Auxiliary Quantile Forecasting with Linear Networks

We propose a novel multi-task method for quantile forecasting with shared Linear layers. Our method is based on the Implicit quantile learning approach, where samples from the Uniform distribution $\mathcal{U}(0, 1)$ are reparameterized to quantile values of the target distribution. We combine the implicit quantile and input time series representations to directly forecast multiple quantile estimations for multiple horizons jointly. Prior works have adopted a Linear layer for the direct estimation of all forecasting horizons in a multi-task learning setup. We show that following similar intuition from multi-task learning to exploit correlations among forecast horizons, we can model multiple quantile estimates as auxiliary tasks for each of the forecast horizon to improve forecast accuracy across the quantile estimates compared to modeling only a single quantile estimate. We show learning auxiliary quantile tasks leads to state-of-the-art performance on deterministic forecasting benchmarks concerning the main-task of forecasting the 50$^{th}$ percentile estimate.