Bayesian Best-Arm Identification with Abstention: A Polynomial-to-Exponential Phase Transition

2026-06-28

AI Summary

This study addresses the problem of incorporating an abstention mechanism into Bayesian fixed-budget best-arm identification, aiming to minimize the risk of recommending a suboptimal arm without abstaining, given a limited abstention budget. The key insight is that, in the Bayesian setting, even an arbitrarily small abstention budget induces a phase transition in the error probability—shifting its decay rate from polynomial to exponential. To characterize this phenomenon, the authors combine information-theoretic lower bounds, Gaussian prior modeling, and a novel adaptive algorithm named PGWS, which leverages the prior density of the top-two gap. They establish tight upper and lower bounds on the optimal error exponent of the form $\exp(-\alpha^2 T / (8\kappa_\nu^2))$. Both theoretical analysis and empirical experiments demonstrate that PGWS asymptotically achieves this optimal bound.

Abstract

We study the Bayesian fixed-budget best-arm identification problem in which a learner can abstain from making a terminal recommendation. Subject to an abstention budget $α$, we analyze the probability of undetected error--the risk of recommending a suboptimal arm without abstaining. Our central finding is that abstention induces a phase transition: without abstention, the error probability decays polynomially in the sampling budget $T$; in contrast, introducing any small positive abstention budget shifts this to an exponential decay. For Gaussian priors and rewards, in the regime $T\to\infty$ followed by $α\downarrow0$, we establish exact matching information-theoretic lower bounds and algorithmic upper bounds on the optimal error exponent, which takes the form $\exp(-\frac{α^{2}T}{8κ_ν^{2}})$. The hardness parameter $κ_ν$ represents the prior density of the top-two gap at zero, highlighting that nearly tied instances drive the fundamental error. We introduce an adaptive algorithm, PGWS, that successfully achieves this optimal exponent by expending its abstention budget on statistically ambiguous instances. We further demonstrate that this polynomial-to-exponential improvement is exclusively a Bayesian phenomenon--in the frequentist setting, abstention only affects lower-order exponent terms. We also extend our results beyond the Gaussian model.

Problem

• Bayesian best-arm identification
• abstention
• error probability
• phase transition
• fixed-budget

Innovation

• Bayesian best-arm identification
• abstention
• phase transition
• error exponent
• adaptive algorithm