Higher-Order Newton Methods with Polynomial Work per Iteration
This work addresses computational efficiency in optimization for researchers and practitioners, but it is incremental as it builds on existing Newton methods with polynomial cost constraints.
The paper tackles the problem of high computational cost in higher-order optimization methods by proposing generalizations of Newton's method that incorporate arbitrary-order derivatives while maintaining polynomial work per iteration, resulting in local convergence of order d and lower oracle complexity compared to classical Newton methods, with numerical examples showing larger basins of attraction as d increases.
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 $d$.