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Adaptive prediction theory combining offline and online learning

Haizheng Li, Lei Guo

TL;DR

The paper develops a rigorous two-stage framework that fuses offline nonlinear-least-squares learning with online meta-LMS adaptation to handle distribution shifts and parameter drift in nonlinear stochastic dynamical systems. It provides a Generalization Lemma giving KL-divergence–based bounds on offline generalization, and introduces a meta-LMS online method to track drifting parameters, with theoretical guarantees showing near-optimal prediction performance under typical conditions. Corollaries cover common time-series and control settings, and the results are complemented by proofs and numerical simulations illustrating improved transient behavior over purely offline or online methods. This work offers a principled foundation for reliable, scalable offline-online learning in safety-critical dynamic environments.

Abstract

Real-world intelligence systems usually operate by combining offline learning and online adaptation with highly correlated and non-stationary system data or signals, which, however, has rarely been investigated theoretically in the literature. This paper initiates a theoretical investigation on the prediction performance of a two-stage learning framework combining offline and online algorithms for a class of nonlinear stochastic dynamical systems. For the offline-learning phase, we establish an upper bound on the generalization error for approximate nonlinear-least-squares estimation under general datasets with strong correlation and distribution shift, leveraging the Kullback-Leibler divergence to quantify the distributional discrepancies. For the online-adaptation phase, we address, on the basis of the offline-trained model, the possible uncertain parameter drift in real-world target systems by proposing a meta-LMS prediction algorithm. This two-stage framework, integrating offline learning with online adaptation, demonstrates superior prediction performances compared with either purely offline or online methods. Both theoretical guarantees and empirical studies are provided.

Adaptive prediction theory combining offline and online learning

TL;DR

The paper develops a rigorous two-stage framework that fuses offline nonlinear-least-squares learning with online meta-LMS adaptation to handle distribution shifts and parameter drift in nonlinear stochastic dynamical systems. It provides a Generalization Lemma giving KL-divergence–based bounds on offline generalization, and introduces a meta-LMS online method to track drifting parameters, with theoretical guarantees showing near-optimal prediction performance under typical conditions. Corollaries cover common time-series and control settings, and the results are complemented by proofs and numerical simulations illustrating improved transient behavior over purely offline or online methods. This work offers a principled foundation for reliable, scalable offline-online learning in safety-critical dynamic environments.

Abstract

Real-world intelligence systems usually operate by combining offline learning and online adaptation with highly correlated and non-stationary system data or signals, which, however, has rarely been investigated theoretically in the literature. This paper initiates a theoretical investigation on the prediction performance of a two-stage learning framework combining offline and online algorithms for a class of nonlinear stochastic dynamical systems. For the offline-learning phase, we establish an upper bound on the generalization error for approximate nonlinear-least-squares estimation under general datasets with strong correlation and distribution shift, leveraging the Kullback-Leibler divergence to quantify the distributional discrepancies. For the online-adaptation phase, we address, on the basis of the offline-trained model, the possible uncertain parameter drift in real-world target systems by proposing a meta-LMS prediction algorithm. This two-stage framework, integrating offline learning with online adaptation, demonstrates superior prediction performances compared with either purely offline or online methods. Both theoretical guarantees and empirical studies are provided.

Paper Structure

This paper contains 24 sections, 10 theorems, 108 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions
  4. Problem Formulation
  5. Notation
  6. Offline-learning phase
  7. Online-adaptation phase
  8. Main Results
  9. Generalization Lemma
  10. Corollaries of Generalization Lemma \ref{['gle']}
  11. Theorems on the prediction ability
  12. Further discussions
  13. Proofs of the main theorems
  14. Proof of Theorem \ref{['thm1']}
  15. Proof of Theorem \ref{['thm1.1']}
  16. ...and 9 more sections

Key Result

Lemma 2.9

\newlabelpredlemma0 If the learning rate $\lambda<\frac{1}{2(M_f+AB+W_{\max})^2}$, then the average prediction error produced by meta-LMS prediction Algorithm pred1 satisfies: and,

Figures (2)

  • Figure 1: Average prediction error
  • Figure 2: Parameter estimation error of $(b,c)$ in Algorithm 2.1

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 2.9
  • Lemma 3.1
  • ...and 16 more