Exploiting higher-order derivatives in convex optimization methods

Abstract

Exploiting higher-order derivatives in convex optimization is known at least since 1970's. In each iteration higher-order (also called tensor) methods minimize a regularized Taylor expansion of the objective function, which leads to faster convergence rates if the corresponding higher-order derivative is Lipschitz-continuous. Recently a series of lower iteration complexity bounds for such methods were proved, and a gap between upper an lower complexity bounds was revealed. Moreover, it was shown that such methods can be implementable since the appropriately regularized Taylor expansion of a convex function is also convex and, thus, can be minimized in polynomial time. Only very recently an algorithm with optimal convergence rate $1/k^{(3p+1)/2}$ was proposed for minimizing convex functions with Lipschitz $p$-th derivative. For convex functions with Lipschitz third derivative, these developments allowed to propose a second-order method with convergence rate $1/k^5$, which is faster than the rate $1/k^{3.5}$ of existing second-order methods.