Leveraging second-order information at the scale of deep networks is one of the main lines of approach for improving the performance of current optimizers for deep learning. Yet, existing approaches for accurate full-matrix preconditioning, such as Full-Matrix Adagrad (GGT) or Matrix-Free Approximate Curvature (M-FAC) suffer from massive storage costs when applied even to medium-scale models, as they must store a sliding window of gradients, whose memory requirements are multiplicative in the model dimension. In this paper, we address this issue via an efficient and simple-to-implement error-feedback technique that can be applied to compress preconditioners by up to two orders of magnitude in practice, without loss of convergence. Specifically, our approach compresses the gradient information via sparsification or low-rank compression \emph{before} it is fed into the preconditioner, feeding the compression error back into future iterations. Extensive experiments on deep neural networks for vision show that this approach can compress full-matrix preconditioners by up to two orders of magnitude without impact on accuracy, effectively removing the memory overhead of full-matrix preconditioning for implementations of full-matrix Adagrad (GGT) and natural gradient (M-FAC). Our code is available at https://github.com/IST-DASLab/EFCP.

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