Common response to all reviewers

We thank the reviewers for their time and effort in reviewing our work. We are glad that our work combining PAC-Bayes + RL + Healthcare setting has received several favourable comments, as expressed in "The application of the PAC-Bayesian framework to reinforcement learning is novel and insightful", "bound has improved constants relative to earlier work on PAC-Bayes bounds in RL", and "The insight of operating the PAC-Bayes value-error bound in Theorem 3.2 for exploration and exploitation is interesting".

We also acknowledge the concerns and criticisms, addressed individually below. While the healthcare setting is part of the novelty in our contribution, we realize that there is a need to better contextualize our work, especially the claims regarding safety, to avoid unintended overclaiming/overselling and generally to improve the clarity. We intend to address this in the revised paper. For instance, to bring the title into a shape that represents the work better, we're considering "PAC-Bayesian Reinforcement Learning Trains Generalizable Treatment Policies" if this sits well. Likewise, we have also streamlined the abstract, and we'll do the corresponding updates in the revised paper to address your feedback.

Should our paper be accepted, we commit to implementing the improvements described in our rebuttal, and to generally shape the paper into a high-quality contribution. We do believe our work contributes an interesting combination of PAC-Bayes, RL, and a healthcare setting demonstration, which deserves the attention of the NeurIPS community; and so we kindly ask the reviewers to reconsider their evaluation to support acceptance.

Response to Reviewer paMC

Thank you for your thorough review and constructive feedback. We appreciate your careful reading and will address each of your concerns.

AQ1, Regarding the title and definitions of "safe" and "generalizable":

Very interesting point was raised and we thank the reviewer for this. Therefore, we should formalize these concepts:

• Generalization: Our PAC-Bayes bound (Theorem 3.2) provides a high-probability guarantee that the deviation of the negative empirical return from its mean (the true expected return) is bounded as per the components of the upper bound. This directly quantifies generalization, how well a policy trained on training trajectories performs on new patient trajectories.
• Safety: For a pre-set (user-chosen and typically problem-dependent) level $\delta$, our framework provides formal certificates that $\mathbb{P}(`return` \ge `critical\\_threshold`) > 1-\delta$ before deployment. For healthcare, this means verifying that treatment policies meet predefined safety standards (e.g., survival probability).

We agree the title could be more precise. A revised title like the one in our common response above may better reflect the content.

AQ2, Regarding writing and space allocation:

We acknowledge that the extensive preliminaries (pages 2-5) do compress our methodological contributions. However, as noted by Reviewer yN38, it helps contextualize the contribution. In the revision, we will aim to find a better balance by condensing Section 2 to make more room for the algorithm presentation and theoretical insights. This is easily doable, we already have a good start on this and the final version will show these preliminaries compressed so that readers can reach the experiments sooner, as requested.

AQ3, Regarding infectious disease control literature:

Thank you for these references. This line of work is indeed relevant, and we will include it in our discussion.

AQ4, Regarding martingale concentration inequalities:

Thank you for pointing out this imprecision. In lines 43-44, we were referring to standard generalization bounds outside the RL domain (hence citations [5,6] to Vapnik-Chervonenkis and Hoeffding). We mistakenly included citation [7] (Azuma) which indeed concerns martingales and should have appeared in our related work discussion where we already mention them. Easy to fix and this will be addressed in the final version.

You are correct in noting that martingale-based concentration inequalities are well established in RL theory as your cited papers exemplify. Our contribution is distinct in different ways: Rather than constructing or identifing martingale sequences from the RL data (e.g., value function errors, Bellman residuals), then applying martingale concentration inequalities (Azuma-Hoeffding, Freedman), we directly leverage the Markov chain structure, and apply a concentration inequality for Markov chains with explicit constants. What drives us is that RL problems often fail to naturally yield martingales, but they all have Markov structure by design. In any way, we'll revise this paragraph to clarify we meant general ML bounds require i.i.d. assumptions, while acknowledging the rich literature on martingale methods in RL in related works section.

AQ5, Elaboration on generalization properties (line 195-196):

The PAC-Bayesian formalism enables generalization reasoning through the KL divergence from the prior to the posterior. Specifically, the bound holds uniformly over all distributions $\rho$, in particular the posterior chosen after seeing data. The KL term penalizes posteriors that deviate from the prior, thus we obtain guarantees for an entire distribution of policies.

