Hierarchical forecasting at scale

Abstract

Hierarchical forecasting techniques allow for the creation of forecasts that are coherent with respect to a pre-specified hierarchy of the underlying time series. This targets a key problem in e-commerce, where we often find millions of products across many product hierarchies, and forecasts must be made for individual products and product aggregations. However, existing hierarchical forecasting techniques scale poorly when the number of time series increases, which limits their applicability at a scale of millions of products.

In this paper, we propose to learn a coherent forecast for millions of products with a single bottom-level forecast model by using a loss function that directly optimizes the hierarchical product structure. We implement our loss function using sparse linear algebra, such that the number of operations in our loss function scales quadratically rather than cubically with the number of products and levels in the hierarchical structure. The benefit of our sparse hierarchical loss function is that it provides practitioners with a method of producing bottom-level forecasts that are coherent to any chosen cross-sectional or temporal hierarchy. In addition, removing the need for a post-processing step as required in traditional hierarchical forecasting techniques reduces the computational cost of the prediction phase in the forecasting pipeline and its deployment complexity.

In our tests on the public M5 dataset, our sparse hierarchical loss function performs up to 10% better as measured by RMSE and MAE than the baseline loss function. Next, we implement our sparse hierarchical loss function within a gradient boosting-based forecasting model at bol.com, a large European e-commerce platform. At bol.com, each day, a forecast for the weekly demand of every product for the next twelve weeks is required. In this setting, our sparse hierarchical loss resulted in an improved forecasting performance as measured by RMSE of about 2% at the product level, compared to the baseline model, and an improvement of about 10% at the product level as measured by MAE. Finally, we found an increase in forecasting performance of about 5%–10% (both RMSE and MAE) when evaluating the forecasting performance across the cross-sectional hierarchies we defined. These results demonstrate the usefulness of our sparse hierarchical loss applied to a production forecasting system at a major e-commerce platform.