REGAIN: REconciliation GAIN-driven Auxiliary Direction Learning

A new framework called REGAIN proposes learning auxiliary measurement directions for forecast reconciliation by directly optimizing the downstream gain they deliver, rather than relying on variance or predictability metrics, according to a paper submitted on 3 June 2026 [1]. The REGAIN framework — short for REconciliation GAIN-driven Auxiliary Direction Learning — reframes how auxiliary series are chosen for forecast reconciliation. Standard approaches start from a fixed measurement system and project forecasts onto a coherent space. REGAIN instead asks which additional linear measurements should be forecast and included in the reconciliation system [1]. The framework learns normalized auxiliary directions, forecasts the induced series using a frozen time-series foundation model as a shared oracle, and selects directions based on their target-weighted loss reduction after augmented generalized least-squares reconciliation [2]. This design isolates the effect of measurement design from artifacts of forecaster adaptation [3]. A key statistical insight underpinning the method is that useful auxiliary directions must supply complementary information about unresolved target uncertainty, not merely be easy to forecast [1]. The paper also details the covariance-risk reduction mechanism, the role of bias changes in realized quadratic risk, and the stability of estimated gain signals [2]. A stagewise learning algorithm with held-out gain screening is developed, along with an optional joint refinement step [4]. The broader idea of selecting auxiliary tasks by their downstream utility has parallels in reinforcement learning. A separate line of work introduced a generate-and-test method for auxiliary task discovery, where new auxiliary tasks are continually generated and only those with high utility — measured by how useful the induced features are for the main task — are retained [5]. That approach significantly outperformed random tasks and learning without auxiliary tasks across multiple environments [5]. REGAIN was evaluated on Beijing PM2.5 air-quality data and Australian Tourism data. The experiments showed 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 [1].