A Bayesian Approach to Contextual Dynamic Pricing using the Proportional Hazards Model with Discrete Price Data

W1 Concern about model misspecification under the Cox PH assumption.

A1 Indeed, every model carries some risk of misspecification. The primary motivation for using a semi-parametric model such as the Cox model is to mitigate this risk. Nevertheless, model misspecification is practically unavoidable, making robustness to such misspecification a critical concern for practitioners. We first offer several general perspectives on this issue, with a particular focus on censored data.

Modeling the hazard rate, rather than the mean of a random variable, has a long-standing tradition in survival analysis. Estimating the mean of a survival random variable is challenging because it is often not directly observable due to censoring. In particular, the tail of the distribution plays a crucial role in determining the mean, yet it is rarely observed in censored data. This issue becomes especially severe in the case of interval-censored data, where no direct observations are available. Consequently, models that directly target the mean, such as linear or log-linear models, are highly vulnerable to model misspecification, particularly when the tail behavior is misspecified. In contrast, models targeting the hazard rate, such as the Cox model, tend to be more robust to model misspecification.

A fully distribution-free approach might be preferable in order to avoid the risk of model misspecification. Unfortunately, however, such an approach does not appear to be suitable in our context, as interval-censored data contains very limited information about the valuation distribution. As a result, estimating $F(\cdot \mid X_t)$ (or $\lambda(\cdot \mid X_t)$) without strong structural assumptions becomes extremely challenging.

In summary, we believe the Cox PH model offers a robust foundation for handling censored observations. Exploring alternative models, such as the proportional odds model, would be an interesting direction for future research.

We thank the reviewer for raising this important point. We will consider including a more detailed discussion of model misspecification issues and possible alternatives in the final version to improve clarity.

W2 & Q2 Question on extending the algorithm to handle discrete subintervals in the price space.

A2 We thank the reviewer for the insightful question. As we understand, you are considering a setting where the price set is a union of small subintervals centered around discrete grid points:

where $g_k - g_{k-1} = \delta$, $\delta \asymp n^{-\gamma}$, and $\delta' \asymp n^{-\gamma'}$. While we do not provide a formal analysis for this generalized case, we offer the following conjecture based on theoretical insights.

We first assume that the subintervals are disjoint, which corresponds to $\gamma' \geq \gamma$. If $\gamma \geq 1/3$, the grid becomes sufficiently dense so that the union of subintervals behaves like a continuous price regime. In this case, we conjecture that the estimation error would match the rate for continuous pricing, i.e., $n^{-1/3}$.

More intriguingly, when $\gamma < 1/3$, the regime becomes more subtle. In the extreme case where $\gamma = \gamma' = 0$, the price set becomes a finite union of non-shrinking subintervals, which resembles a continuous regime. Building on insights from Tang et al. (2012), the confidence interval width around each grid point in the discrete regime is roughly $n^{-(1-\gamma)/2}$. If each subinterval has width $\delta' \asymp n^{-\gamma'}$ satisfying $\gamma' \geq (1-\gamma)/2$, then each subinterval may lie entirely within the confidence region. This suggests that the algorithm could still behave similarly to the discrete regime.

We emphasize that this remains a conjecture, and we leave the formal theoretical verification as an exciting direction for future work.

Q1 Question on why the regret bound shows better dimension dependence then Choi et al. (2023) when $\gamma = 1$.

A3 The key reason lies in the tightness of the entropy (log-covering number) bound used in the convergence analysis. In Appendix B.1 of Choi et al. (2023), the authors leverage an entropy bound derived from Huang (1996), which states that the $\epsilon$-covering number of the relevant function class is of the order $(1/\epsilon^d) (e^{1/\epsilon})$. Choi et al. (2023) approximated the log-covering number as $d/\epsilon$, but this approximation is loose. A more accurate bound is given by $d \log(1/\epsilon) + 1/\epsilon$, which yields tighter control in the empirical process analysis. If this refined estimate is applied to their regret analysis, this leads to a tighter dependency on $d$. In particular, it would match the regret bound $\tilde{O}(T^{2/3} + \sqrt{dT})$ that we obtain in our setting when $\gamma = 1$. This suggests that the dimension dependence in Choi et al. (2023) is not intrinsic, but rather an artifact of a conservative entropy approximation.

minor1 Suggestion to highlight and formalize the extension to nonuniform grids

A4 We thank the reviewer for this helpful suggestion. As correctly noted, the derived regret bounds can be extended to the case of nonuniform grids, provided a certain sparsity condition is satisfied. Due to space constraints, we did not emphasize the nonuniform case in the main text. We agree that highlighting this generalization could enhance the paper’s practical relevance. We will revise the manuscript to emphasize this point more clearly, and we will consider formally presenting the result in Appendix D as a corollary in the final version.

minor2 Suggestion to include direct benchmark comparisons in the experimental results

A5 We thank the reviewer for the suggestion. As noted in Section 6, our current experiments are primarily designed to highlight the benefits of leveraging discrete support information. To demonstrate this aspect, we conducted a comparative evaluation with Choi et al. (2023), which also uses the Cox PH model under a well-specified setting.

