Pareto Optimal Risk-Agnostic Distributional Bandits with Heavy-Tail Rewards

We are grateful for your careful reading and constructive suggestions. Our detailed responses are provided below.

Q1) Generality of Lower Bounds: We acknowledge that our heavy‑tail lower bounds presently cover only Spectral Risk Measures (SRM) and Distortion Risk Measures (DRM), so UBSR and CERM remain open. Nonetheless, SRM and DRM are families, not single functionals: they subsume virtually all coherent risks used in practice, including CVaR / Expected Shortfall (Rockafellar & Uryasev 2000; Acerbi & Tasche 2002), power‑ and exponential‑spectral measures (Dowd, K., Cotter, J., & Sorwar, G. 2008), and the Wang, Proportional‑Hazard, Dual‑Power, and Beta distortions (Wang 1995; Wang et al. 1997). Establishing heavy‑tail lower bounds for these two broad classes therefore already gives our results considerable practical reach, while extending the theory to utility‑based risks such as UBSR and CERM, whose proofs require additional regularity assumptions, remains an important direction for future work.

Q2) Sensitivity to Tail Index: Thank you for pointing this out. We agree that showing results for additional tail indices would be informative. Because of space constraints we focused on the extreme heavy‑tail case $p=1.2$ (close to the limit case $p\to1$) to highlight the algorithm’s robustness where existing methods struggle. During the discussion period we will run and report supplementary experiments for additional indices between $1$ and $1.5$ (e.g. $p=1.1$ and $p=1.5$) to illustrate how performance evolves as the distribution approaches the light‑tailed regime.

Q3) Heavy‑Tail Difficulty: Thank you for the suggestion. Appendix B (“Detailed Comparison of Prior Work”) already contrasts light‑ and heavy‑tailed settings, but we will tighten that subsection by adding a brief, reader‑friendly note clarifying that light‑tailed rewards admit sub‑Gaussian concentration, so classic UCB analyses suffice, whereas heavy tails lack such exponential tails and thus require robust estimators like MoEQ. This addition should make the paper more accessible to readers new to risk‑aware bandits.

I thank the authors for their effort in preparing the rebuttal. The rebuttal addressed my questions well. To be specific, the rebuttal clarifies the generality of the lower bound, and comments that the current lower bound, though being power, covers the SRM and DRM family but not the UBSR and CERM. I do not think this is an issue and I thank the authors for the transparency here. It is also great that the authors have promised to add additional experimental results regarding sensitivity of the tail index. Please make sure to do so. Overall, I conclude that the rebuttal addressed my questions.

Thank you for the thoughtful feedback. Our detailed responses are provided below.

Q1) Optimality of MR-DistLCB: We expect the Pareto regret lower bound for the multi‑risk bandit to have the same asymptotic order as the single‑risk lower bound. Any single‑risk instance can be embedded in a multi‑risk environment by copying that one risk measure across all $k$ objectives; under this construction, the Pareto regret is identical to the single‑risk regret, so the single‑risk lower bound carries over unchanged. Because our upper bound in Theorem 6.7 matches that rate, we regard MR‑DistLCB as order‑optimal. At present, heavy‑tail single‑risk lower bounds exist only for SRM and DRM (Theorems 6.1 and 6.3), so a matching multi‑risk lower bound can currently be proved only for these two families (see Remark 6.4). Extending the lower‑bound machinery to utility‑based risks such as UBSR and CERM remains an open problem, which we will emphasise in the revised manuscript.

Q2) Theoretical Challenges: Our work closes three previously open gaps. First, we establish the first exponential‑type deviation bound in $W_{1}$ for a median‑based distribution estimator under heavy‑tail distributions (Theorem 4.4), extending median inequalities from scalar means to full CDFs. Second, we introduce partial‑bootstrap MoEQ, which updates with just one new sample, rather than an entire independent block, yet still enjoys exponential guarantees under heavy‑tail distributions, making it the first median method that is both online‑practical and theoretically sound in the infinite‑variance regime. Third, we derive the first heavy‑tail information‑theoretic lower bounds for SRM and DRM; the core difficulty here is to find an explicit, tractable expression for the gap $\Delta_{2}^{\rho}\ge\Omega(c)$ that links the risk difference to a mean (or KL) difference so the change‑of‑measure argument can go through. We achieve this for SRM/DRM using their spectral/distortion structure.

