Legendre Jumper Martingales improve online exchangeability testing

This paper by Johan Hallberg Szabadváry presents a new family of conformal test martingales designed for online exchangeability testing — a key technique for detecting distributional shifts in streaming data. The core innovation is the use of shifted Legendre polynomials to construct betting functions that can capture not just mean shifts (as in the classic Simple Jumper) but also variance, skewness, and higher-order deviations from uniformity. The Simple Legendre Jumper generalizes the linear betting function to polynomials of arbitrary degree, allowing more sensitive detection of complex changes. However, combining multiple polynomial degrees naively (the Product Legendre Jumper) causes the state space to grow exponentially, a cost the author terms the 'jumping tax.'

To overcome the exponential blowup, the author introduces the Variational Legendre Jumper, which factorises the joint adaptation via a mean-field approximation, reducing the computational scaling from exponential to linear with minimal loss in statistical power. Additionally, the Composite Legendre Jumper incorporates multiple jumping rates to maintain a wealth floor under exchangeability and automatically adapts to the unknown timescale of the shift. Empirical results from a real-world classification task show that these combined methods consistently outperform any single-degree martingale when distributional shift occurs. The composite variant is recommended as the default choice when the shift timescale is unknown, making this work a practical advancement for online anomaly detection and adaptive systems.