That said, we agree with the reviewer that providing direct comparisons with benchmark models such as linear and log-linear approaches would significantly improve the readability and clarity of our experimental section, rather than referring to external results. We will incorporate these comparisons into the main text in future revisions by adding additional experiments.

References

• Choi et al. Semi-parametric contextual pricing algorithm using Cox proportional hazards model. ICML, 2023.
• Huang. Efficient estimation for the proportional hazards model with interval censoring. Annals of Statistics, 1996.
• Tang et al. Likelihood based inference for current status data on a grid: A boundary phenomenon and an adaptive inference procedure. Annals of Statistics, 2012.

Q1 Technical Novelty

A1 We appreciate the reviewer’s thoughtful reading and for highlighting the need to better articulate the technical contributions beyond prior work.

Our key contribution lies in conveying the message that, in dynamic pricing problems with a discrete price set, one can leverage the size of the price support to improve revenue. While our work is inspired by Chae (2023) and Choi et al. (2023), achieving the above goal required the following important technical developments.

One of our key technical contributions is the realization that the exploration rate $\eta_l$ must also depend on the support size $|\mathcal{G}|$ to attain the optimal regret rate. Therefore, we utilize the support size information not only to construct the estimator $\hat\theta^{l-1}$ but also to determine the exploration rate, as specified in Eq (6) of the manuscript. This enables our algorithm to adapt to all values of $\gamma \in (0,1]$ and achieve optimal regret uniformly across this range. Balancing estimation accuracy and exploration in this discrete price setting is a technically crucial contribution not addressed in prior work.

Second, although it is natural to expect that the result of Chae (2023) can be extended to survival models incorporating covariates, developing a rigorous extension without strong assumptions requires substantial non-trivial techniques and effort. This challenge is particularly pronounced in the Bayesian setting, where many parts of the proof are highly prior-specific in infinite-dimensional models. In our case, we would like to emphasize that the prior for the baseline hazard function is different from that used in Chae (2023), mainly for computational convenience, and this difference necessitates additional prior-specific technical developments.

To the best of our knowledge, no prior work has established regret bounds for Bayesian contextual pricing algorithms under discrete support, nor have they addressed the interplay between grid sparsity, exploration scheduling, and estimation error. We thank the reviewer again for this important question, which gave us the opportunity to better clarify and articulate the core technical contributions of our work. We believe that these developments collectively address the important technical gaps not covered in existing works.

Q2 Modeling Assumptions and Practicality

A2-(1) We would like to clarify that the i.i.d. assumption is not used for modeling the entire sequential process of dynamic pricing, but only within each epoch for the purpose of theoretical analysis. As described in Section 4, our algorithm adopts an epoch-based design, which partitions the time horizon $T$ into multiple epochs. Within each epoch, a fixed pricing policy is applied. This design allows us to treat the data $\{(X_t, P_t, Y_t)\}_{t \in \mathcal{E}_l}$ as i.i.d. conditioned on past observations up to epoch $1, ..., l-1$. This structure provides significant technical convenience, enabling us to analyze the estimation error of point estimator $\hat\theta^{l-1}$ at each epoch. The use of such an epoch-based design is a common strategy in the contextual pricing literature and serves as a practical bridge between theory and sequential decision-making.

A2-(2) We would like to emphasize that our setting assumes $\mathcal{G}$ depends on the level of the entire time horizon $T$, not varying across rounds $t$. This is a practically motivated assumption that sellers in real-world typically construct human-friendly discrete sets (e.g., $\mathcal{G}$={\$5, \\$10, \$15,...\}) given$T$, and$\mathcal{G}$should be understood as a prescribed set of all possible prices from which a seller may choose. For instance, one seller may plan to sell products over$T=10000$rounds using a fine grid of$K=1000$, while another seller may consider a coarser grid with only$K=10$. In this context, the former seller corresponds to a larger grid sparsity level$\gamma$. Hence,$\gamma$can be viewed as a latent property that reflects the seller's preference in designing the price grid set$\mathcal{G}$. Since this latent$\gamma$is typically unknown in advance, our key contribution is to develop an algorithm that adapts to all possible values of$\gamma \in (0,1]$, achieving optimal regret uniformly over this range.

