Online Nonstochastic Prediction: Logarithmic Regret via Predictive Online Least Squares

TL;DR

This paper develops an online least squares prediction method for linear systems that achieves logarithmic regret even with unbounded trajectories, using stabilizing hints and model knowledge.

Contribution

It introduces an unconstrained online least squares approach with stabilizing hints, enabling logarithmic regret in marginally stable systems with or without model knowledge.

Findings

Logarithmic regret achieved with stabilizing hints in systems with unbounded trajectories.

Universal hints enable model-free prediction with logarithmic regret.

Method adapts to nonstochastic disturbances, outperforming classical fixed-gain observers.

Abstract

We study online prediction for marginally stable, partially observed linear dynamical systems under nonstochastic disturbances. Our objective is to minimize the cumulative squared prediction loss and compete with the best-in-hindsight Luenberger predictor. Standard online learning methods typically rely on bounded domains/gradients, and thus their guarantees may fail to deal with potentially unbounded trajectories in marginally stable systems. In this paper, we introduce an unconstrained online least squares method that stabilizes the learning process via tailored predictive hints. With model knowledge, we prove that hints constructed from any stabilizing Luenberger predictor render the hint residuals uniformly bounded, achieving logarithmic regret despite unbounded trajectory growth. We also discuss model-free prediction and introduce a simple universal hint for symmetric systems, under which logarithmic regret is maintained without model knowledge. Our results provide an adaptive, instance-wise optimal online predictor compared to classical fixed-gain observers under nonstochastic disturbances.