Title: REGAIN: REconciliation GAIN-driven Auxiliary Direction Learning

Abstract:

Traditional forecast reconciliation typically begins with a predetermined measurement framework, focusing on how predictions should be mapped into a coherent space. In contrast, this study poses a distinct inquiry: which supplementary linear measurements ought to be predicted and integrated into the reconciliation process? To address this, we introduce REGAIN, a framework designed to learn normalized auxiliary directions. This method forecasts the resulting series using a fixed forecasting oracle and prioritizes these directions based on their ability to reduce target-weighted loss following augmented generalized least-squares reconciliation.

Diverging from approaches that rely on variance components or predictability metrics for auxiliary selection, REGAIN focuses on optimizing the downstream impact of an auxiliary measurement on the final reconciled forecasts. Our statistical analysis reveals that effective auxiliary directions must offer complementary insights into unresolved target uncertainty, rather than simply being easy to predict. The study further elucidates the mechanism behind covariance-risk reduction, the influence of bias shifts on realized quadratic risk, and the robustness of the estimated gain signals.

We present a stagewise learning algorithm featuring held-out gain screening, alongside an optional joint refinement procedure. Empirical evaluations using Beijing PM2.5 and Australian Tourism datasets demonstrate that measurements selected via gain optimization enhance both standard multivariate and hierarchical forecasting models. These improvements are particularly pronounced when the selected measurements expose residual uncertainties that were previously unaccounted for by the original measurement system.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC