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Gradient-Based Adaptive Prediction and Control for Nonlinear Dynamical Systems

Yujing Liu, Xin Zheng, Zhixin Liu, Lei Guo

TL;DR

This work addresses adaptive prediction and control for nonlinear stochastic dynamical systems under a weak convexity condition on the prediction-based loss, removing the need for persistent excitation. It introduces a gradient-based adaptive predictor with a modified step-size $\mu_k$ that accelerates convergence, and proves global convergence with rates faster than classical stochastic gradient methods. Building on this, the authors derive a convergence rate for the closed-loop control error under a nonlinear minimum-phase condition with linearly growing nonlinearities. The approach is validated on a real judicial-sentencing dataset and through a nonlinear simulation, showing faster decay of prediction error and improved tracking compared with standard SGD, and highlighting applicability to a wide class of nonlinear activations and losses through the weak-convexity framework.

Abstract

This paper investigates gradient-based adaptive prediction and control for nonlinear stochastic dynamical systems under a weak convexity condition on the prediction-based loss. This condition accommodates a broad range of nonlinear models in control and machine learning such as saturation functions, sigmoid, ReLU and tanh activation functions, and standard classification models. Without requiring any persistent excitation of the data, we establish global convergence of the proposed adaptive predictor and derive explicit rates for its asymptotic performance. Furthermore, under a classical nonlinear minimum-phase condition and with a linear growth bound on the nonlinearities, we establish the convergence rate of the resulting closed-loop control error. Finally, we demonstrate the effectiveness of the proposed adaptive prediction algorithm on a real-world judicial sentencing dataset. The adaptive control performance will also be evaluated via a numerical simulation.

Gradient-Based Adaptive Prediction and Control for Nonlinear Dynamical Systems

TL;DR

This work addresses adaptive prediction and control for nonlinear stochastic dynamical systems under a weak convexity condition on the prediction-based loss, removing the need for persistent excitation. It introduces a gradient-based adaptive predictor with a modified step-size that accelerates convergence, and proves global convergence with rates faster than classical stochastic gradient methods. Building on this, the authors derive a convergence rate for the closed-loop control error under a nonlinear minimum-phase condition with linearly growing nonlinearities. The approach is validated on a real judicial-sentencing dataset and through a nonlinear simulation, showing faster decay of prediction error and improved tracking compared with standard SGD, and highlighting applicability to a wide class of nonlinear activations and losses through the weak-convexity framework.

Abstract

This paper investigates gradient-based adaptive prediction and control for nonlinear stochastic dynamical systems under a weak convexity condition on the prediction-based loss. This condition accommodates a broad range of nonlinear models in control and machine learning such as saturation functions, sigmoid, ReLU and tanh activation functions, and standard classification models. Without requiring any persistent excitation of the data, we establish global convergence of the proposed adaptive predictor and derive explicit rates for its asymptotic performance. Furthermore, under a classical nonlinear minimum-phase condition and with a linear growth bound on the nonlinearities, we establish the convergence rate of the resulting closed-loop control error. Finally, we demonstrate the effectiveness of the proposed adaptive prediction algorithm on a real-world judicial sentencing dataset. The adaptive control performance will also be evaluated via a numerical simulation.
Paper Structure (11 sections, 2 theorems, 54 equations, 3 figures)

This paper contains 11 sections, 2 theorems, 54 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notations
  4. Model and Assumptions
  5. Main Results
  6. Adaptive Prediction
  7. Adaptive Control
  8. Experiments
  9. Prediction of Judicial Sentencing based on Real-Data
  10. Prediction and Control for a Simulation Example
  11. Conclusion

Key Result

Theorem 1

Under Assumptions asm1--asm2, if $\beta_3>1$ and either $\beta_1=\tfrac{1}{2},\, \beta_2>\tfrac{1}{2}$ or $\beta_1 \in (\tfrac{1}{2},1]$, $\, \beta_2=0$, then there exists a finite random variable $c$ such that where $\tilde{\theta}_k=\theta_k-\theta^*$. Moreover, if for any compact set $D$, there exists a non-decreasing positive and deterministic sequence $\{d_n\}$ such that then we have

Figures (3)

  • Figure 1: Comparison of prediction accuracy.
  • Figure 2: Comparison of average regret.
  • Figure 3: Comparison of control performance.

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Theorem 2