  In this paper, we study the fundamental open question of finding the optimal
high-order algorithm for solving smooth convex minimization problems. Arjevani
et al. (2019) established the lower bound
$\Omega\left(\epsilon^{-2/(3p+1)}\right)$ on the number of the $p$-th order
oracle calls required by an algorithm to find an $\epsilon$-accurate solution
to the problem, where the $p$-th order oracle stands for the computation of the
objective function value and the derivatives up to the order $p$. However, the
existing state-of-the-art high-order methods of Gasnikov et al. (2019b); Bubeck
et al. (2019); Jiang et al. (2019) achieve the oracle complexity
$\mathcal{O}\left(\epsilon^{-2/(3p+1)} \log (1/\epsilon)\right)$, which does
not match the lower bound. The reason for this is that these algorithms require
performing a complex binary search procedure, which makes them neither optimal
nor practical. We fix this fundamental issue by providing the first algorithm
with $\mathcal{O}\left(\epsilon^{-2/(3p+1)}\right)$ $p$-th order oracle
complexity.

Optimization and Control;Machine Learning