REGAIN: REconciliation GAIN-driven Auxiliary Direction Learning

• URL: http://arxiv.org/abs/2606.04380v1
• Date: Wed, 03 Jun 2026 02:53:27 GMT
• Title: REGAIN: REconciliation GAIN-driven Auxiliary Direction Learning
• Abstract summary: We propose a reconciliation-gain framework that learns normalized auxiliary directions.<n>We show that useful auxiliary directions must provide complementary information about unresolved target uncertainty.<n> Experiments on Beijing PM2.5 and Australian Tourism data show that gain-selected measurements can improve both ordinary multivariate and hierarchical forecasts.
• Score: 12.590383160848141
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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