Sign-Perturbed Sum (SPS) is a powerful finite-sample system identification algorithm which can construct confidence regions for the true data generating system with exact coverage probabilities, for any finite sample size. SPS was developed in a series of papers and it has a wide range of applications, from general linear systems, even in a closed-loop setup, to nonlinear and nonparametric approaches. Although several theoretical properties of SPS were proven in the literature, the sample complexity of the method was not analysed so far. This paper aims to fill this gap and provides the first results on the sample complexity of SPS. Here, we focus on scalar linear regression problems, that is we study the behaviour of SPS confidence intervals. We provide high probability upper bounds, under three different sets of assumptions, showing that the sizes of SPS confidence intervals shrink at a geometric rate around the true parameter, if the observation noises are subgaussian. We also show that similar bounds hold for the previously proposed outer approximation of the confidence region. Finally, we present simulation experiments comparing the theoretical and the empirical convergence rates.

翻译：符号扰动和（SPS）是一种强大的有限样本系统辨识算法，能够在任意有限样本量下，以精确覆盖概率构建真实数据生成系统的置信区域。SPS 在一系列论文中得到发展，具有广泛的应用范围，涵盖一般线性系统（甚至在闭环设置下），直至非线性与非参数方法。尽管文献中已证明了 SPS 的多项理论性质，但该方法的样本复杂度此前尚未得到分析。本文旨在填补这一空白，首次给出 SPS 样本复杂度的相关结果。在此，我们聚焦于标量线性回归问题，即研究 SPS 置信区间的行为。我们在三组不同假设条件下，给出了高概率上界，表明若观测噪声服从次高斯分布，则 SPS 置信区间的尺寸会以几何速率围绕真实参数缩小。我们还证明，先前提出的置信区域外部逼近方法也满足类似的上界。最后，我们通过仿真实验比较了理论收敛速率与经验收敛速率。