  Existing hierarchical forecasting techniques scale poorly when the number of
time series increases. We propose to learn a coherent forecast for millions of
time series with a single bottom-level forecast model by using a sparse loss
function that directly optimizes the hierarchical product and/or temporal
structure. The benefit of our sparse hierarchical loss function is that it
provides practitioners 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. On the public M5 dataset, our
sparse hierarchical loss function performs up to 10% (RMSE) better compared to
the baseline loss function. We implement our sparse hierarchical loss function
within an existing forecasting model at bol, a large European e-commerce
platform, resulting in an improved forecasting performance of 2% at the product
level. Finally, we found an increase in forecasting performance of about 5-10%
when evaluating the forecasting performance across the cross-sectional
hierarchies that we defined. These results demonstrate the usefulness of our
sparse hierarchical loss applied to a production forecasting system at a major
e-commerce platform.
