Iterative Hypothesis Pruning and Distribution-based Early Labeling for Sequential Hypothesis Testing

We consider the problem where an active Decision-Maker (DM) is tasked to identify the true hypothesis using as few samples as possible while maintaining accuracy. The DM collects samples according to its determined actions and knows the distributions under each hypothesis. We propose the $\Phi$-$\Delta$ algorithm, a deterministic and adaptive multi-stage hypothesis-elimination algorithm where the DM selects an action, applies it repeatedly, and discards hypotheses in light of its obtained samples. The DM selects actions based on maximal separation expressed by the maximal minimal Total Variation Distance (TVD) between each two possible output distributions. To further optimize the search (in terms of the mean number of samples required to separate hypotheses), close distributions (in TVD) are clustered, and the algorithm eliminates whole clusters rather than individual hypotheses. We extensively analyze our algorithm and show it is asymptotically optimal as the desired error probability approaches zero. Our analysis also includes identifying instances when the algorithm is asymptotically optimal in the number of hypotheses, bounding the mean number of samples per-stage and in total, characterizing necessary and sufficient conditions for vanishing error rates when clustering hypotheses, evaluating algorithm complexity, and discussing its optimality in finite regimes.