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 r_t, an OCO learner selects a predictable normal w_t and produces q_t=\langle w_t,r_t\rangle-h_S(w_t). We prove the exact pathwise identity $ \dist(\bar r_T,S) =\frac1T\sum_{t=1}^Tq_t+\frac{\Reg_T}{T}. When |q_t|\leq B, composing this identity with one-sided betting yields a finite-time transfer: if the OCO and log-wealth regrets are at most a_T and \ell_T, respectively, then a target gap exceeding [ \frac{a_T}{T} +2B\sqrt{\frac{\log(1/α)+\ell_T}{T}} ] 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 w_t 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 δ^2/(4B^2)$. 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.