Conformal prediction is a statistical tool for producing prediction regions of machine learning models that are valid with high probability. However, applying conformal prediction to time series data leads to conservative prediction regions. In fact, to obtain prediction regions over $T$ time steps with confidence $1-\delta$, {previous works require that each individual prediction region is valid} with confidence $1-\delta/T$. We propose an optimization-based method for reducing this conservatism to enable long horizon planning and verification when using learning-enabled time series predictors. Instead of considering prediction errors individually at each time step, we consider a parameterized prediction error over multiple time steps. By optimizing the parameters over an additional dataset, we find prediction regions that are not conservative. We show that this problem can be cast as a mixed integer linear complementarity program (MILCP), which we then relax into a linear complementarity program (LCP). Additionally, we prove that the relaxed LP has the same optimal cost as the original MILCP. Finally, we demonstrate the efficacy of our method on a case study using pedestrian trajectory predictors.

翻译：保形预测是一种统计工具，用于生成机器学习模型在高概率下有效的预测区域。然而，将保形预测应用于时间序列数据会导致保守的预测区域。事实上，为了在$T$个时间步上以置信度$1-\delta$获得预测区域，先前的工作要求每个单独预测区域以置信度$1-\delta/T$有效。我们提出了一种基于优化的方法来减少这种保守性，从而在使用学习型时间序列预测器时实现长时域规划与验证。我们不单独考虑每个时间步的预测误差，而是考虑多个时间步上的参数化预测误差。通过在一个额外数据集上优化参数，我们得到了非保守的预测区域。我们证明该问题可转化为混合整数线性互补规划（MILCP），随后将其松弛为线性互补规划（LCP）。此外，我们还证明了松弛后的线性规划具有与原始MILCP相同的最优成本。最后，我们通过一个使用行人轨迹预测器的案例研究展示了该方法的有效性。