Optimal distribution-free sequential change detection with conformal martingales

원문 초록

We study the challenging problem of distribution-free sequential changepoint detection for independent observations with unknown and \emph{completely unrestricted} pre- and post-change laws. For over two decades, the primary existing solution for this problem has employed Vovk's conformal test martingales and associated e-detectors, which control the probability of false alarm (PFA) and the average run length (ARL) respectively. The majority of these works focus on validity, with statistical efficiency usually left for simulations. We develop a comprehensive theory of how conformal p-values behave under non-exchangeable data with a changepoint at an unknown time $T$. We use this to analyze the post-change growth and resulting detection delays and find that the standard existing methods are provably suboptimal for PFA and ARL control, and can lead to delays that are $\Omega(T)$ and $\Omega(\sqrt{\text{ARL}})$ respectively. We construct new conformal e-processes and e-detectors with learned scores and a fixed, distribution-independent betting dictionary. For any distribution pair with positive finite KL divergence, our methods asymptotically achieve $\log T/\mathrm{KL}$ delay for PFA control and $\log\mathrm{ARL}/\mathrm{KL}$ delay (with a sharp leading constant of 1), which are both proven to be minimax optimal in a suitable double-array framework. Simulations illustrate the large empirical benefits of our new approach.