factorizes gradients as the sum of a low-rank tensor and a sparse one to efficiently capture information within optimizer states, including outliers. Additionally, we provide a recipe for mixed precision training of TensorGRaD, achieving further memory savings without sacrificing accuracy. We showcase the effectiveness of TensorGRaD on Fourier Neural Operators, a class of models crucial for solving partial differential equations (PDE). We provide theoretical guarantees for TensorGRaD, demonstrating its fundamental advantage over matrix-based gradient compression methods. We empirically demonstrate large improvements across various PDE tasks, including the challenging turbulent Navier-Stokes case at a Reynolds number of $10^5$. TensorGRaD reduces total memory usage by over $50\%$ while maintaining and sometimes even improving accuracy.

• The paper introduces a gradient tensor decomposition method that uses Tucker decomposition to preserve multidimensional structure and reduce memory usage by up to 75%.
• The paper's experiments on PDE tasks like Navier–Stokes and Darcy flow confirm improved convergence rates and overall model performance compared to traditional methods.
• The paper offers a scalable solution for memory-constrained scientific computing, paving the way for efficient deep learning in high-dimensional settings.

Memory-Efficient Training via Gradient Tensor Decomposition

The paper "Tensor-GaLore: Memory-Efficient Training via Gradient Tensor Decomposition" proposes a novel tensor decomposition technique to enhance the memory efficiency of deep neural networks, especially those using higher-order tensor weights such as Fourier Neural Operators (FNOs). The paper introduces Tensor-GaLore, an innovative approach aiming to address the significant memory demands encountered during the optimization process of neural models employed in scientific computing.

Summary of the Approach

The authors identify the inefficiencies of current matrix-based gradient projection methods like GaLore, particularly when applied to models with inherent tensor structures. These models often require managing gradients that exhibit multidimensional relationships, crucial for capturing physical phenomena and complex data structures. Flattening these into matrices for memory optimization may result in loss of critical information and reduce compression efficacy. Tensor-GaLore circumvents these limitations by employing Tucker decomposition to directly project gradient tensors onto low-rank subspaces. This method maintains the structural integrity of the original tensor dimensions, thereby enabling substantial memory savings without sacrificing model accuracy.

Key Findings and Results

The theoretical underpinnings of Tensor-GaLore are rigorously proven, demonstrating convergence and revealing the low-rank nature of gradient tensors during FNO training. Experimentally, Tensor-GaLore showcases its effectiveness across various partial differential equation (PDE) tasks—such as the Navier-Stokes and Darcy flow equations—achieving up to 75% reduction in optimizer memory usage. This notable achievement is compared directly to GaLore and demonstrates Tensor-GaLore's superiority in both memory savings and model performance.

A detailed memory profiling analysis delineates how Tensor-GaLore consistently reduces the memory required for optimizer states while improving convergence rates. The robustness of the method is evident as it maintains performance across increasing problem complexities, upholding the implicit regularization advantages and leading to enhanced model generalization.

Practical and Theoretical Implications

The introduction of Tensor-GaLore opens pathways to more efficient and scalable deep learning models in scientific computing. By maintaining the multidimensional relationships in gradient projections and optimizing memory usage, this technique democratizes access to high-performance computing. It enables researchers with limited computational resources to pursue complex scientific modeling tasks previously considered computationally prohibitive. Moreover, the method lays a theoretical foundation for future exploration into tensor-based gradient optimization, potentially influencing a broader range of applications where high-dimensional data intricacies are prevalent.

Future Directions

Future research could explore extending the application of Tensor-GaLore beyond scientific computing to areas involving large-scale tensor computations, such as deep learning architectures in natural language processing and computer vision. Investigating hardware-specific optimizations and integration with mixed precision training could yield further improvements. Additionally, adaptive algorithms that dynamically adjust tensor ranks during training could enhance efficiency and adapt better to the evolving landscape of neural network architectures.

In conclusion, Tensor-GaLore presents a compelling advancement in neural network optimization, underscoring the significance of preserving high-order tensor structures in achieving memory efficiency and performance enhancement. This work provides a fertile ground for future innovations in memory-efficient model training and deployment.