DRSOM: A Dimension Reduced Second-Order Method and Preliminary Analyses

We introduce a Dimension-Reduced Second-Order Method (DRSOM) for convex and nonconvex unconstrained optimization. Under a trust-region-like framework our method preserves the convergence of the second-order method while using only Hessian-vector products in two directions. Moreover, the computational overhead remains comparable to the first-order such as the gradient descent method. We show that the method has a complexity of $O(\epsilon^{-3/2})$ to satisfy the first-order and second-order conditions in the subspace. The applicability and performance of DRSOM are exhibited by various computational experiments in logistic regression, $L_2-L_p$ minimization, sensor network localization, and neural network training. For neural networks, our preliminary implementation seems to gain computational advantages in terms of training accuracy and iteration complexity over state-of-the-art first-order methods including SGD and ADAM.