  Several applications in time series forecasting require predicting multiple
steps ahead. Despite the vast amount of literature in the topic, both classical
and recent deep learning based approaches have mostly focused on minimising
performance averaged over the predicted window. We observe that this can lead
to disparate distributions of errors across forecasting steps, especially for
recent transformer architectures trained on popular forecasting benchmarks.
That is, optimising performance on average can lead to undesirably large errors
at specific time-steps. In this work, we present a Constrained Learning
approach for long-term time series forecasting that aims to find the best model
in terms of average performance that respects a user-defined upper bound on the
loss at each time-step. We call our approach loss shaping constraints because
it imposes constraints on the loss at each time step, and leverage recent
duality results to show that despite its non-convexity, the resulting problem
has a bounded duality gap. We propose a practical Primal-Dual algorithm to
tackle it, and demonstrate that the proposed approach exhibits competitive
average performance in time series forecasting benchmarks, while shaping the
distribution of errors across the predicted window.
