Learning to Bet for Horizon-Aware Anytime-Valid Testing

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 $(t, \log W_t)$, where $t$ is the time and $W_t$ 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 $(t, \log W_t)$ 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.

We study hypothesis testing for experiments that are monitored as data arrives but must stop by a fixed deadline, such as a budget or participant limit. Standard methods can either raise too many false alarms when results are checked repeatedly, or become too cautious when designed for infinite horizon. We develop a strategy inspired by betting: bet steadily when evidence is progressing, take bigger risks near the deadline, and act more cautiously when enough evidence has already accumulated. Using simulated experiments, we train a machine learning system to choose these actions in real time. The learned method detects real effects more often by the deadline and produces tighter plausible ranges, while still controlling false alarms. This can improve A/B tests, adaptive experiments, and resource-limited studies.