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Logarithmic Regret and Polynomial Scaling in Online Multi-step-ahead Prediction

Jiachen Qian, Yang Zheng

TL;DR

This work studies online, model-free multi-step-ahead prediction for unknown linear stochastic systems. It derives that the optimal $\ ext{H}$-step-ahead predictor is a linear function of past inputs, past outputs, and future inputs, enabling an online ridge-learning algorithm that estimates the predictor weights. The main theoretical contribution is a regret guarantee: the online predictor achieves logarithmic regret in the time horizon $N$, with a constant that scales polynomially with the prediction horizon $\mathrm{H}$ according to the largest Jordan block size $\kappa$ of eigenvalue $1$ in $A$. The results advance data-driven forecasting and predictive control by providing explicit, almost-sure performance guarantees for multi-step predictions in linear stochastic systems.

Abstract

This letter studies the problem of online multi-step-ahead prediction for unknown linear stochastic systems. Using conditional distribution theory, we derive an optimal parameterization of the prediction policy as a linear function of future inputs, past inputs, and past outputs. Based on this characterization, we propose an online least-squares algorithm to learn the policy and analyze its regret relative to the optimal model-based predictor. We show that the online algorithm achieves logarithmic regret with respect to the optimal Kalman filter in the multi-step setting. Furthermore, with new proof techniques, we establish an almost-sure regret bound that does not rely on fixed failure probabilities for sufficiently large horizons $N$. Finally, our analysis also reveals that, while the regret remains logarithmic in $N$, its constant factor grows polynomially with the prediction horizon $H$, with the polynomial order set by the largest Jordan block of eigenvalue 1 in the system matrix.

Logarithmic Regret and Polynomial Scaling in Online Multi-step-ahead Prediction

TL;DR

This work studies online, model-free multi-step-ahead prediction for unknown linear stochastic systems. It derives that the optimal -step-ahead predictor is a linear function of past inputs, past outputs, and future inputs, enabling an online ridge-learning algorithm that estimates the predictor weights. The main theoretical contribution is a regret guarantee: the online predictor achieves logarithmic regret in the time horizon , with a constant that scales polynomially with the prediction horizon according to the largest Jordan block size of eigenvalue in . The results advance data-driven forecasting and predictive control by providing explicit, almost-sure performance guarantees for multi-step predictions in linear stochastic systems.

Abstract

This letter studies the problem of online multi-step-ahead prediction for unknown linear stochastic systems. Using conditional distribution theory, we derive an optimal parameterization of the prediction policy as a linear function of future inputs, past inputs, and past outputs. Based on this characterization, we propose an online least-squares algorithm to learn the policy and analyze its regret relative to the optimal model-based predictor. We show that the online algorithm achieves logarithmic regret with respect to the optimal Kalman filter in the multi-step setting. Furthermore, with new proof techniques, we establish an almost-sure regret bound that does not rely on fixed failure probabilities for sufficiently large horizons . Finally, our analysis also reveals that, while the regret remains logarithmic in , its constant factor grows polynomially with the prediction horizon , with the polynomial order set by the largest Jordan block of eigenvalue 1 in the system matrix.

Paper Structure

This paper contains 23 sections, 7 theorems, 91 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and problem statement
  3. Linear stochastic system and the Kalman filter
  4. Optimal one-step prediction policy
  5. Problem statement
  6. Main results
  7. Multi-step-ahead regression model
  8. Online learning and regret guarantee
  9. Technical Proofs
  10. Proof of \ref{['proposition:innovation']}
  11. Proof of \ref{['thm1']}
  12. Decomposition of regret
  13. Term-wise Analysis of the Regret
  14. Proof of \ref{['lemma:accumulation-factor']}
  15. Numerical experiments
  16. ...and 8 more sections

Key Result

Lemma 1

Consider the linear stochastic system eq: LinearSystem and the optimal $H$-step ahead prediction problem eq:MMSEProblemHstep. The optimal $H$-step ahead prediction $\bar{y}_{k+H}$ can be obtained recursively as where $\hat{x}_{k+1}\triangleq \mathbb{E}\left\{x_{k+1}\mid \mathcal{F}_{k}\right\}$ is the standard Kalman state estimation from eq: KalmanPredictor.

Figures (1)

  • Figure 1: Comparison of regret $\mathcal{R}_N$ across prediction horizons $H\in \{2,4,5,6\}$. The regret remains logarithmic for all $H$. The multiplicative constant increases nonlinearly with $H$.

Theorems & Definitions (8)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof