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

arXiv: https://arxiv.org/abs/2608.09450 · HTML: https://mlsanity.com/paper/2608.09450/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 23 references that are in the map.
- First posted on arXiv 10 Aug 2026.

## Builds on (most cited first, 7 of 7)

- [Coin Betting and Parameter-Free Online Learning](https://mlsanity.com/paper/1602.04128/) (2016)
- [Black-Box Reductions for Parameter-free Online Learning in Banach Spaces](https://mlsanity.com/paper/1802.06293/) (2018)
- [Nonparametric Two-Sample Testing by Betting](https://mlsanity.com/paper/2112.09162/) (2021)
- [Approachability, Regret and Calibration; implications and equivalences](https://mlsanity.com/paper/1301.2663/) (2013)
- [Approachability in unknown games: Online learning meets multi-objective optimization](https://mlsanity.com/paper/1402.2043/) (2014)
- [An Online Convex Optimization Approach to Blackwell's Approachability](https://mlsanity.com/paper/1503.00255/) (2015)
- [Rate-Preserving Reductions for Blackwell Approachability](https://mlsanity.com/paper/2406.07585/) (2024)

## Abstract (arXiv)

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.