符号扰动和方法的样本复杂度
Sample Complexity of the Sign-Perturbed Sums Method
AI总结：
本文研究符号扰动和（SPS）方法在一般线性回归问题中的样本复杂度，建立了有限样本下置信区域直径的高概率上界，证明其收缩速率与经典渐近置信椭球相同且达到最优，并通过实验比较了理论界与经验尺寸的差异。
摘要：
我们研究了符号扰动和（SPS）方法的样本复杂度，该方法在温和的统计假设下（例如独立且对称的噪声项）为真实系统参数构造精确的、非渐近的置信区域。SPS 的标准版本处理线性回归问题，但它可以推广到随机线性（动态）系统，甚至包括闭环设置，以及非线性和非参数问题。尽管该方法的强一致性已被严格证明，但该算法的样本复杂度此前仅针对标量线性回归问题进行了分析。在本文中，我们研究了一般线性回归问题的 SPS 样本复杂度。我们为有限样本量下 SPS 置信区域的直径建立了高概率上界，并证明 SPS 区域以与经典渐近置信椭球相同的最优速率收缩。最后，我们通过实验研究了 SPS 置信区域的理论界与经验尺寸之间的差异。
英文摘要：
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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