有限和单调包含问题的最优预言机复杂度
Optimal Oracle Complexity for Finite-Sum Monotone Inclusions
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中文总结 AI 辅助
本文提出一种切换正则化方法，在均方Lipschitz条件下实现有限和单调包含问题的最优预言机复杂度，达到$\mathcal{O}(n+\sqrt{n}LR/\varepsilon)$的期望分量评估次数，并证明其匹配下界。
中文摘要 AI 辅助
我们提出了一种在均方Lipschitz连续性条件下，针对有限和单调包含问题的预言机最优方法。我们的切换正则化方法找到一个点$y$和一个证书$g\in G(y)$，使得$(\mathbb{E}\\|F(y)+g\\|^2)^{1/2}\le\varepsilon$，并使用$\mathcal{O}(n+\sqrt{n}LR/\varepsilon)$次期望分量评估和预解评估。该方法通过在正则化强度$L/\sqrt{n}$下切换到中心化随机近端迭代，消除了在每个正则化阶段重启方差缩减求解器所带来的附加$n\log n$成本。在剩余阶段之间携带算子估计，将其总成本限制为$\mathcal{O}(n)$。对于具有自适应停止和期望查询预算的随机线性跨度分量预言机算法，存在匹配的$\Omega(n+\sqrt{n}LR/\varepsilon)$下界。因此，对于$0<\varepsilon\le LR/2$，我们的方法在该预言机模型中达到了最优的最坏情况期望分量复杂度，直至通用常数。
英文摘要
We present an oracle-optimal method for finite-sum monotone inclusions under mean-square Lipschitz continuity. Our switching regularization method finds a point $y$ and a certificate $g\in G(y)$ with $(\mathbb{E}\|F(y)+g\|^2)^{1/2}\le\varepsilon$ using $\mathcal{O}(n+\sqrt{n}LR/\varepsilon)$ expected component evaluations and resolvent evaluations. It removes the additive $n\log n$ cost of restarting a variance-reduced solver at every regularization stage by switching to a centered stochastic proximal iteration at regularization strength $L/\sqrt{n}$. Carrying an operator estimate between the remaining stages limits their total cost to $\mathcal{O}(n)$. A matching $Ω(n+\sqrt{n}LR/\varepsilon)$ lower bound holds for randomized linear-span component-oracle algorithms with adaptive stopping and expected query budgets. Thus, for $0<\varepsilon\le LR/2$, our method attains the optimal worst-case expected component complexity in this oracle model, up to universal constants.