In this light, we believe that our discrete price setting is more realistic than previous works assuming a fully continuous price setting. We agree with the reviewer that extending the framework to allow for time- or context-dependent price sets (e.g., $K$ is a function of both $T$ and the context dimension $d$) is an interesting direction for future work. We thank the reviewer again for encouraging us for pointing out a promising avenue for future extensions.

A2-(3) We agree with the reviewer that hyperparameter tuning in online learning is inherently non-trivial, which remains an active area of research (refer to Cha and Cho (2025); Lee et al. (2024)). Among recent works in contextual dynamic pricing, Fan et al. (2024) propose tuning hyperparameters at the start of each epoch based on data from previous epochs. In contrast, Choi et al. (2023) adopt a more practical approach, which is similar to ours, where hyperparameters are tuned only once using data collected during an initial period $T_0$. This approach is more computationally efficient, particularly when the total horizon $T$ is large.

To assess the practical cost of tuning, we measured the average computation time required per hyperparameter configuration in our simulation setup, where the initial period length is $T_0=3000$ and the grid size is $K=100$. The average time was 23.4 seconds, making the total tuning procedure computationally efficient in practice.

In setting where a validation set is unavailable, we suggest that practitioners select hyperparameters based on prior beliefs about the desired level of exploration. If stronger exploration is preferred, larger values for $n_1$, $\eta_1$, and $\eta_2$ should be used; for more aggressive exploitation, smaller values are appropriate. As a practical guideline, we suggest choosing $\eta_1$ and $\eta_2$ below 1 to avoid excessive exploration.

Q3 Estimation Methodology

A3 We thank the reviewer for raising this important point. In our theoretical analysis, the regret bounds are derived under the assumption that the estimator $\hat\theta^{l-1}$ corresponds to the posterior mean of the true Bayesian posterior, which contracts around the ground truth at the rate characterized in Theorems 3.1 and 3.2.

In practice, we employ a variational Bayes (VB) method to approximate this posterior due to its computational efficiency in high-dimensional and nonparametric settings. While the regret guarantees are not derived explicitly for the VB approximation, our implementation uses a variational family that is sufficiently expressive to closely match the posterior mean in all experiments. As empirically demonstrated in Liu et al. (2024), the considered VB approach performs comparably to, and sometimes even better than, traditional MCMC methods in terms of estimation accuracy.

From a theoretical perspective, as the reviewer points out, the use of a VB estimator (i.e., a point estimator based on the variational posterior distribution) naturally raises questions about its impact on the regret guarantees. Although a formal regret analysis for VB-based estimation is beyond the scope of the present work, we can offer a general theoretical perspective.

According to our regret analysis, the regret bound depends directly on the convergence rate of the estimator $\hat{\theta}^{l-1}$ (see lines 296–297). Therefore, if the VB estimator attains the same convergence rate as required in our analysis, the regret guarantees continue to hold.

In this regard, several general theoretical frameworks have been developed for analyzing the contraction rates of VB posteriors (e.g., Zhang and Gao (2020); Alquier and Ridgway (2020); Yang et al. (2020)). These results provide sufficient conditions under which the VB posterior achieves the same contraction rate as the full posterior, thereby preserving the regret guarantees in our setting. While a rigorous theoretical treatment of VB methods in our specific setting remains an open question, we greatly appreciate the reviewer’s insightful comment and will include a detailed discussion of this point in the final version of the paper.

References

• Alquier and Ridgway. Concentration of tempered posteriors and of their variational approximations. Annals of Statistics, 2020.
• Cha and Cho. Hyperparameters in continual learning: A reality check. TMLR, 2025.
• Chae. Adaptive Bayesian inference for current status data on a grid. Bernoulli, 2023.
• Choi et al. Semi-parametric contextual pricing algorithm using Cox proportional hazards model. ICML, 2023.
• Fan et al. Policy optimization using semiparametric models for dynamic pricing. JASA, 2024.
• Lee et al. Hyperparameter selection in continual learning. arXiv, 2024.
• Liu et al. Variational Bayesian approach for analyzing interval-censored data under the proportional hazards model. Computational Statistics & Data Analysis, 2024
• Yang et al. $\alpha$-variational inference with statistical guarantees. Annals of Statistics, 2020.
• Zhang and Gao. Convergence rates of variational posterior distributions. Annals of Statistics, 2020.

