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

Betting-based sequential tests and Blackwell approachability are linked by a rate-explicit reduction through support-function residuals. For a compact convex target S and vector observations rt, an OCO learner selects a predictable normal wt and produces qt=langle wt,rtrangle-hS(wt). We prove the exact pathwise identity dist(bar rT,S) =frac1Tsumt=1Tqt+fracRegTT. When |qt|≤ B, composing this identity with one-sided betting yields a finite-time transfer: if the OCO and log-wealth regrets are at most aT and ℓT, respectively, then a target gap exceeding [ fracaTT +2B√fraclog(1/α)+ℓTT ] forces rejection by time T, while non-rejection certifies the converse radius. We then formulate a controlled stochastic experiment in which an action selected after wt satisfies Blackwell's supporting-halfspace condition for every null mean payoff. The resulting wealth is an e-process under adaptive nulls; sublinear OCO regret gives stochastic approachability, whereas persistent mean separation under an alternative gives exponential wealth at rate at least δ2/(4B2). Deterministic Blackwell games and passive tests are, respectively, the noise-free and singleton-action cases of this protocol. Bounded two-sample means, kernel MMD, and active heterogeneous data sources instantiate the reduction. The resulting connection is exact algebraically, quantitative at finite time, and operational when experiments are controlled.