The First Optimal Acceleration of High-Order Methods in Smooth Convex Optimization

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 Omegaleft(ε-2/(3p+1)right) on the number of the p-th order oracle calls required by an algorithm to find an ε-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 Oleft(ε-2/(3p+1) log (1/ε)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 Oleft(ε-2/(3p+1)right) p-th order oracle complexity.