Learning with Imperfect Models: When Multi-step Prediction Mitigates Compounding Error

Compounding error, where small prediction mistakes accumulate over time, presents a major challenge in learning-based control. For example, this issue often limits the performance of model-based reinforcement learning and imitation learning. One common approach to mitigate compounding error is to train multi-step predictors directly, rather than relying on autoregressive rollout of a single-step model. However, it is not well understood when the benefits of multi-step prediction outweigh the added complexity of learning a more complicated model. In this work, we provide a rigorous analysis of this trade-off in the context of linear dynamical systems. We show that when the model class is well-specified and accurately captures the system dynamics, single-step models achieve lower 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 thus outperform single-step approaches. These theoretical results are supported by numerical experiments, wherein we also (a) empirically evaluate an intermediate strategy which trains a single-step model using a multi-step loss and (b) evaluate performance of single step and multi-step predictors in a closed loop control setting.