Computing the Newton-step faster than Hessian accumulation

Abstract

Computing the Newton-step of a generic function with $N$ decision variables takes $O(N^3)$ flops. In this paper, we show that given the computational graph of the function, this bound can be reduced to $O(m\tau^3)$, where $\tau, m$ are the width and size of a tree-decomposition of the graph. The proposed algorithm generalizes nonlinear optimal-control methods based on LQR to general optimization problems and provides non-trivial gains in iteration-complexity even in cases where the Hessian is dense.