Conformal Prediction Regions for Time Series using Linear Complementarity Programming

TL;DR

The paper addresses the conservatism of conformal prediction when applied to time-series by introducing a parameterized, multi-step nonconformity score $R=\max(\alpha_1 R_1,\ldots,\alpha_T R_T)$ and learning the horizon weights $\alpha_t$ from calibration data. It formulates the alpha-learning as a mixed-integer linear complementarity program (MILCP) and shows how to relax it to a linear complementarity program (LCP) with the same optimal value, leveraging a Koenker-based LP for the quantile and its KKT conditions to obtain a tractable solver framework. The method yields substantially tighter conformal prediction regions than prior union-bound approaches while maintaining valid coverage, as demonstrated on ORCA pedestrian trajectories and F16 altitude prediction. This approach enables non-conservative, verifiable uncertainty quantification for learning-enabled time-series predictors in safety-critical settings and supports longer-horizon planning and verification.

Abstract

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 $1-δ$, {previous works require that each individual prediction region is valid} with confidence $1-δ/T$. 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 case studies using pedestrian trajectory predictors and F16 fighter jet altitude predictors.

Figures (7)

• Figure 1: Sample trajectory of a pedestrian (teal stars), the pedestrian predictions using a social LSTM from alahi2016social (blue stars), the conformal prediction regions for our approach (green circles), and the conformal prediction regions for the approach from lindemann2022safe (red circles). Notably, our approach is less conservative.
• Figure 2: Comparison of our approach and the union bound approach over $100$ trials for the ORCA case study.
• Figure 3: Comparison of our approach and the union bound approach over $100$ trials for the F16 case study.
• Figure 4: Non-conformity score quantiles for the calibration and testing data for the two case studies.
• Figure 5: Conformal prediction regions and errors for various prediction horizons. The green circles show the regions produced by our approach and the red circles show the regions produced by the union bound approach. Notably, our approach produces much tighter valid predictions regions.

Theorems & Definitions (11)

• Remark 1
• Theorem 1
• Remark 2
• Theorem 2
• Proposition 1
• Proposition 2
• Theorem 3
• Corollary 1
• proof : Proof of Theorem \ref{['thm:alphasNonZero']}
• proof : Proof of Theorem \ref{['thm:1']}