Exploring Accuracy Law for Deep Time Series Forecasters: An Empirical Study

Deep time series forecasting has emerged as a rapidly growing field in recent years. Despite the exponential growth of community interests, progress on standard benchmarks is often limited to marginal improvements. A common consensus of the community is that time series forecasting inherently faces a non-zero error lower bound due to its partially observable and uncertain nature. However, a fundamental question arises: how to estimate the performance upper bound of deep time series forecasters? We delve into univariate time series forecasting, a prevalent forecasting paradigm spanning traditional statistical models to advanced time series foundation models. Going beyond classical series-wise predictability metrics, we realize that the forecasting performance is highly related to window-wise properties due to the sequence-to-sequence forecasting paradigm of deep time series models and introduce a quantitative measurement of window-wise pattern complexity. Through rigorous statistical analyses over more than 4700 newly trained deep forecasting models, we discover a consistent empirical relationship between the minimum attainable forecasting error of deep models and the complexity of window-wise series patterns, which is termed the accuracy law. We further demonstrate that this empirical finding successfully guides us to identify saturated tasks from widely used benchmarks and derive an effective training strategy for time series foundation models, offering valuable insights for future research.