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：时间序列预测中的若干应用需要预测多步未来值。尽管该主题已有大量文献，无论是经典方法还是近期基于深度学习的方法，大多侧重于最小化预测窗口内的平均性能。我们观察到，这可能导致不同预测步数之间误差分布不均，特别是在流行预测基准上训练的近期Transformer架构中。也就是说，优化平均性能可能导致特定时间步出现不可接受的大误差。在本工作中，我们提出了一种用于长期时间序列预测的约束学习方法，旨在找到在满足用户定义的每个时间步损失上界的前提下，具有最佳平均性能的模型。我们称该方法为损失整形约束，因为它对每个时间步的损失施加约束，并利用近期对偶性结果表明，尽管问题非凸，但所得问题具有有界对偶间隙。我们提出了一种实用的原对偶算法来解决该问题，并证明所提方法在时间序列预测基准中展现出具有竞争力的平均性能，同时能重塑预测窗口内的误差分布。