REGAIN: REconciliation GAIN-driven Auxiliary Direction Learning
2026-06-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how to improve forecast accuracy by adding new helpful measurements, rather than just adjusting existing ones. They introduce REGAIN, a method that picks extra measurements based on how much they reduce errors after combining forecasts in a smart way. Their analysis shows that these new measurements should provide new, useful information, not just be easy to predict. Experiments with pollution and tourism data demonstrate that this approach can improve forecast quality by capturing uncertainties missed by original measurements.
forecast reconciliationlinear measurementsauxiliary directionsgeneralized least squaresquadratic riskhierarchical forecastingforecast error reductionstagewise learningresidual uncertaintytime series forecasting
Authors
Weijia Li, Shun Hu, Yanfei Kang
Abstract
Forecast reconciliation usually starts from a fixed measurement system and asks how forecasts should be projected onto a coherent space. We ask a different question: which additional linear measurements should be forecast and included in the reconciliation system? We propose REGAIN, a reconciliation-gain framework that learns normalized auxiliary directions, forecasts the induced series with a frozen forecasting oracle, and selects directions by their target-weighted loss reduction after augmented generalized least-squares reconciliation. Unlike variance-based components or predictability-based auxiliary selection, REGAIN optimizes the downstream effect of an auxiliary measurement on the final reconciled forecasts. We provide a statistical characterization showing that useful auxiliary directions must provide complementary information about unresolved target uncertainty, rather than merely being easy to forecast. The analysis also clarifies the covariance-risk reduction mechanism, the role of bias changes in realized quadratic risk, and the stability of estimated gain signals. A stagewise learning algorithm with held-out gain screening is developed, together with an optional joint refinement step. Experiments on Beijing PM2.5 and Australian Tourism data show that gain-selected measurements can improve both ordinary multivariate and hierarchical forecasts, especially when they reveal residual uncertainty not captured by the original measurement system.