Shampoo: Preconditioned Stochastic Tensor Optimization (1802.09568v2)

Abstract: Preconditioned gradient methods are among the most general and powerful tools in optimization. However, preconditioning requires storing and manipulating prohibitively large matrices. We describe and analyze a new structure-aware preconditioning algorithm, called Shampoo, for stochastic optimization over tensor spaces. Shampoo maintains a set of preconditioning matrices, each of which operates on a single dimension, contracting over the remaining dimensions. We establish convergence guarantees in the stochastic convex setting, the proof of which builds upon matrix trace inequalities. Our experiments with state-of-the-art deep learning models show that Shampoo is capable of converging considerably faster than commonly used optimizers. Although it involves a more complex update rule, Shampoo's runtime per step is comparable to that of simple gradient methods such as SGD, AdaGrad, and Adam.

• The paper introduces Shampoo, a preconditioning algorithm that uses separate tensor-dimension preconditioners to decrease memory and computational costs.
• The paper provides theoretical convergence guarantees with an O(√T) regret bound using matrix trace inequalities.
• The paper validates Shampoo with experiments showing faster convergence than SGD, AdaGrad, and Adam while keeping runtime efficiency.

Shampoo: Preconditioned Stochastic Tensor Optimization

The paper introduces Shampoo, a novel structure-aware preconditioning algorithm designed for efficient optimization in tensor spaces. Shampoo addresses challenges in preconditioned gradient methods, which often require handling prohibitively large matrices. Traditional preconditioning methods employ large matrices for transforming gradients, which leads to significant computational and storage requirements. Shampoo circumvents these limitations by maintaining a separate preconditioner for each dimension of a tensor, significantly reducing memory demands and computational overhead.

Key Contributions and Findings

1. Algorithm Design and Implementation: Shampoo leverages the multi-dimensional structure of tensors, common in contemporary machine learning models like neural networks and convolutional layers, to optimize parameter updates. By utilizing separate preconditioners along each tensor dimension, Shampoo performs update steps efficiently, especially when compared to full-matrix methods that can be computationally prohibitive.
2. Convergence Analysis: The paper provides theoretical guarantees for Shampoo's convergence in a stochastic convex optimization setting. It employs matrix trace inequalities to establish these guarantees, ensuring that the convergence rate of Shampoo is competitive with classical methods. Specifically, the algorithm achieves a regret bound of $O(\sqrt{T})$, matching the optimal rate attainable by conventional optimization techniques under similar assumptions.
3. Experimental Validation: Through experiments on state-of-the-art deep learning models, Shampoo demonstrates superior convergence speeds relative to widely used optimizers, such as SGD, AdaGrad, and Adam. Despite employing a more complex preconditioning scheme, Shampoo's runtime per step remains comparable to simpler methods, greatly enhancing its practical applicability.

Numerical Results and Implications

The experiments highlight Shampoo's robust performance, showing faster convergence in practice without introducing prohibitive computational costs. This transforms how preconditioned methods can be applied in large-scale models, particularly those employing tensors with high-dimensional representations.

Theoretical Implications

Shampoo enriches the theoretical landscape of online convex optimization by incorporating matrix trace inequalities and extending them into the tensor domain. This broadens the applicability of existing online optimization frameworks and offers new pathways for enhancing optimization algorithms across various applications using high-dimensional data.

Future Directions

Shampoo's design is particularly suited for tensor-structured data, prevalent in machine learning applications such as image recognition, natural language processing, and deep network training. The algorithm's ability to balance efficient computation with the benefits of preconditioning presents a compelling case for further research in its scalability across diverse architectures. Potential directions include exploring adaptations for specific model architectures and extending Shampoo's application to non-convex optimization tasks, aligning closely with recent trends in reinforcement learning and generative modeling.

Shampoo represents a significant advance in the practical and theoretical aspects of tensor optimization. Its approach, grounded in both well-established and emerging mathematical concepts, provides the machine learning community with a powerful tool for efficiently tackling high-dimensional optimization challenges.