A New Perspective on Shampoo's Preconditioner (2406.17748v1)

Abstract: Shampoo, a second-order optimization algorithm which uses a Kronecker product preconditioner, has recently garnered increasing attention from the machine learning community. The preconditioner used by Shampoo can be viewed either as an approximation of the Gauss--Newton component of the Hessian or the covariance matrix of the gradients maintained by Adagrad. We provide an explicit and novel connection between the $\textit{optimal}$ Kronecker product approximation of these matrices and the approximation made by Shampoo. Our connection highlights a subtle but common misconception about Shampoo's approximation. In particular, the $\textit{square}$ of the approximation used by the Shampoo optimizer is equivalent to a single step of the power iteration algorithm for computing the aforementioned optimal Kronecker product approximation. Across a variety of datasets and architectures we empirically demonstrate that this is close to the optimal Kronecker product approximation. Additionally, for the Hessian approximation viewpoint, we empirically study the impact of various practical tricks to make Shampoo more computationally efficient (such as using the batch gradient and the empirical Fisher) on the quality of Hessian approximation.

• The paper establishes that Shampoo’s squared preconditioner is equivalent to one power iteration step, clarifying its theoretical basis.
• The authors validate their claims across various architectures, demonstrating high-fidelity Hessian approximations using cosine similarity metrics.
• Practical implications include enhanced efficiency in second-order optimization, paving the way for improved neural network training.

A New Perspective on Shampoo's Preconditioner

The paper "A New Perspective on Shampoo's Preconditioner," authored by Depen Morwani, Itai Shapira, Nikhil Vyas, Eran Malach, Sham Kakade, and Lucas Janson, presents a thorough examination and novel insights into the optimization algorithm known as Shampoo. The primary focus of the paper is the exploration and explanation of the Kronecker product preconditioner used by Shampoo, providing new theoretical connections and empirical evidence.

Shampoo is a second-order optimization algorithm that leverages the Kronecker product to approximate the Hessian or the gradient covariance matrix utilized in Adagrad. The core contribution of the paper is the identification and elucidation of the relationship between Shampoo's approximation and the optimal Kronecker product approximation of these matrices.

Core Contributions

The authors present several key contributions to our understanding of Shampoo and its preconditioner:

1. Theoretical Connection to Power Iteration:
   • The paper establishes that the square of the approximation used in Shampoo is equivalent to a single iteration of the power iteration algorithm for computing the optimal Kronecker product approximation. This connection elucidates the previously known, yet misunderstood, approximation properties of Shampoo.
2. Empirical Validation:
   • Extensive empirical studies demonstrate that for various datasets and architectures, the single step of the power iteration yields an approximation very close to the optimal Kronecker product. This empirical validation strengthens the theoretical claims and provides practical relevance.
3. Hessian Approximation:
   • From the Hessian viewpoint, the authors explore the computational efficiency tricks employed by Shampoo, such as using batch gradients and the empirical Fisher information matrix. These practical considerations are analyzed for their impact on the quality of the Hessian approximation.

Numerical Results

The empirical analysis spans multiple datasets and neural network architectures including logistic regression on a subsampled MNIST dataset, ResNet18 on CIFAR-5M, and ConvNeXt-T on ImageNet. Key findings include:

• Shampoo's squared approximation tracks the optimal Kronecker product approximation significantly better than the original Shampoo approximation.
• The cosine similarity between the Shampoo squared approximation and the true matrices demonstrates high fidelity in various settings, underscoring the robustness of the single-step power iteration approach.
• Approximation quality deteriorates with larger batch sizes, indicating a nuanced dependency on the underlying data distribution and batch processing.

Implications and Future Developments

Theoretically, the paper contributes to a more refined understanding of the applicability and limitations of the Kronecker product approximation in second-order optimization methods. Practically, these insights could lead to more efficient implementations of Shampoo and potentially stimulate further developments in second-order optimization techniques.

Future research could explore several avenues:

• Adapting to Different Architectures:
  • Investigating the applicability of the theoretical framework to more complex neural architectures, such as Transformers and modern convolutional networks.
• Extending Power Iteration:
  • Evaluating the implications of using multiple iterations of power iteration and its potential trade-offs in terms of computational efficiency and approximation quality.
• Alternative Approximations:
  • Exploring other efficient approximations that could complement or enhance the Kronecker product approach, thereby improving the scalability and utility of second-order methods in large-scale deep learning.

Conclusion

In summary, this paper advances our theoretical and practical understanding of Shampoo's preconditioner by highlighting the connection to the power iteration method for Kronecker product approximation. It provides not only robust theoretical insight but also significant empirical validation, thus paving the way for more effective and efficient second-order optimization methods in neural network training. The implications are broad, touching both the theoretical foundations and practical applications of machine learning optimization.