# REGAIN: REconciliation GAIN-driven Auxiliary Direction Learning

arXiv: https://arxiv.org/abs/2606.04380 · HTML: https://mlsanity.com/paper/2606.04380/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 42 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [Chronos: Learning the Language of Time Series](https://mlsanity.com/paper/2403.07815/) (2024)
- [A decoder-only foundation model for time-series forecasting](https://mlsanity.com/paper/2310.10688/) (2023)
- [One Fits All:Power General Time Series Analysis by Pretrained LM](https://mlsanity.com/paper/2302.11939/) (2023)
- [Large Language Models Are Zero-Shot Time Series Forecasters](https://mlsanity.com/paper/2310.07820/) (2023)
- [TimeXer: Empowering Transformers for Time Series Forecasting with Exogenous Variables](https://mlsanity.com/paper/2402.19072/) (2024)
- [Tiny Time Mixers (TTMs): Fast Pre-trained Models for Enhanced Zero/Few-Shot Forecasting of Multivariate Time Series](https://mlsanity.com/paper/2401.03955/) (2024)
- [ForecastPFN: Synthetically-Trained Zero-Shot Forecasting](https://mlsanity.com/paper/2311.01933/) (2023)
- [Revitalizing Multivariate Time Series Forecasting: Learnable Decomposition with Inter-Series Dependencies and Intra-Series Variations Modeling](https://mlsanity.com/paper/2402.12694/) (2024)
- [BasisFormer: Attention-based Time Series Forecasting with Learnable and Interpretable Basis](https://mlsanity.com/paper/2310.20496/) (2023)
- [Towards a General Time Series Forecasting Model with Unified Representation and Adaptive Transfer](https://mlsanity.com/paper/2405.17478/) (2024)
- [Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives](https://mlsanity.com/paper/2006.08570/) (2020)
- [Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues](https://mlsanity.com/paper/2303.17277/) (2023)
- [A New Method for Performance Analysis in Nonlinear Dimensionality Reduction](https://mlsanity.com/paper/1711.06252/) (2017)
- [Constructing hierarchical time series through clustering: Is there an optimal way for forecasting?](https://mlsanity.com/paper/2404.06064/) (2024)
- [Forecast Linear Augmented Projection (FLAP): A free lunch to reduce forecast error variance](https://mlsanity.com/paper/2407.01868/) (2024)

## Abstract (arXiv)

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.