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Broad stochastic configuration residual learning system for norm-convergent universal approximation

Han Su, Zhongyan Li, Wanquan Liu

TL;DR

The paper addresses the fragility of universal approximation in randomized broad learning networks, where BRLS relies on probabilistic error convergence rather than norm convergence. It introduces Broad stochastic configuration residual learning system (BSCRLS), integrating a supervisory mechanism to adaptively constrain random parameters so that $\lim_{m\to\infty} \|\mathbf{F}-\mathbf{F}_m\|_2 = 0$, and provides a formal theorem validating norm-convergent universal approximation. Three incremental BSCRLS variants enable enhancement, feature, or data-driven updates, broadening applicability to dynamic tasks. Empirical results on the SPDD solar-panel dust dataset show BSCRLS achieving higher accuracy and competitive training times compared with BRLS and 13 baselines, demonstrating practical robustness and scalability. Overall, BSCRLS offers a rigorous, fast alternative to deep networks with provable universal approximation guarantees in norm, enabling reliable performance in real-world monitoring applications.

Abstract

Universal approximation serves as the foundation of neural network learning algorithms. However, some networks establish their universal approximation property by demonstrating that the iterative errors converge in probability measure rather than the more rigorous norm convergence, which makes the universal approximation property of randomized learning networks highly sensitive to random parameter selection, Broad residual learning system (BRLS), as a member of randomized learning models, also encounters this issue. We theoretically demonstrate the limitation of its universal approximation property, that is, the iterative errors do not satisfy norm convergence if the selection of random parameters is inappropriate and the convergence rate meets certain conditions. To address this issue, we propose the broad stochastic configuration residual learning system (BSCRLS) algorithm, which features a novel supervisory mechanism adaptively constraining the range settings of random parameters on the basis of BRLS framework, Furthermore, we prove the universal approximation theorem of BSCRLS based on the more stringent norm convergence. Three versions of incremental BSCRLS algorithms are presented to satisfy the application requirements of various network updates. Solar panels dust detection experiments are performed on publicly available dataset and compared with 13 deep and broad learning algorithms. Experimental results reveal the effectiveness and superiority of BSCRLS algorithms.

Broad stochastic configuration residual learning system for norm-convergent universal approximation

TL;DR

The paper addresses the fragility of universal approximation in randomized broad learning networks, where BRLS relies on probabilistic error convergence rather than norm convergence. It introduces Broad stochastic configuration residual learning system (BSCRLS), integrating a supervisory mechanism to adaptively constrain random parameters so that , and provides a formal theorem validating norm-convergent universal approximation. Three incremental BSCRLS variants enable enhancement, feature, or data-driven updates, broadening applicability to dynamic tasks. Empirical results on the SPDD solar-panel dust dataset show BSCRLS achieving higher accuracy and competitive training times compared with BRLS and 13 baselines, demonstrating practical robustness and scalability. Overall, BSCRLS offers a rigorous, fast alternative to deep networks with provable universal approximation guarantees in norm, enabling reliable performance in real-world monitoring applications.

Abstract

Universal approximation serves as the foundation of neural network learning algorithms. However, some networks establish their universal approximation property by demonstrating that the iterative errors converge in probability measure rather than the more rigorous norm convergence, which makes the universal approximation property of randomized learning networks highly sensitive to random parameter selection, Broad residual learning system (BRLS), as a member of randomized learning models, also encounters this issue. We theoretically demonstrate the limitation of its universal approximation property, that is, the iterative errors do not satisfy norm convergence if the selection of random parameters is inappropriate and the convergence rate meets certain conditions. To address this issue, we propose the broad stochastic configuration residual learning system (BSCRLS) algorithm, which features a novel supervisory mechanism adaptively constraining the range settings of random parameters on the basis of BRLS framework, Furthermore, we prove the universal approximation theorem of BSCRLS based on the more stringent norm convergence. Three versions of incremental BSCRLS algorithms are presented to satisfy the application requirements of various network updates. Solar panels dust detection experiments are performed on publicly available dataset and compared with 13 deep and broad learning algorithms. Experimental results reveal the effectiveness and superiority of BSCRLS algorithms.

Paper Structure

This paper contains 17 sections, 4 theorems, 40 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Broad residual learning system
  4. Stochastic configuration network
  5. Broad stochastic configuration residual learning system
  6. Limitations of universal approximation property in BRLS
  7. Universal approximation property of BSCRLS
  8. BSCRLS algorithm description
  9. Basic BSCRLS
  10. Incremental BSCRLS
  11. Experiments: Solar Panels Dust Detection
  12. Experimental introduction and dataset
  13. Data cleaning and preprocessing
  14. Experiment 1: Performance comparison of BSCRLS with other algorithms on SPDD dataset
  15. Experiment 2: Performance comparison of BSCRLS with universal approximation property converging in norm and BRLS converging in probability measure
  16. ...and 2 more sections

Key Result

Theorem 2.1

For any compact set $\boldsymbol{S}\subset\boldsymbol{I}^d$ and any vector-valued continuous function $\boldsymbol{f}=[f_1,f_2,\dots,f_c]$, $f_r\in C(\boldsymbol{I}^d)$, $r=1,\dots,c$, there exists a sequence of $\{\boldsymbol{f}_{\boldsymbol{\omega}_{n,m}}\}$ in BRLS denoted as $\boldsymbol{f}_{\bo where $\rho_{\boldsymbol{S}}$ is the distance between $\boldsymbol{f}$ and $\boldsymbol{f}_{\boldsy

Figures (2)

  • Figure 1: Some sample images of solar panels dust detection
  • Figure 2: Performance comparison of BSCRLS with convergence in norm and BRLS with convergence in probability measure

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2