W1 Justification of our setting

A1 We clarify that $\delta = \kappa n^{-\gamma}$ is only used in the i.i.d. setup discussed in Sections 2 and 3, where $n$ denotes the number of observations. In contrast, our sequential contextual pricing setting is introduced in Section 4, where the grid resolution is defined as $\delta = \kappa T^{-\gamma}$ given time horizon $T$. Importantly, under this formulation, the discrete price set (action space) $\mathcal{G}$ remains unchanged across all epochs and rounds.

In our setup, given $T$, the discrete price set $\mathcal{G}$ should be interpreted as a pre-specified set of all possible prices from which a seller may choose. In practice, prices are often selected from human-friendly discrete sets (e.g., $\mathcal{G}$={\$5, \\$10, …}), and multiple observations are typically available for each. For instance, one seller may plan to sell a product over $T=10000$ rounds using a fine grid of $K=1000$, while another seller may consider a coarser grid with only $K=10$. In this example, the former seller corresponds to a larger value of $\gamma$, whereas the latter corresponds to a smaller $\gamma$. Thus, $\gamma$ can be interpreted as a latent property that captures the seller's preference in designing the price set $\mathcal{G}$. Since this latent $\gamma$ is typically unknown in advance, our key contribution is to develop an algorithm that adapts to all possible values of $\gamma \in (0, 1]$, achieving optimal regret uniformly over this range.

We emphasize that our framework does not allow the price set $\mathcal{G}$ to change over time as $t$ increases; it remains fixed throughout all epochs and rounds. Of course, it is possible that, up to a certain point, some prices in $\mathcal{G}$ may not have been observed. Exploring such settings would be an interesting direction for future research.

W2 Request for a proof sketch or roadmap

A2 We sincerely thank the reviewer for carefully engaging with the technical sections and for this insightful suggestion. We appreciate your effort to understand the proof structure and agree that providing a clear roadmap can improve the accessibility of our results.

Below, we provide a high-level proof roadmap outlining how each of the main lemmas and theorems are logically connected.

Posterior Convergence Rate (Theorem 3.1, 3.2) We analyze the convergence rate of the posterior under both discrete ($\gamma < 1/3$) and continuous ($\gamma \geq 1/3$) regimes. The roadmap is as follows:

• Lemma A.6, A.7: Provide covering number bounds in the discrete and continuous regimes, respectively.
• Lemma A.1: Establish posterior consistency under mild assumptions.
• Theorem 3.1, 3.2: Derive the posterior convergence rates.
• Lemma C.1-C.4: Provide the decomposition of the Hellinger distance and the prior mass condition required for Lemma A.1, Theorem 3.1, and Theorem 3.2.

Regret Analysis (Theorem 5.2) The regret analysis builds upon the posterior convergence results and decomposes regret across epochs:

• Lemma 5.1: Bound the estimation error of the point estimator $\widehat{\theta}^{l-1}$ using posterior concentration.
• Lemma B.1: Decompose the regret into estimation error and exploration terms.
• Lemma B.2: Derive the regret during each epoch.
• Theorem 5.2: Aggregate per-epoch regret to derive a total regret upper bound.
• Lemma C.2, C.6, C.7: Technical lemmas required for Lemma 5.1 and Lemma B.2.

We will incorporate this roadmap in the final version to improve the clarity and accessibility of our technical proofs.

W3 Request for discussion on more general censoring mechanism.

A3 We thank the reviewer for the detailed and thoughtful comment. We would like to clarify why the Cox PH model is both a principled and appropriate choice, particularly in the presence of censored observations.

The key distinction between the Cox PH model and standard linear models lies in the use of the hazard function, which is a central concept in survival analysis. Unlike the linear model, which typically models the conditional mean of a random variable, the Cox PH model focuses on modeling the hazard rate, a quantity that fully characterizes the survival function. This is especially important in censored data settings, where direct observations of event times (in our case, customer valuations) are often unavailable or incomplete. In addition, the hazard function is particularly useful since it is only required to be non-negative and integrable. This flexibility makes hazard-based modeling especially well-suited to censored data.

In our work, we adopt the Cox PH model, which takes the form: $\lambda(v \mid x) = \lambda_0(v) \exp(x^\top \beta)$. A key advantage of this model is that it permits separate analysis of $\lambda_0$ and $\beta$, enabling theoretical development under minimal assumptions (e.g., non-negativity) on the functional form of $\lambda_0$. This separability facilitates both flexibility in modeling and tractability in analysis.

