REGAIN: REconciliation GAIN-driven Auxiliary Direction Learning

2026-06-02

AI Summary

This work addresses the limitation of existing forecast reconciliation methods, which rely on fixed observation systems and struggle to capture unmodeled residual uncertainty in target variables. The authors propose REGAIN, a novel framework that uniquely optimizes reconciliation by maximizing post-reconciliation forecast performance gains. REGAIN learns normalized auxiliary directions, leverages a frozen forecaster to generate induced sequences, and selects optimal auxiliary measurements based on weighted loss reduction. Crucially, it prioritizes auxiliary information that provides complementary uncertainty rather than merely high predictability. The approach integrates augmented generalized least squares reconciliation, staged learning, and a holdout-set gain selection mechanism—optionally followed by joint fine-tuning. Evaluated on Beijing PM2.5 and Australian tourism datasets, REGAIN substantially improves both multivariate and hierarchical forecast reconciliation accuracy, particularly when the original observation system fails to account for residual uncertainty.

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.

Problem

• forecast reconciliation
• auxiliary directions
• measurement selection
• uncertainty reduction
• coherent forecasting

Innovation

• forecast reconciliation
• auxiliary direction learning
• reconciliation gain
• uncertainty complementarity
• augmented generalized least squares