DRSOM: A Dimension Reduced Second-Order Method

In this paper, we propose 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 curvature information in a few directions. Consequently, the computational overhead of our method remains comparable to the first-order such as the gradient descent method. Theoretically, we show that the method has a local quadratic convergence and a global convergence rate of $O(\epsilon^{-3/2})$ to satisfy the first-order and second-order conditions if the subspace satisfies a commonly adopted approximated Hessian assumption. We further show that this assumption can be removed if we perform one \emph{corrector step} (using a Krylov method, for example) periodically at the end stage of the algorithm. The applicability and performance of DRSOM are exhibited by various computational experiments, particularly in machine learning and deep learning. 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 such as SGD and ADAM.