Q3) Generality of Lower Bounds: Regarding the scope of Theorem 6.3, the restriction to Spectral Risk Measures (SRM) and Distortion Risk Measures (DRM) is far less limiting than it may appear. These two families already subsume the risk measures that dominate practical finance and insurance work (Jokhadze & Schmidt 2020), for example:

• CVaR / Expected Shortfall (Rockafellar & Uryasev 2000; Acerbi & Tasche 2002)
• Power‑spectral and Exponential‑spectral measures (Dowd, K., Cotter, J., & Sorwar, G. 2008)
• Wang transform (Gaussian distortion), Proportional‑Hazard transform $g(u)=u^{\beta}$, Dual‑Power and Beta distortion families (Wang 1995; Wang et al. 1997)

These SRM and DRM families already embrace most coherent risks used in practice so establishing heavy‑tail lower bounds for them gives the theory broad practical reach. That said, an important next step is to extend the analysis to utility‑based risks such as UBSR and CERM; handling those measures will require additional regularity assumptions and is therefore left for future work.

References:

1. Acerbi, C., & Tasche, D. (2002). Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking & Finance, 26(7), 1505‑1518.

2. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of Conditional Value‑at‑Risk. Journal of Risk, 2, 21‑42.

3. Dowd, K., Cotter, J., & Sorwar, G. (2008). Spectral risk measures: properties and limitations. Journal of Financial Services Research, 34(1), 61-75.

4. Jokhadze, V. & Schmidt, W. M. (2020). Measuring model risk in financial risk management and pricing. International Journal of Theoretical and Applied Finance, 23(2), 2050012.

5. Wang, S. S. (1995). Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics, 17, 43‑54.

6. Wang, S. S., Young, V. R., & Panjer, H. H. (1997). A class of distortion operators for pricing financial and insurance risks. Journal of Risk and Insurance, 64(2), 211‑232.

I thank the authors for the detailed responses that clarified my questions / concerns. I will maintain my positive assessment.

We are grateful for your careful reading and constructive suggestions.

Typos and Readability: Thank you for catching the typo; we will correct it. We also agree that theorems start too abruptly; space constraints led to this. In the revision, we will insert brief lead‑in sentences and intuitive remarks before each result to improve readability.

Q1) Lower‑bound extension to other Lipschitz risks: As noted in Remark 6.4, our proof on the lower bounds relies on the explicit gap condition $\Delta_{2}^{\rho} ≥ \Omega(c)$, which links the risk difference to the mean difference. For SRM/DRM cases, this condition holds because they are defined by spectral/distortion weights. For UBSR/CERM cases with a linear utility, our result is still applicable. However, for nonlinear utility functions, the current change-of-measure argument is nontrivial unless we impose a specific characterization on utility functions. Hence, extending the lower bound to a general family of utility-based risk measures remains an open question, so we leave it for future work.

We appreciate the reviewer’s insightful comments.

Q1) Clarify “risk‑agnostic”: We thank the reviewer for highlighting this point and fully agree with the concern. To avoid ambiguity, we will replace the term “risk‑agnostic” with “risk‑measure‑agnostic (RMA)” throughout the paper (title, abstract, and §5). Risk‑measure‑agnostic explicitly means that the estimator and confidence index remain valid, without modification, for any Lipschitz risk functional.

Q2) Large‑n condition in Cor. 4.5 (FB‑MoEQ): We agree the large sample requirement in Cor. 4.5 looks strong, but it appears only in the proof of the full‑bootstrap (FB) setting, where a conservative union‑bound on the resampled maximum inflates the sample size; it is not inherent to MoEQ. The FB bound could be relaxed with sharper heavy‑tail tools (e.g., Bennett‑type inequalities) or other types of bootstraps (e.g., m‑out‑of‑n/Poisson bootstraps), yet tightening the FB analysis lies beyond this paper’s scope. We focus on establishing the first exponential‑decay guarantee for MoEQ under heavy tails, and we leave refinement of the FB analysis to future work. All experiments therefore employ the partial‑bootstrap (PB) MoEQ, which achieves the same exponential concentration without any oversampling.

Q3) More direct comparisons: Thank you for the suggestion. Our MR‑Trunc (L326-327) baseline is the multi‑risk extension of TED, so TED‑style truncation is already evaluated on both the synthetic and S&P‑500 datasets (see Figures 2 and 3 in Section 7). In those results, MR‑DistLCB consistently outperforms MR‑Trunc/TED, especially as the number of risk measures grows. If the reviewer has a different comparison in mind, such as another evaluation metric, please let us know and we will be happy to include it.

Q4) Lower‑bound extension to other Lipschitz risks: As noted in Remark 6.4, our proof on the lower bounds relies on the gap condition $\Delta_{2}^{\rho} ≥ \Omega(c)$, which we can express explicitly for SRM/DRM because their risk difference decomposes into a weighted integral of the CDFs (see Proof of Theorem 6.1.). For utility‑based risks like UBSR or CERM, such a decomposition is missing; the risk is defined through a non‑linear utility function, so the current KL‑based argument cannot be applied directly. While there exist some trivial cases, such as a linear utility function, a new characterization of the utility term in UBSR or CERM would be required to obtain the similar lower bound, making this an open problem we leave for future work.