Billions-Scale Forecast Reconciliation

2026-02-04

AI Summary

This work addresses the challenge of enforcing aggregation consistency in large-scale hierarchical time series forecasting by proposing an optimization-based point forecast reconciliation method. By efficiently solving a constrained weighted least squares problem, the approach accommodates multi-level and overlapping hierarchies while achieving scalable computation at a scale exceeding 4 billion forecast dimensions. Theoretical analysis reveals an equivalence between weighted least squares reconciliation and proportion-based reconciliation under specific conditions, thereby elucidating their generalized relationship. Notably, this study presents the first successful deployment of an optimization-driven reconciliation framework in a real-world retail setting at the billion-scale, establishing a new record for the largest application of such methods to date and demonstrating both the computational efficiency and practical viability of the proposed approach.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The problem of combining multiple forecasts of related quantities that obey expected equality and additivity constraints, often referred to a hierarchical forecast reconciliation, is naturally stated as a simple optimization problem. In this paper we explore optimization-based point forecast reconciliation at scales faced by large retailers. We implement and benchmark several algorithms to solve the forecast reconciliation problem, showing efficacy when the dimension of the problem exceeds four billion forecasted values. To the best of our knowledge, this is the largest forecast reconciliation problem, and perhaps on-par with the largest constrained least-squares-problem ever solved. We also make several theoretical contributions. We show that for a restricted class of problems and when the loss function is weighted appropriately, least-squares forecast reconciliation is equivalent to share-based forecast reconciliation. This formalizes how the optimization based approach can be thought of as a generalization of share-based reconciliation, applicable to multiple, overlapping data hierarchies.

Problem

• forecast reconciliation
• hierarchical forecasting
• constrained least-squares
• large-scale optimization
• additivity constraints

Innovation

• forecast reconciliation
• constrained least squares
• large-scale optimization
• hierarchical forecasting
• share-based reconciliation