Statistical Efficiency of Single- and Multi-step Models for Forecasting and Control
This work addresses a key trade-off in forecasting and control for linear dynamical systems, providing theoretical insights that are incremental but specific to model specification scenarios.
The paper tackles the problem of compounding error in learning-based control by analyzing when multi-step predictors outperform single-step models, showing that single-step models have lower asymptotic error in well-specified cases, but multi-step predictors reduce bias and improve accuracy under misspecification due to partial observability.
Compounding error, where small prediction mistakes accumulate over time, presents a major challenge in learning-based control. A common remedy is to train multi-step predictors directly instead of rolling out single-step models. However, it is unclear when the benefits of multi-step predictors outweigh the difficulty of learning a more complex model. We provide the first quantitative analysis of this trade-off for linear dynamical systems. We study three predictor classes: (i) single step models, (ii) multi-step models, and (iii) single step models trained with multi-step losses. We show that when the model class is well-specified and accurately captures the system dynamics, single-step models achieve the lowest asymptotic prediction error. On the other hand, when the model class is misspecified due to partial observability, direct multi-step predictors can significantly reduce bias and improve accuracy. We provide theoretical and empirical evidence that these trade-offs persist when predictors are used in closed-loop control.