Instantiating Bayesian CVaR lower bounds in Interactive Decision Making Problems

Abstract: Recent work established a generalized-Fano framework for lower bounding prior-predictive (Bayesian) CVaR in interactive statistical decision making. In this paper, we show how to instantiate that framework in concrete interactive problems and derive explicit Bayesian CVaR lower bounds from its abstract corollaries. Our approach compares a hard model with a reference model using squared Hellinger distance, and combines a lower bound on a reference hinge term with a bound on the distinguishability of the two models. We apply this approach to canonical examples, including Gaussian bandits, and obtain explicit bounds that make the dependence on key problem parameters transparent. These results show how the generalized-Fano Bayesian CVaR framework can be used as a practical lower-bound tool for interactive learning and risk-sensitive decision making.

• The paper introduces an explicit two-point Hellinger-based lower bound for Bayesian CVaR, bridging expected loss and tail risk.
• It instantiates the framework on passive Gaussian mean estimation and two-armed Gaussian bandits with closed-form expressions.
• The results reveal that worst-case Bayesian CVaR scales as Ω(n^(-1/2)) and √T, offering explicit risk-level calibration for tail performance.

Instantiating Bayesian CVaR Lower Bounds in Interactive Statistical Decision Making

Introduction: Context and Motivation

This work addresses the instantiation of information-theoretic lower bounds for Bayesian risk-sensitive criteria in interactive statistical decision making (ISDM), with a specific focus on Conditional Value-at-Risk (CVaR) under the prior-predictive (Bayesian) law. The CVaR criterion quantifies the mean performance in the upper tail at risk level $\alpha$, and thus is widely used in risk management, finance, and robust control. Existing lower-bound methods for statistical decision making (classical Fano, Le Cam, Assouad) typically target expected loss under both minimax and Bayesian protocols, and their generalizations to sequential/interactive ISDM settings (such as in bandits and RL) have elucidated the fundamental limits of sample complexity and regret.

Recently, a generalized-Fano-type framework for Bayesian CVaR was established, providing abstract lower-bound templates for tail-sensitive performance criteria in ISDM (Bongole et al., 17 Jan 2026). However, these templates are non-explicit and require instantiation in concrete problems, notably the selection of reference models and explicit evaluation of Hellinger/KL divergences and hinge terms. This paper provides a detailed methodology for making these lower bounds concrete by extracting a fully explicit and computationally tractable two-point Hellinger-based bound, and by demonstrating its instantiation in canonical passive and interactive learning settings.

Technical Contribution

The central result is a reusable, explicit two-point lower-bound template for Bayesian CVaR in ISDM settings, cast in terms of the squared Hellinger distance between Bayesian mixture laws. The methodology follows the generalized-Fano approach but provides step-by-step resolution of the minimization and inversion sub-tasks and calibration between the tail risk lower bound, the diversity induced by hard instance pair selection, and the contraction induced by interactive observation protocols. The paper then demonstrates this framework concretely on (a) passive Gaussian mean estimation and (b) two-armed Gaussian bandits. Each example yields closed-form, explicit CVaR lower bounds as functions of the instance separation, sample size/horizon, and risk level $\alpha$.

A central claim, made explicit in the technical results, is that the minimax rate under worst-case priors for Bayesian CVaR matches the classical Bayes-risk-oriented rates (e.g., $\Omega(n^{-1/2})$ for mean estimation, $\Omega(\sqrt{T})$ for two-armed bandits), but with explicit monotonic dependence of the constant on $\alpha$, tightly controlling the behavior in the tail as $\alpha \rightarrow 1$.

