Shampoo: Preconditioned stochastic tensor optimization

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. Surprisingly, although it involves a more complex update rule, Shampoo's runtime per step is comparable in practice to that of simple gradient methods such as SGD, AdaGrad, and Adam.