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$\ell_1$-Based Adaptive Identification under Quantized Observations with Applications

Xin Zheng, Yifei Jin, Yujing Liu, Lei Guo

TL;DR

This work addresses adaptive identification with quantized observations by introducing an $\ll_1$-norm based algorithm that does not rely on persistent excitation. The proposed ADA method employs a two-step, projection-based, quasi-Newton update to achieve global convergence of the parameter estimates and vanishing average regret, even under non-PE data conditions. Theoretical results quantify convergence rates and regret behavior, while a real-world probation decision dataset demonstrates its practical advantage over $\ll_2$-based approaches. The approach broadens reliable identification to systems with quantized measurements and feedback, with immediate implications for data-driven sentencing and other quantized-data applications.

Abstract

Quantized observations are ubiquitous in a wide range of applications across engineering and the social sciences, and algorithms based on the $\ell_1$-norm are well recognized for their robustness to outliers compared with their $\ell_2$-based counterparts. Nevertheless, adaptive identification methods that integrate quantized observations with $\ell_1$-optimization remain largely underexplored. Motivated by this gap, we develop a novel $\ell_1$-based adaptive identification algorithm specifically designed for quantized observations. Without relying on the traditional persistent excitation condition, we establish global convergence of the parameter estimates to their true values and show that the average regret asymptotically vanishes as the data size increases. Finally, we apply our new identification algorithm to a judicial sentencing problem using real-world data, which demonstrates its superior performance and practical significance.

$\ell_1$-Based Adaptive Identification under Quantized Observations with Applications

TL;DR

This work addresses adaptive identification with quantized observations by introducing an -norm based algorithm that does not rely on persistent excitation. The proposed ADA method employs a two-step, projection-based, quasi-Newton update to achieve global convergence of the parameter estimates and vanishing average regret, even under non-PE data conditions. Theoretical results quantify convergence rates and regret behavior, while a real-world probation decision dataset demonstrates its practical advantage over -based approaches. The approach broadens reliable identification to systems with quantized measurements and feedback, with immediate implications for data-driven sentencing and other quantized-data applications.

Abstract

Quantized observations are ubiquitous in a wide range of applications across engineering and the social sciences, and algorithms based on the -norm are well recognized for their robustness to outliers compared with their -based counterparts. Nevertheless, adaptive identification methods that integrate quantized observations with -optimization remain largely underexplored. Motivated by this gap, we develop a novel -based adaptive identification algorithm specifically designed for quantized observations. Without relying on the traditional persistent excitation condition, we establish global convergence of the parameter estimates to their true values and show that the average regret asymptotically vanishes as the data size increases. Finally, we apply our new identification algorithm to a judicial sentencing problem using real-world data, which demonstrates its superior performance and practical significance.
Paper Structure (10 sections, 11 theorems, 60 equations, 1 figure, 1 algorithm)

This paper contains 10 sections, 11 theorems, 60 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Quantized Observation Model
  4. Notations and Assumptions
  5. The Main Results
  6. Adaptive Algorithm
  7. The Main Theorems
  8. Application to Probation Decision Problems
  9. Conclusion
  10. Proof of The Theorems

Key Result

Theorem 1

If Assumptions threshold--symmetry hold, then the estimation error has the following upper bound as $k \rightarrow \infty$: where $\tilde{\theta}_{k+1} = \theta - \theta_{k+1}$, $\lambda_{\min}(k)$ and $\lambda_{\max}(k)$ are the smallest and largest eigenvalues of the matrix $P_0^{-1} + \sum\limits_{i=0}^{k} \phi_i \phi_i^{\mathrm{T}}$, respectively.

Figures (1)

  • Figure 1: Comparison of adaptive prediction accuracy.

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Definition 1
  • Remark 3
  • Theorem 1
  • Remark 4
  • Theorem 2
  • Remark 5
  • Lemma 1
  • Lemma 2
  • ...and 7 more