Statistical Efficiency of Single- and Multi-step Models for Forecasting and Control

TL;DR

This paper analyzes the trade-offs between single-step and multi-step models for forecasting and control in linear systems, showing when each approach is preferable based on model accuracy and observability.

Contribution

It provides the first quantitative analysis of the benefits and limitations of single- and multi-step predictors in linear dynamical systems, including theoretical and empirical insights.

Findings

Single-step models excel with well-specified dynamics.

Multi-step predictors reduce bias under partial observability.

Trade-offs persist in closed-loop control scenarios.

Abstract

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.