用于有限和求根问题的方差缩减快速Krasnoselkii-Mann方法
Variance-Reduced Fast Krasnoselkii-Mann Methods for Finite-Sum Root-Finding Problems
AI总结：
提出一类带方差缩减的快速Krasnoselkii-Mann方法，求解有限和求根问题，实现O(1/k^2)收敛速率及最优预言机复杂度，并扩展至包含问题。
摘要：
我们提出了一类新的带有方差缩减的快速Krasnoselkii--Mann方法，用于求解有限和余强制方程$Gx = 0$。我们的算法是单循环的，并利用了一族新的无偏方差缩减估计器，这些估计器专门为更广泛的求根算法类别而设计。我们的方法在$\mathbb{E}[\\| Gx^k\\|^2]$方面实现了$\mathcal{O}(1/k^2)$和$o(1/k^2)$的末次迭代收敛速率，其中$k$是迭代计数器，$\mathbb{E}[\\.]$是总期望。我们还建立了几乎必然的$o(1/k^2)$收敛速率以及迭代序列$\{x^k\}$几乎必然收敛到$Gx=0$的解。我们将框架实例化用于两种著名的估计器：SVRG和SAGA。通过适当选择参数，两种变体在达到$\\.epsilon$-解时均获得$\\.mathcal{O}(n + n^{2/3}\epsilon^{-1})$的预言机复杂度，其中$n$表示有限和算子$G$中的项数。此外，在$\sigma$-强拟单调性下，我们的方法实现了线性收敛速率和$\\.mathcal{O}(n+ \max\{n, n^{2/3}\kappa\} \log(\frac{1}{\epsilon}))$的预言机复杂度，其中$\kappa:= L/\sigma$。我们将方法扩展到求解一类有限和包含问题（可能非单调），证明了我们的方案在方程设置中保持相同的理论保证。最后，数值实验验证了我们的算法，并展示了与最先进方法相比其有前景的性能。
英文摘要：
We propose a new class of fast Krasnoselkii--Mann methods with variance reduction to solve a finite-sum co-coercive equation $Gx = 0$. Our algorithm is single-loop and leverages a new family of unbiased variance-reduced estimators specifically designed for a wider class of root-finding algorithms. Our method achieves both $\mathcal{O}(1/k^2)$ and $o(1/k^2)$ last-iterate convergence rates in terms of $\mathbb{E}[\| Gx^k\|^2]$, where $k$ is the iteration counter and $\mathbb{E}[\cdot]$ is the total expectation. We also establish almost sure $o(1/k^2)$ convergence rates and the almost sure convergence of iterates $\{x^k\}$ to a solution of $Gx=0$. We instantiate our framework for two prominent estimators: SVRG and SAGA. By an appropriate choice of parameters, both variants attain an oracle complexity of $\mathcal{O}(n + n^{2/3}ε^{-1})$ to reach an $ε$-solution, where $n$ represents the number of summands in the finite-sum operator $G$. Furthermore, under $σ$-strong quasi-monotonicity, our method achieves a linear convergence rate and an oracle complexity of $\mathcal{O}(n+ \max\{n, n^{2/3}κ\} \log(\frac{1}ε))$, where $κ:= L/σ$. We extend our approach to solve a class of finite-sum inclusions (possibly nonmonotone), demonstrating that our schemes retain the same theoretical guarantees as in the equation setting. Finally, numerical experiments validate our algorithms and demonstrate their promising performance compared to state-of-the-art methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