Two-Point Hellinger–Bayesian CVaR Lower Bound

The heart of the technique is the following. Consider a prior supported on two hard-to-distinguish models $M_1$ and $M_2$, with symmetric prior $\mu = 1/2(\delta_{M_1}+\delta_{M_2})$. For a bounded loss $L(\cdot,\cdot)$, if for all transcripts $\alpha$0, $\alpha$1, and the Hellinger divergence between $\alpha$2 and $\alpha$3 is bounded by $\alpha$4, then the prior-predictive Bayesian CVaR at risk level $\alpha$5 is

$\alpha$6

The minimization in $\alpha$7 can be carried out in closed form; explicit expressions and tight constants are provided. This bound is independent of the algorithm and hence applies to all decision rules.

Instantiation in Canonical Examples

Passive Gaussian Mean Estimation

For $\alpha$8 i.i.d. samples $\alpha$9 for $\Omega(n^{-1/2})$0, and loss $\Omega(n^{-1/2})$1, the framework yields

$\Omega(n^{-1/2})$2

where $\Omega(n^{-1/2})$3 is explicitly defined in terms of the risk level $\Omega(n^{-1/2})$4 and a generalized hinge-minimization. Optimization over $\Omega(n^{-1/2})$5 yields the worst-case lower-bound scaling, giving

$\Omega(n^{-1/2})$6

with $\Omega(n^{-1/2})$7 an explicit function of $\Omega(n^{-1/2})$8. This matches the classical Bayes risk lower bound in rate and introduces risk-level calibration absent from expected-loss analysis.

Two-Armed Gaussian Bandits

With horizon $\Omega(n^{-1/2})$9, gap $\Omega(\sqrt{T})$0, and symmetric prior on two reward environments, under an arbitrary bandit algorithm,

$\Omega(\sqrt{T})$1

and optimizing over $\Omega(\sqrt{T})$2 yields

$\Omega(\sqrt{T})$3

up to constants. The analysis makes explicit that the worst-case CVaR grows as $\Omega(\sqrt{T})$4, consistent with minimax regret rates, but again with an explicit and monotone calibration in risk level $\Omega(\sqrt{T})$5.

Bayesian CVaR versus Bayes Risk Benchmarks

A structural guarantee is formalized via: $\Omega(\sqrt{T})$6 So any lower bound for Bayes risk automatically lower bounds Bayesian CVaR, but the presented framework in fact computes the dependency of the lower bound itself on $\Omega(\sqrt{T})$7 structurally, offering sharper characterization of the tail, and thus going beyond expectation-based statements. This addresses applications in risk-averse learning and safety-critical settings, where tail behavior is operationally more relevant than mean loss.

Implications and Future Directions

• Theoretical Impact: The methodology bridges the gap between information-theoretic lower bounds for expected loss and those for tail-sensitive risk criteria such as CVaR in both passive and interactive (adaptive) settings. The two-point Hellinger template provides a reusable, computationally verifiable tool for lower-bounding prior-predictive CVaR in a variety of learning and control problems. Notably, the proofs and constructions are algorithm-agnostic and hence universal.
• Practical Relevance: For practitioners designing algorithms under robustness or risk constraints, these lower bounds quantify the irreducible tail risk for any Bayesian learning agent, informing benchmark design and the impossibility of uniformly risk-sensitive, low-tail-regret learning in certain adversarial/model-uncertainty regimes.
• Research Trajectory: The explicit two-point construction invites generalization to $\Omega(\sqrt{T})$8-point and continuum-model classes (Assouad/Le Cam generalizations) and to adaptive and nonparametric settings. Comparing these lower bounds to upper bounds for specific risk-sensitive algorithms (including risk-aware bandit and RL algorithms, e.g. CVaR/ESRL) offers direction for tight characterizations of risk/robustness trade-offs in interactive/online learning.

Conclusion

This work demonstrates that information-theoretic tools for expected-risk lower bounds, suitably generalized and instantiated, yield concrete, tight, and explicit lower bounds for prior-predictive Bayesian CVaR in interactive decision making problems. The framework is made practical via a reusable two-point Hellinger template, normalized to canonical problems, and retains sharp risk-level dependence. Future work includes extending these templates to richer model classes and connecting to algorithmic upper bounds for risk-constrained learning (2604.12519).