In contrast, linear models such as $v = x^\top \beta + \epsilon$, require explicit assumptions on the error distribution, with common choices including the Gaussian and extreme value distributions. However, these assumptions can be quite restrictive, particularly in censored data where estimating the tail behavior of $\epsilon$ is difficult. These limitations were a primary reason we use the Cox PH model in our framework.

That said, we emphasize that our methodology is not inherently restricted to the Cox PH model. Other semi-parametric models commonly used in survival analysis, such as the Proportional Odds model, or the Accelerated Failure Time model, can also be considered within our framework. Extending our contextual dynamic pricing problem to these models under a discrete price setting is a promising direction for future work.

In implementing the Cox PH model, we place a Gamma prior on the baseline hazard function mainly for computational convenience. We would like to clarify that this choice does not reduce the inherent complexity of the model, nor does it restrict the model’s generality.

We again thank the reviewer for raising this important point and will incorporate a summary of the above discussion into the final version.

Q1 Extensions to other semi-parametric models under interval-censored data

A1 We thank the reviewer for highlighting the connection between statistical estimation results and pricing algorithms.

Our key theoretical insight is that once the price is discretely supported, one can construct a better pricing algorithm by leveraging this support information, which leads to improved regret rates. While our work focuses on the Cox PH model, we agree that extensions to other semi-parametric models under the case 1 interval-censored data structure are both plausible and worth exploring. For example, models such as the Proportional Odds (PO) model or the Accelerated Failure Time (AFT) model can be considered within the same framework.

If these models permit the incorporation of discrete support information, their performance could potentially be improved in a similar fashion. From a technical perspective, the Cox PH model permits separate analysis of the baseline function $F_0$ and the regression parameter $\beta$ (see Theorem 3.3 in Huang (1996)). Once such separability is established, the theoretical analysis of $F_0$ with discrete support information becomes more tractable. Notably, Theorem 3.3 in Huang (1995) also implies that such separability holds in the PO model, suggesting that discrete support information may be exploited for pricing algorithm design in that setting.

A detailed extension to such directions across a broader class of models would be interesting for future work.

Q2 Technical challenges and justification for regret behavior under nonuniform grids

A2 In the main text, we focus on uniform grids where the grid spacing satisfies $|g_{k+1} - g_k| = \delta$ for all $k$. In contrast, Appendix D discusses nonuniform grids where the spacing satisfies $a\delta \leq |g_{k+1} - g_k| \leq b\delta$ for some constants $0 < a \leq b < \infty$. In both cases, the grid spacing satisfies the asymptotic equivalence $|g_{k+1} - g_k| \asymp \delta$, ensuring the grid resolution is still characterized by $\delta$.

As a result, using nonuniform grids affects only the constant factors in the regret bound through their dependence on $a$ and $b$, while the key dependence on the sparsity level $\gamma$ remains unchanged. Hence, this generalization does not fundamentally change the regret behavior.

Q3 Challenges in generalizing posterior convergence rates to contextual settings

A3 Our first key technical contribution lies in extending the convergence rate analysis from the non-contextual case in Chae (2023) to the Cox PH model with covariates. This extension requires substantial non-trivial techniques and effort without strong assumptions. This challenge is particularly pronounced in the Bayesian setting, where many parts of the proof are highly prior-specific in infinite-dimensional models. In our case, we would like to emphasize that the prior for the baseline hazard function is different from that used in Chae (2023), mainly for computational convenience, and this difference necessitates additional prior-specific technical developments.

A second technical difficulty lies in bounding the Hellinger distance when incorporating covariates. In our setting, we show that the Hellinger distance can be bounded by a separable and additive combination of the distances between the baseline functions and the regression parameters:

as shown in Lemmas C.1 and C.2. This additive structure is crucial for controlling the Hellinger metric entropy in the posterior contraction analysis (see Lemma A.6 and A.7).

To enable such analysis, it is essential that the posterior distribution concentrates around the true parameter, i.e., we require posterior consistency (Lemma A.1). Establishing this result under a semi-parametric model like the Cox PH model is technically demanding.

A third technical point is related to Assumption (A4), which is required for proving Lemma A.1. As discussed in Remark A.2, one could impose a stronger condition than (A4) to simplify the proof and derive results as in (8) uniformly for all $\gamma$. However, such a restrictive condition would require larger values of the exploration parameter $\eta_l$, leading to higher regret due to excessive exploration (see lines 296–297). Hence, adopting the weaker assumption (A4) is essential in the contextual pricing problem, and proving posterior consistency under this condition is a delicate and non-trivial contribution. For further technical details, we refer the reviewer to Appendix A.1.

