  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.
