Characterizing the gap between fixed-horizon and fixed-confidence settings in Best Arm Identification

Characterize the gap between fixed-horizon and fixed-confidence settings in Best Arm Identification, establishing a formal understanding of their differences and clarifying how these settings diverge in complexity and algorithmic design.

Background

The authors note that, while fixed-confidence BAI is relatively well understood and supported by instance-dependent algorithms and stopping rules, the fixed-horizon (fixed-budget) setting remains less developed and harder to analyze. Their experiments suggest practical differences between the two settings—for example, the presence of a stopping action appears to help learning by inducing a form of curriculum—but a rigorous theoretical characterization is lacking.

This open question concerns the formal delineation and quantification of the differences between fixed-horizon and fixed-confidence frameworks in BAI, which could guide algorithm design and complexity analyses for identification tasks under budget constraints.

References

We believe our work makes a fundamental contribution to active testing, and in particular to the sub-field of best-arm identification, where key questions—such as the gap between fixed-horizon and fixed-confidence settings—remain open.

— Learning to Explore: An In-Context Learning Approach for Pure Exploration  (2506.01876 - Russo et al., 2 Jun 2025) in Section 6, Conclusions