Computing the Newton-step faster than Hessian accumulation

• URL: http://arxiv.org/abs/2108.01219v1
• Date: Mon, 2 Aug 2021 11:22:08 GMT
• Title: Computing the Newton-step faster than Hessian accumulation
• Authors: Akshay Srinivasan, Emanuel Todorov
• Abstract summary: We show that given the computational graph of the function, this bound can be reduced to $O(mtau3)$, 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.
• Score: 8.147652597876862
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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