Second order stochastic optimizers allow parameter update step size and direction to adapt to loss curvature, but have traditionally required too much memory and compute for deep learning. Recently, Shampoo [Gupta et al., 2018] introduced a Kronecker factored preconditioner to reduce these requirements: it is used for large deep models [Anil et al., 2020] and in production [Anil et al., 2022]. However, it takes inverse matrix roots of ill-conditioned matrices. This requires 64-bit precision, imposing strong hardware constraints. In this paper, we propose a novel factorization, Kronecker Approximation-Domination (KrAD). Using KrAD, we update a matrix that directly approximates the inverse empirical Fisher matrix (like full matrix AdaGrad), avoiding inversion and hence 64-bit precision. We then propose KrADagrad$^\star$, with similar computational costs to Shampoo and the same regret. Synthetic ill-conditioned experiments show improved performance over Shampoo for 32-bit precision, while for several real datasets we have comparable or better generalization.

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