  We present EGN, a stochastic second-order optimization algorithm that
combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank
linear algebra to compute the descent direction. Leveraging the Duncan-Guttman
matrix identity, the parameter update is obtained by factorizing a matrix which
has the size of the mini-batch. This is particularly advantageous for
large-scale machine learning problems where the dimension of the neural network
parameter vector is several orders of magnitude larger than the batch size.
Additionally, we show how improvements such as line search, adaptive
regularization, and momentum can be seamlessly added to EGN to further
accelerate the algorithm. Moreover, under mild assumptions, we prove that our
algorithm converges to an $\epsilon$-stationary point at a linear rate.
Finally, our numerical experiments demonstrate that EGN consistently exceeds,
or at most matches the generalization performance of well-tuned SGD, Adam, and
SGN optimizers across various supervised and reinforcement learning tasks.
