From Approachability Residuals to Anytime-Valid Evidence: The Online Convex Geometry of Testing by Betting

This paper investigates the fundamental mathematical connection between two seemingly distinct fields: Blackwell approachability (a game-theoretic framework for reaching a target set) and betting-based sequential testing (a statistical framework for anytime-valid hypothesis testing). The author seeks to determine if these two protocols share a common geometric structure that can be exploited to create more robust and efficient sequential tests.

The author derives a pathwise identity that relates the distance of an average vector observation to a compact convex target set $S$ with the average support-function residual produced by an Online Convex Optimization (OCO) learner. By treating the OCO learner's output as a predictable direction, the author constructs a scalar payoff that serves as the basis for a betting-based test. This approach allows the author to transform the geometric problem of approaching a target set into a statistical problem of accumulating wealth against a null hypothesis.

The study proves that the average support residual is exactly equal to the distance to the target set plus the average regret of the OCO learner. When this residual is used in a betting-based test, it provides a finite-time transfer: if the OCO regret and log-wealth regret are controlled, any target gap exceeding a specific threshold forces the rejection of the null hypothesis. The author demonstrates that this framework generalizes existing methods, such as bounded two-sample mean testing and kernel Maximum Mean Discrepancy (MMD) tests, and extends them to controlled stochastic experiments where the statistician can influence the data-generating process.

This work provides a unified, modular framework for sequential inference. By separating the geometric task (learning a direction to the target) from the statistical task (compounding wealth to reject a null), researchers can design more flexible tests for complex, heterogeneous data sources. The exactness of the residual-regret identity ensures that the resulting tests are not just asymptotically consistent but also quantitatively precise at finite time horizons.