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Sample Complexity of the Sign-Perturbed Sums Method

Szabolcs Szentpéteri, Balázs Csanád Csáji

TL;DR

This work provides a non-asymptotic, distribution-free analysis of the Sign-Perturbed Sums (SPS) method for general linear regression with subgaussian, symmetric noises. It derives high-probability upper bounds on the diameters of SPS confidence regions that shrink at the optimal rate $O(1/\sqrt{n})$, matching classical asymptotic ellipsoids in rate while remaining valid for finite samples. The results rely on a two-step decomposition of SPS regions into ellipsoidal forms and leverages concentration inequalities for both the numerator and denominator terms. Simulations demonstrate that the theoretical bounds capture the correct decay trend, though they are conservative due to their data-agnostic, concentration-based nature. The paper lays groundwork for extending SPS to ellipsoidal outer-approximations and to dynamic, potentially closed-loop, systems.

Abstract

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.

Sample Complexity of the Sign-Perturbed Sums Method

TL;DR

This work provides a non-asymptotic, distribution-free analysis of the Sign-Perturbed Sums (SPS) method for general linear regression with subgaussian, symmetric noises. It derives high-probability upper bounds on the diameters of SPS confidence regions that shrink at the optimal rate , matching classical asymptotic ellipsoids in rate while remaining valid for finite samples. The results rely on a two-step decomposition of SPS regions into ellipsoidal forms and leverages concentration inequalities for both the numerator and denominator terms. Simulations demonstrate that the theoretical bounds capture the correct decay trend, though they are conservative due to their data-agnostic, concentration-based nature. The paper lays groundwork for extending SPS to ellipsoidal outer-approximations and to dynamic, potentially closed-loop, systems.

Abstract

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.
Paper Structure (12 sections, 7 theorems, 81 equations, 1 figure, 2 algorithms)

This paper contains 12 sections, 7 theorems, 81 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Data Generation
  4. Assumptions
  5. The Sign-Perturbed Sums Algorithm
  6. Sample Complexity of SPS
  7. Simulation Experiments
  8. Conclusion
  9. Proof of Lemma \ref{['lemma:M_proj']}
  10. Proof of Lemma \ref{['lemma:numerator_indicator']}
  11. Proof of Lemma \ref{['lemma:denominator_indicator']}
  12. Proof of Lemma \ref{['lemma:sample_complex_indicator_m2_q1']}

Key Result

Theorem 1

Assuming the noise sequence $\{W_t\}$contains independent random variables that are symmetric about zero and $\bar{R}_n$ is nonsingular, the confidence probability of the confidence region $\mathcal{C}_{p,n}$ is exactly $p$,

Figures (1)

  • Figure 1: Comparison of the empirical sizes and the theoretical upper bound on the sizes of the SPS-Indicator region for $m=2$, $\delta=0.1$, $t_0=250$, $n=2000$ and $s =100$.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • proof
  • proof : Proof of Theorem \ref{['thm:sample_complex_indicator']}
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 1