用随机采样优化依赖参数的矩阵迹，高效且有理论保障。
Stochastic Trace Optimization of Parameter Dependent Matrices Based on Statistical Learning Theory
- 基于统计学习理论设计蒙特卡洛估计器，可高效估算矩阵迹。
- 采样量小，尤其对非对角线质量小的矩阵和小参数空间更优。
- 理论严谨，适合关注优化算法可靠性的研究者。
我们研究依赖参数 θ（来自紧参数空间 Θ）的方阵 A(θ) ∈ R^{m×m}，提出一种蒙特卡洛估计器来最小化 trace(A(θ))，并确定采样数量，使得估计器的后向误差在高概率下有界。推导出两类界：基于 ε-网和通用链式方法。两类界均表明，当矩阵非对角线质量小、参数空间 Θ 小时，所需采样量较少，且对矩阵维数 m 的依赖弱或不明显。ε-网界的计算较简单，常数明确；而链式界依赖难以计算的 Talagrand 函数，仅在极特殊情况下可求解。两类界比较困难，但文献表明链式界可能更优。
原文摘要 · Abstract (English)
We consider matrices $\boldsymbol{A}(\boldsymbolθ)\in\mathbb{R}^{m\times m}$ that depend, possibly nonlinearly, on a parameter $\boldsymbolθ$ from a compact parameter space $Θ$. We present a Monte Carlo estimator for minimizing $\text{trace}(\boldsymbol{A}(\boldsymbolθ))$ over all $\boldsymbolθ\inΘ$, and determine the sampling amount so that the backward error of the estimator is bounded with high probability. We derive two types of bounds, based on epsilon nets and on generic chaining. Both types predict a small sampling amount for matrices $\boldsymbol{A}(\boldsymbolθ)$ with small offdiagonal mass, and parameter spaces $Θ$ of small ``size.'' Dependence on the matrix dimension~$m$ is only weak or not explicit. The bounds based on epsilon nets are easier to evaluate and come with fully specified constants. In contrast, the bounds based on chaining depend on the Talagrand functionals which are difficult to evaluate, except in very special cases. Comparisons between the two types of bounds are difficult, although the literature suggests that chaining bounds can be superior.
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