发表机构
Yandex Research（扬德克斯研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对凸有限和优化问题，提出一种随机二阶算法，其期望二阶 oracle 复杂度为 $\tilde{\mathcal{O}}(n + n^{6/7}/\epsilon^{2/7})$，并证明该复杂度在满足线性跨度假设的随机二阶算法类中是最优的，解决了该问题类的开放性问题。
AI 中文摘要
本文考虑一个有限和优化问题，其中目标函数是凸的，$n$ 个分量函数是二次连续可微的，且它们的 Hessian 矩阵是均方 Lipschitz 连续的。我们的贡献有两方面。首先，我们开发了一种随机算法，该算法在期望上需要 $\tilde{\mathcal{O}}(n + n^{6/7}/\epsilon^{2/7})$ 次二阶 oracle 调用，以找到期望精度为 $\epsilon$ 的近似解。其次，我们为任何满足一阶线性跨度假设的随机二阶版本的随机二阶算法建立了 $\Omega(n + n^{6/7}/\epsilon^{2/7})$ 的下界复杂度，该假设在优化文献中被广泛采用。我们的下界和上界复杂度在忽略对数因子后相匹配，从而解决了建立该问题类最优复杂性的一个重要开放问题。
英文摘要
In this paper, we consider a finite-sum optimization problem, where the objective function is convex, the $n$ component functions are twice continuously differentiable and their Hessians are mean-square Lipschitz. Our contribution is twofold. First, we develop a stochastic algorithm, which requires $\tilde{\mathcal{O}}(n + n^{6/7}/ε^{2/7})$ second-order oracle calls in expectation to find an approximate solution to the problem with the expected accuracy $ε$. Second, we establish a lower complexity bound of $Ω(n + n^{6/7}/ε^{2/7})$ for any stochastic second-order algorithm satisfying a stochastic second-order version of the first-order linear span assumption, which is widely adopted in the optimization literature. Our lower and upper complexity bounds match up to logarithmic factors and thus resolve an important open question of establishing the optimal complexity of this problem class.
