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K-FAC (arXiv:1503.05671, arXiv:1602.01407) is a tractable implementation ofNatural Gradient (NG) for Deep Learning (DL), whose bottleneck is computing theinverses of the so-called ``Kronecker-Factors'' (K-factors). RS-KFAC(arXiv:2206.15397) is a K-FAC improvement which provides a cheap way ofestimating the K-factors inverses.  In this paper, we exploit the exponential-average construction paradigm ofthe K-factors, and use online numerical linear algebra techniques to propose aneven cheaper (but less accurate) way of estimating the K-factors inverses. Inparticular, we propose a K-factor inverse update which scales linearly in layersize. We also propose an inverse application procedure which scales linearly aswell (the one of K-FAC scales cubically and the one of RS-KFAC scalesquadratically). Overall, our proposed algorithm gives an approximate K-FACimplementation whose preconditioning part scales linearly in layer size(compare to cubic for K-FAC and quadratic for RS-KFAC). Importantly however,this update is only applicable in some circumstances (typically for all FClayers), unlike the RS-KFAC approach (arXiv:2206.15397).  Numerical results show RS-KFAC's inversion error can be reduced with minimalCPU overhead by adding our proposed update to it. Based on the proposedprocedure, a correction to it, and RS-KFAC, we propose three practicalalgorithms for optimizing generic Deep Neural Nets. Numerical results show thattwo of these outperform RS-KFAC for any target test accuracy on CIFAR10classification with a slightly modified version of VGG16_bn. Our proposedalgorithms achieve 91$\%$ test accuracy faster than SENG (the state of artimplementation of empirical NG for DL; arXiv:2006.05924) but underperform itfor higher test-accuracy.

Brand New K-FACs: Speeding up K-FAC with Online Decomposition Updates