Temporal horizons in forecasting: a performance-learnability trade-off

TL;DR

This paper investigates the trade-off in forecasting horizon length for autoregressive models, revealing how it affects training difficulty and forecast accuracy in chaotic and periodic systems, with implications for hyperparameter tuning.

Contribution

It formalizes the relationship between training horizon length and loss landscape geometry, providing theoretical insights and validation for optimal horizon selection in forecasting models.

Findings

Loss landscape roughness increases exponentially with horizon in chaotic systems.

Models trained on long horizons generalize better to short-term forecasts.

Practical guidelines for selecting training horizons based on system dynamics.

Abstract

When training autoregressive models to forecast dynamical systems, a critical question arises: how far into the future should the model be trained to predict? Too short a horizon may miss long-term trends, while too long a horizon can impede convergence due to accumulating prediction errors. In this work, we formalize this trade-off by analyzing how the geometry of the loss landscape depends on the training horizon. We prove that for chaotic systems, the loss landscape's roughness grows exponentially with the training horizon, while for limit cycles, it grows linearly, making long-horizon training inherently challenging. However, we also show that models trained on long horizons generalize well to short-term forecasts, whereas those trained on short horizons suffer exponentially (resp. linearly) worse long-term predictions in chaotic (resp. periodic) systems. We validate our theory through numerical experiments and discuss practical implications for selecting training horizons. Our results provide a principled foundation for hyperparameter optimization in autoregressive forecasting models.