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.

翻译：在时间序列预测的诸多应用中，需要预测未来多个时间步的结果。尽管已有大量相关文献，但无论是经典方法还是近期基于深度学习的方法，大多聚焦于最小化预测窗口内的平均性能。我们观察到，这可能导致预测误差在不同时间步上的分布不均，尤其是在基于热门预测基准训练的现代Transformer架构中。换言之，优化平均性能可能导致特定时间步出现异常大的误差。为此，本文提出一种面向长期时间序列预测的约束学习方法，旨在寻找在平均性能最优且满足用户定义的各时间步损失上限的模型。我们将该方法称为损失整形约束，因其对每个时间步的损失施加约束；我们利用对偶理论的最新成果证明，尽管原问题非凸，但其对偶间隙有界。我们设计了一种实用的原-对偶算法进行求解，并在时间序列预测基准上验证了该方法在保持竞争性平均性能的同时，有效调控了预测窗口内的误差分布。