Higher-Order Newton Methods with Polynomial Work per Iteration

• URL: http://arxiv.org/abs/2311.06374v2
• Date: Wed, 12 Jun 2024 17:30:24 GMT
• Title: Higher-Order Newton Methods with Polynomial Work per Iteration
• Authors: Amir Ali Ahmadi, Abraar Chaudhry, Jeffrey Zhang,
• Abstract summary: We present generalizations of oracles method that incorporate derivatives of an arbitrary $d$ but maintain a dependence on basins in their iteration dimension. We show on numerical examples that of attraction around local as $d$ can get larger around local as additional assumptions are made.
• Score: 0.7568448369029973
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present generalizations of Newton's method that incorporate derivatives of an arbitrary order $d$ but maintain a polynomial dependence on dimension in their cost per iteration. At each step, our $d^{\text{th}}$-order method uses semidefinite programming to construct and minimize a sum of squares-convex approximation to the $d^{\text{th}}$-order Taylor expansion of the function we wish to minimize. We prove that our $d^{\text{th}}$-order method has local convergence of order $d$. This results in lower oracle complexity compared to the classical Newton method. We show on numerical examples that basins of attraction around local minima can get larger as $d$ increases. Under additional assumptions, we present a modified algorithm, again with polynomial cost per iteration, which is globally convergent and has local convergence of order $d$.

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