Higher-Order Newton Methods with Polynomial Work per Iteration

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 dth-order method uses semidefinite programming to construct and minimize a sum of squares-convex approximation to the dth-order Taylor expansion of the function we wish to minimize. We prove that our dth-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.