2022/05/19 by Dmitry Kovalev, Kovalev, Dmitry, Alexander Gasnikov +1 · 5 citations
Engineering · Mathematics · Computer Science · #Sparse and Compressive Sensing Techniques #Advanced Optimization Algorithms Research #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2205.09647
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 Ω(ε-2/(3p+1)) 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 O(ε-2/(3p+1) log (1/ε)), 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 O(ε-2/(3p+1)) p-th order oracle complexity.