# Learning to Bet for Horizon-Aware Anytime-Valid Testing

arXiv: https://arxiv.org/abs/2603.19551 · HTML: https://mlsanity.com/paper/2603.19551/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 24 references that are in the map.
- First posted on arXiv 20 Mar 2026.

## Builds on (most cited first, 13 of 13)

- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [Deep Reinforcement Learning with Double Q-learning](https://mlsanity.com/paper/1509.06461/) (2015)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio](https://mlsanity.com/paper/2110.14099/) (2021)
- [On the near-optimality of betting confidence sets for bounded means](https://mlsanity.com/paper/2310.01547/) (2023)
- [On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies](https://mlsanity.com/paper/2207.12382/) (2022)
- [On the optimality of coin-betting for mean estimation](https://mlsanity.com/paper/2412.02640/) (2024)
- [Anytime Validity is Free: Inducing Sequential Tests](https://mlsanity.com/paper/2501.03982/) (2025)
- [Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2502.07774/) (2025)
- [STaR-Bets: Sequential Target-Recalculating Bets for Tighter Confidence Intervals](https://mlsanity.com/paper/2505.22422/) (2025)

## Built on by (most cited first, 1 of 1)

- [Confidence Horizons](https://mlsanity.com/paper/2608.03889/) (2026)

## Abstract (arXiv)

We develop horizon-aware anytime-valid tests and confidence sequences for bounded means under a strict deadline N. Using the betting/e-process framework, we cast horizon-aware betting as a finite-horizon optimal control problem with state space (t, log Wt), where t is the time and Wt is the test martingale value. We first show that in certain interior regions of the state space, policies that deviate significantly from Kelly betting are provably suboptimal, while Kelly betting reaches the threshold with high probability. We then identify sufficient conditions showing that outside this region, more aggressive betting than Kelly can be better if the bettor is behind schedule, and less aggressive can be better if the bettor is ahead. Taken together these results suggest a simple phase diagram in the (t, log Wt) plane, delineating regions where Kelly, fractional Kelly, and aggressive betting may be preferable. Guided by this phase diagram, we introduce a Deep Reinforcement Learning approach based on a universal Deep Q-Network (DQN) agent that learns a single policy from synthetic experience and maps simple statistics of past observations to bets across horizons and null values. In limited-horizon experiments, the learned DQN policy yields state-of-the-art results.