提出SPS方法在一般线性回归中的样本复杂度分析，证明其置信区域收缩率达最优。
Sample Complexity of the Sign-Perturbed Sums Method
- 基于符号扰动法构造非渐近精确置信区域，适用于广义线性模型
- 在有限样本下建立置信区域直径的高概率上界，收敛速度达理论最优
- 实验验证理论边界与实际区域大小差异，支持方法可靠性
我们研究了符号扰动和（Sign-Perturbed Sums, SPS）方法的样本复杂度，该方法在较弱统计假设（如独立对称噪声）下可构建精确的非渐近置信区域。标准SPS用于线性回归问题，但可推广至随机线性系统（含闭环设置）及非线性、非参数问题。尽管该方法的强一致性已严格证明，其样本复杂度仅在标量线性回归中被分析。本文研究了通用线性回归下的SPS样本复杂度，建立了有限样本下置信区域直径的高概率上界，表明SPS置信区域以与经典渐近置信椭球相同的最优速率收缩。最后，通过实验分析了理论边界与实际置信区域大小之间的差异。
原文摘要 · Abstract (English)
We study the sample complexity of the Sign-Perturbed Sums (SPS) method, which constructs exact, non-asymptotic confidence regions for the true system parameters under mild statistical assumptions, such as independent and symmetric noise terms. The standard version of SPS deals with linear regression problems, however, it can be generalized to stochastic linear (dynamical) systems, even with closed-loop setups, and to nonlinear and nonparametric problems, as well. Although the strong consistency of the method was rigorously proven, the sample complexity of the algorithm was only analyzed so far for scalar linear regression problems. In this paper we study the sample complexity of SPS for general linear regression problems. We establish high probability upper bounds for the diameters of SPS confidence regions for finite sample sizes and show that the SPS regions shrink at the same, optimal rate as the classical asymptotic confidence ellipsoids. Finally, the difference between the theoretical bounds and the empirical sizes of SPS confidence regions is investigated experimentally.
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