Sketchy: Memory-efficient Adaptive Regularization with Frequent Directions (2302.03764v2)

Abstract: Adaptive regularization methods that exploit more than the diagonal entries exhibit state of the art performance for many tasks, but can be prohibitive in terms of memory and running time. We find the spectra of the Kronecker-factored gradient covariance matrix in deep learning (DL) training tasks are concentrated on a small leading eigenspace that changes throughout training, motivating a low-rank sketching approach. We describe a generic method for reducing memory and compute requirements of maintaining a matrix preconditioner using the Frequent Directions (FD) sketch. While previous approaches have explored applying FD for second-order optimization, we present a novel analysis which allows efficient interpolation between resource requirements and the degradation in regret guarantees with rank $k$: in the online convex optimization (OCO) setting over dimension $d$, we match full-matrix $d^2$ memory regret using only $dk$ memory up to additive error in the bottom $d-k$ eigenvalues of the gradient covariance. Further, we show extensions of our work to Shampoo, resulting in a method competitive in quality with Shampoo and Adam, yet requiring only sub-linear memory for tracking second moments.

• The paper demonstrates that a low-rank sketch of gradient covariance via Frequent Directions can yield memory efficiency while maintaining adaptive regularization performance.
• It leverages spectral analysis in online convex optimization to justify the dynamic low-rank approach that achieves regret bounds similar to full-matrix AdaGrad.
• Empirical evaluations on models like ResNet and Conformer confirm reduced memory usage and competitive performance, enabling scalable deep learning training.

Analysis of "#Sketchy: Memory-efficient Adaptive Regularization with Frequent Directions"

This paper introduces a novel approach to memory-efficient adaptive regularization in deep learning optimization through the use of the Frequent Directions (FD) sketch, focusing on efficiently managing the Kronecker-factored gradient covariance matrix. The authors propose a dynamic low-rank sketching method adapted to second-order optimization, incorporating a novel regret analysis specifically tailored for the online convex optimization (OCO) setting. This approach seeks to improve upon standard adaptive gradient methods, like Adam or classical AdaGrad, by reducing the memory footprint tied to maintaining a dense matrix preconditioner, while also capitalizing on the spectral properties observed in gradient covariance matrices during training.

Main Contributions

1. Spectral Analysis of Gradient Covariance: The authors provide evidence that the gradient covariance matrix's spectrum in deep learning models is concentrated in a small leading eigenspace. This observation underpins the choice of using a low-rank matrix sketching approach, as it suggests only a minor portion of the eigenspace needs to be tracked for effective adaptive regularization.
2. Frequent Directions in Online Convex Optimization: Employing the FD sketch, the paper presents a memory-efficient approach that achieves regret bounds similar to the full-matrix AdaGrad, with substantially less memory usage. The authors demonstrate this through a novel analysis method in OCO, merging FD with dynamic diagonal regularization.
3. Algorithm Development and Evaluation: The paper extends the use of FD to various adaptive optimization algorithms, including Shampoo and variations that utilize exponential moving averages. These algorithms are evaluated across several modern deep learning settings, demonstrating competitive performance with traditional methods that require at least linear memory relative to parameter count.
4. Practical Implementations: Practical applications of the proposed algorithms were tested in neural network training settings, notably with architectures like ResNet and the Conformer model, showcasing the applicability of the Sketchy approach in real-world scenarios. Their experiments emphasized reductions in memory consumption while retaining competitive performance metrics when compared to existing optimizers such as Adam.
5. Empirical Results: The rigorous empirical investigations show that the Sketchy approach can improve the overall memory-quality tradeoff efficiently. The experimental analysis highlights a Pareto improvement, leveraging a higher-rank approximation rather than resorting to rank-1 preconditioning.

Implications and Future Directions

The use of the FD sketch for managing memory efficiency in deep learning optimization presents significant advantages, particularly as model sizes continue to grow. From a theoretical standpoint, the paper advances a novel application of spectral analysis in gradient covariance matrices, providing a foundation for future work in optimizing resource usage in model training without sacrificing convergence performance.

Practically, the proposed methods help mitigate memory bandwidth issues that have become increasingly prominent due to hardware limitations, contributing crucial insights to the efficiency of large-scale model training environments. This gap between increasing logical performance and slower memory bandwidth growth is crucial for researchers and practitioners when devising strategies for deploying large models efficiently.

For future developments, potential research could explore optimizations beyond the current rank tuning restrictions, adaptive FC-based schedules, or leveraging these spectral properties across diverse architectures and domains. Additionally, further investigation into the equilibrium between memoization and computational trade-offs under varying network and architectural constraints will remain an area of vital interest.

In conclusion, the proposed Sketchy algorithms provide a sound contribution towards more memory-efficient deep learning optimizations, balancing the nuanced relationship between memory constraints and algorithmic performance in high-demand AI applications.