Temporal horizons in forecasting: a performance-learnability trade-off
This provides a principled foundation for hyperparameter optimization in autoregressive forecasting models, addressing a critical trade-off for practitioners in time-series analysis.
The paper tackles the problem of selecting training horizons for autoregressive models in forecasting dynamical systems, showing that for chaotic systems, loss landscape roughness grows exponentially with horizon, while for limit cycles it grows linearly, and models trained on long horizons generalize better to short-term forecasts.
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.