Lower bounds for Bayesian CVaR in interactive problems with local differential privacy
Abstract
We develop an information-theoretic method for lower bounding the prior-predictive conditional value-at-risk (CVaR) of loss in interactive statistical decision making (ISDM). A generalized Fano inequality replaces a single error event by a randomized binary statistic constructed from a bounded transform of the loss, reducing the problem to a Bernoulli f-divergence inequality. A bounded hinge transform yields a direct lower bound on Bayesian CVaR under a prior fixed in advance. We derive reverse-KL specializations and obtain explicit bounds for Gaussian mean estimation and Gaussian stochastic bandits. We then impose local differential privacy (LDP) at the observation stage of the same ISDM model. LDP contraction inequalities combined with the adaptive KL chain rule give explicit Bayesian CVaR lower bounds for Gaussian mean estimation with heterogeneous privacy parameters and Gaussian K-armed bandits.
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BibTeX
@misc{bongole2026lower,
title = {{Lower bounds for Bayesian CVaR in interactive problems with local differential privacy}},
author = {Raghav Bongole and Amirreza Zamani and Tobias J. Oechtering and Mikael Skoglund},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28153},
eprint = {2610.28153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28153},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lower bounds for Bayesian CVaR in interactive problems with local differential privacy
AU - Raghav Bongole
AU - Amirreza Zamani
AU - Tobias J. Oechtering
AU - Mikael Skoglund
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28153
UR - https://arxiv.org/abs/2610.28153
AB - We develop an information-theoretic method for lower bounding the prior-predictive conditional value-at-risk (CVaR) of loss in interactive statistical decision making (ISDM). A generalized Fano inequality replaces a single error event by a randomized binary statistic constructed from a bounded transform of the loss, reducing the problem to a Bernoulli f-divergence inequality. A bounded hinge transform yields a direct lower bound on Bayesian CVaR under a prior fixed in advance. We derive reverse-KL specializations and obtain explicit bounds for Gaussian mean estimation and Gaussian stochastic bandits. We then impose local differential privacy (LDP) at the observation stage of the same ISDM model. LDP contraction inequalities combined with the adaptive KL chain rule give explicit Bayesian CVaR lower bounds for Gaussian mean estimation with heterogeneous privacy parameters and Gaussian K-armed bandits.
ER -