Q4 Phase transition in covergence and regret rates across $\gamma < 1/3$ and $\gamma \geq 1/3$

A4 We build upon a key insight from Tang et al. (2012), who first observed this phenomenon in non-contextual survival analysis. The underlying intuition is closely related to isotonic regression problems, where one aims to estimate a non-decreasing function such as the baseline function $F_0$.

When $\gamma < 1/3$, the grid of prices is sufficiently sparse, meaning that grid points are far enough. In this case, the estimation problem at each grid point becomes nearly independent of the others. As a result, simple pointwise estimators are already well-ordered and directly serve as valid solutions to the isotonic regression problem. This leads to improved estimation efficiency and faster convergence rates.

In contrast, when $\gamma \geq 1/3$, the grid becomes denser and neighboring points are more closely spaced. Consequently, the pointwise estimators can fluctuate and may violate the monotonicity. In such cases, local pooling over neighboring grid points becomes necessary to construct a valid non-decreasing estimator, which limits the convergence rate.

This transition in estimation behavior naturally extends to the regret bounds in the dynamic pricing problem. Notably, regret in our setting is influenced not only by estimation error but also by the degree of exploration (see lines 296-297). When $\gamma < 1/3$, sparse grids allow accurate estimation at each grid point with minimal exploration. This enables the algorithm to prioritize exploitation, as reflected in the decreasing exploration parameter $\eta_l$ defined in Eq (6).

On the other hand, for $\gamma \geq 1/3$, denser grids require more frequent exploration to maintain learning efficiency across the entire price space. This increased exploration effort leads to a higher regret bound. Thus, both estimation error and the degree of exploration jointly explain the phase transition observed in our regret analysis.

Q5 Validity of regret comparison with Choi et al. (2023) under different benchmarks

A5 We appreciate the reviewer's concern regarding the validity of comparing regret bounds under different benchmarks, and we would like to clarify that our discrete price setting naturally interploates between discrete and continuous regimes.

In particular, this setting is parameterized by the grid size $K$, or equivalently the sparsity level $\gamma$. At one extreme, when $K = c$ (i.e., $\gamma = 0$) for some constant $c > 0$, the price set is fixed and highly sparse. At the other extreme, when $K=T$ (i.e., $\gamma = 1$), the grid becomes dense enough to cover the continuous price setup. Our algorithm is designed to adapt to any $\gamma \in (0,1]$, and thus the derived regret bounds naturally extend to the continuous regime, making them comparable to those in Choi et al. (2023). The main point in lines 309-311 is to emphasize that our algorithm can achieve faster regret rates by leveraging the discreteness when the grid is sparse.

Moreover, we conducted the comparison between both algorithms under the same discrete pricing benchmark by varying the grid size. As shown in Figure 1, our algorithm significantly outperforms the method in Choi et al. (2023) when the grid is sparse, due to its ability to utilize the structure of the discrete support.

Q6 More discussions on the choice of $\eta_l$ and satisfaction of Assumption (A4)

A6 We appreciate the reviewer's attention to the role of $\eta_l$ and assumption (A4), both of which are indeed central to our algorithm.

As noted in lines 296–297, the choice of $\eta_l$ directly influences the regret bound by balancing exploration and exploitation across epochs. Specifically, $\eta_l$ controls the degree of uniform exploration over the grid, which is reflected in $q(\cdot \mid x)$. Since assumption (A4) requires a lower bound on $q(\cdot \mid x)$, an appropriate choice of $\eta_l$ is essential to ensure this condition is satisfied. This connection is formally established in Lemma C.5.

We will revise the manuscript to better highlight the role of $\eta_l$ and its relationship to (A4).

References

• Chae. Adaptive Bayesian inference for current status data on a grid. Bernoulli, 2023.
• Choi et al. Semi-parametric contextual pricing algorithm using Cox proportional hazards model. ICML, 2023.
• Huang. Maximum likelihood estimation for proportional odds regression model with current status data. Analysis of Censored Data, 1995.
• Huang. Efficient estimation for the proportional hazards model with interval censoring. Annals of Statistics, 1996.
• Tang et al. Likelihood based inference for current status data on a grid: A boundary phenomenon and an adaptive inference procedure. Annals of Statistics, 2012.