Title: A Computationally Efficient Sparsified Online Newton Method. Abstract: Second-order methods have huge potential in improving the convergence of deep neural network (DNN) training, but are prohibitive due to their large memory and compute requirements. Furthermore, computing the matrix inverse or the Newton direction, which is needed in second-order methods, requires high precision computation for stable training as the preconditioner could have a large condition number. This paper provides a first attempt at developing computationally efficient sparse preconditioners for DNN training which can also tolerate low precision computation. Our new Sparsified Online Newton (SONew) algorithm emerges from the novel use of the so-called LogDet matrix divergence measure; we combine it with sparsity constraints to minimize the regret in the online convex optimization framework. Our mathematical analysis allows us to reduce the condition number of our sparse preconditioning matrix, thus improving the stability of training with low precision. We conduct experiments on a feed-forward neural-network autoencoder benchmark, where we compare training loss of optimizers when run for a fixed number of epochs. In the float32 experiments, our methods outperform the best-performing first-order optimizers and perform comparably to Shampoo, a state-of-the-art second-order optimizer. However, our method is even more effective in low-precision, where SONew finishes training considerably faster while performing comparably with Shampoo on training loss.