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arXiv · 2608.09450

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

Jinze Zhao
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Abstract

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.

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Jinze Zhao. 2026-08-10. From Approachability Residuals to Anytime-Valid Evidence: The Online Convex Geometry of Testing by Betting. https://arxiv.org/abs/2608.09450

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