aiqbench analysis
This research paper investigates the theoretical conditions under which Gaussian inference remains valid for variance-aware Upper Confidence Bound (UCB) policies in multi-armed bandit settings. The authors derive a sharp criterion linking reward gaps and variances to determine when deterministic approximations of arm counts hold uniformly, offering rigorous insights into statistical stability without relying on heuristic assumptions.
Allocation Stability and Wald Inference under Variance-Aware UCB
arXiv:2412.08843v3 Announce Type: replace-cross Abstract: Allocation stability is often used to justify Gaussian inference from bandit data, but when is it necessary? In this paper, we address this question for a two-armed, fixed-horizon variance-aware UCB policy with bounded reward distributions that may vary with the horizon. We find a sharp criterion in terms of the reward gap and variances that determines whether the optimal-arm count admits a deterministic approximation with vanishing relative error, while the suboptimal-arm count is always stable. Despite the possible instability of the optimal-arm count, we show that the ordinary Wald statistic for a linear combination of the arm means has a standard normal limit for every fixed nonzero coefficient vector, provided the product of the pull count and reward variance diverges in probability for each arm. Under the same condition, however, this Gaussian approximation holds uniformly over deterministic nonzero coefficient vectors if and only if the optimal-arm count is stable. The analysis relies on two main ingredients: (i) a pathwise comparison with an auxiliary policy whose final optimal-arm count is asymptotically equivalent to the original count and independent of the optimal-arm reward sequence; and (ii) joint limits for the rescaled optimal-arm count and the two studentized sample-mean errors under the original policy, which yield nonstandard Wald limits for certain linear combinations of the arm means with coefficients that vary with the horizon.
Abstract:Allocation stability is often used to justify Gaussian inference from bandit data, but when is it necessary? In this paper, we address this question for a two-armed, fixed-horizon variance-aware UCB policy with bounded reward distributions that may vary with the horizon. We find a sharp criterion in terms of the reward gap and variances that determines whether the optimal-arm count admits a deterministic approximation with vanishing relative error, while the suboptimal-arm count is always stable. Despite the possible instability of the optimal-arm count, we show that the ordinary Wald statistic for a linear combination of the arm means has a standard normal limit for every fixed nonzero coefficient vector, provided the product of the pull count and reward variance diverges in probability for each arm. Under the same condition, however, this Gaussian approximation holds uniformly over deterministic nonzero coefficient vectors if and only if the optimal-arm count is stable. The analysis relies on two main ingredients: (i) a pathwise comparison with an auxiliary policy whose final optimal-arm count is asymptotically equivalent to the original count and independent of the optimal-arm reward sequence; and (ii) joint limits for the rescaled optimal-arm count and the two studentized sample-mean errors under the original policy, which yield nonstandard Wald limits for certain linear combinations of the arm means with coefficients that vary with the horizon.
Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
arXiv:2412.08843 [stat.ML]
(or arXiv:2412.08843v3 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2412.08843 Focus to learn more arXiv-issued DOI via DataCite
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