论文标题
平滑凸优化中高级方法的第一个最佳加速度
The First Optimal Acceleration of High-Order Methods in Smooth Convex Optimization
论文作者
论文摘要
In this paper, we study the fundamental open question of finding the optimal high-order algorithm for solving smooth convex minimization problems. Arjevani等。 (2019) established the lower bound $Ω\left(ε^{-2/(3p+1)}\right)$ 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 $p$.但是，Gasnikov等人的现有最新高级方法。 （2019b）; Bubeck等。 （2019）;江等。 （2019）实现Oracle复杂度$ \ MATHCAL {O} \ left（ε^{ - 2/（3p+1）} \ log（1/ε）\ right）$，这与下限不匹配。 The reason for this is that these algorithms require performing a complex binary search procedure, which makes them neither optimal nor practical.我们通过提供$ \ MATHCAL {O} \ left（ε^{ - 2/（3p+1）} \ right）$ p $ $ p $ ther Oracle oracle Complactity的第一个算法来解决此基本问题。
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 $Ω\left(ε^{-2/(3p+1)}\right)$ 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 $\mathcal{O}\left(ε^{-2/(3p+1)} \log (1/ε)\right)$, 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 $\mathcal{O}\left(ε^{-2/(3p+1)}\right)$ $p$-th order oracle complexity.
