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

Authors: Anne Somalwar, Bruce D. Lee, George J. Pappas, Nikolai Matni Date: 2026-03-24 Paper ID: openalex:2603.23465

Summary

This paper analyzes the statistical efficiency trade-off between single-step and multi-step predictive models, particularly in the context of learning-based control where compounding error is a major concern. The authors compare three model classes: single-step, direct multi-step, and single-step models trained with multi-step losses, focusing on linear dynamical systems. They theoretically establish that when the system dynamics are perfectly captured, single-step models yield the lowest asymptotic error. Conversely, for partially observable systems where the model is misspecified, direct multi-step predictors offer substantial improvements by reducing prediction bias. These observed trade-offs are shown to be relevant even when the predictors are integrated into closed-loop control structures.

Key Contributions

• Provided the first quantitative analysis of the trade-off between single-step and multi-step predictors for linear dynamical systems.
• Showed that well-specified single-step models achieve the lowest asymptotic prediction error.
• Demonstrated that direct multi-step predictors significantly reduce bias and improve accuracy when the model class suffers from partial observability (model misspecification).
• Provided theoretical and empirical evidence that these trade-offs hold when predictors are deployed in closed-loop control.

Limitations

The theoretical analysis is primarily focused on linear dynamical systems. The persistence of these trade-offs in highly nonlinear systems remains an open question.

Open Questions & Future Work

• lqr-cost-comparison-multi-step-predictor
• rigorous-nonlinear-system-analysis
• closed-loop-control-beyond-lqr

Archivist Review

Archivist review kept only candidates judged central to the paper and reusable across future work. Approved 0 concept(s), 3 open question(s), and 0 dataset(s), with 0 rejected candidate note(s).

Approved Open Questions

• Analyze closed-loop LQR comparison: This comparison is crucial for practitioners to fully understand the trade-offs in control performance across all three model classes, especially since the prediction error ranking (Prop II.4) contradicted the LQR cost ranking observed between the intermediate and multi-step predictors (Section IV-A).
• Rigorous nonlinear system analysis: Extending the rigorous statistical efficiency guarantees to nonlinear systems would validate the general applicability of the findings beyond linear time-invariant models, which is a significant limitation of the current theoretical work.
• Analyze control performance beyond LQR: Moving beyond the LQR objective allows the findings to inform model design choices in the broader and more complex domain of policy optimization via model-based reinforcement learning.

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