AQ6, Relation between LHS and prior $\mu$ in Theorem 3.2:

The expectation in LHS is over the posterior $\rho$. The prior $\mu$ appears on the RHS through $\mathrm{KL}(\rho \| \mu)$. This is a direct result of applying the change-of-measure inequality in Appendix B.5.2.

AQ7, Regarding posterior fitting:

Indeed, this requires clarification. For obtaining $\rho$, we cannot directly optimize the lower bound in (9) because it's non-convex, there are two known solutions: I. construct an inverted KL version of it and optimize, but the inverse KL can only be estimated using bisection for example, and the method is sensitive to gradient explosion. II. construct a quazi-convex version (Thiemann et al. 2017, Masegosa et al. 2020) and use alternated optimization (Masegosa et al. 2020 Appendix H). Using this method, the RHS of Eq(9) becomes of the form $-\underset{\theta \sim \rho}{\mathbb{E}}\left[\hat{\mathcal{L}}\_{\mathfrak{D}}(\theta)\right] - f(\lambda)\mathrm{KL}(\rho\Vert\mu)$ This optimization scheme can also be slow and impractical for RL (optimize $\lambda$ for a fixed $\rho$ and vice-versa). That's why we took a practical decision of absorbing $f(\lambda)$ into a scalar $\beta$ that intuitively tells how much of KL regularization we want to enforce. Finally, the term $-\underset{\theta \sim \rho}{\mathbb{E}}\left[\hat{\mathcal{L}}\_{\mathfrak{D}}(\theta)\right]$ is just approximated with a reliable empirical estimate taken from the critic network $\mathbb{E}\_{\theta \sim \rho}[-Q(s, \pi\_\theta(s))]$.

AQ8, Experimental results and validation:

We acknowledge that PB-SAC doesn't dramatically outperform baselines in terms of return. However, the significance of our work lies in providing both competitive performance and formal guarantees. (Please refer to our response to Reviewer zDTS regarding clinical relevance.)

Regarding the seeds: we used 5-10 seeds (line 308)

Minor Issues:

Thank you for highlighting those typos, we corrected all of them. For the last one, That condition from prior works depend on the total number of observed samples (they denote it with $n$) and in our work, it should be $HT$ (or $N$ which is defined in line 185) instead of $H$.

We hope this addresses all your concerns and clarifies our contributions. We greatly appreciate your detailed feedback and we remain open to further discussion if any points require additional clarification.

Common response to all reviewers

We thank the reviewers for their time and effort in reviewing our work. We are glad that our work combining PAC-Bayes + RL + Healthcare setting has received several favourable comments, as expressed in "The application of the PAC-Bayesian framework to reinforcement learning is novel and insightful", "bound has improved constants relative to earlier work on PAC-Bayes bounds in RL", and "The insight of operating the PAC-Bayes value-error bound in Theorem 3.2 for exploration and exploitation is interesting".

We also acknowledge the concerns and criticisms, addressed individually below. While the healthcare setting is part of the novelty in our contribution, we realize that there is a need to better contextualize our work, especially the claims regarding safety, to avoid unintended overclaiming/overselling and generally to improve the clarity. We intend to address this in the revised paper. For instance, to bring the title into a shape that represents the work better, we're considering "PAC-Bayesian Reinforcement Learning Trains Generalizable Treatment Policies" if this sits well. Likewise, we have also streamlined the abstract, and we'll do the corresponding updates in the revised paper to address your feedback.

Should our paper be accepted, we commit to implementing the improvements described in our rebuttal, and to generally shape the paper into a high-quality contribution. We do believe our work contributes an interesting combination of PAC-Bayes, RL, and a healthcare setting demonstration, which deserves the attention of the NeurIPS community; and so we kindly ask the reviewers to reconsider their evaluation to support acceptance.

Response to Reviewer zDTS

Thank you for your thoughtful review. We appreciate your recognition of the importance of our problem and would like to address your concerns.

Regarding novelty and the IID-to-mixing extension:

While we appreciate the perspective, we respectfully note that the extension is not as immediate as it might appear. Although it is true that after $t_{mix}$ steps, the distribution of $X_{t_{mix}}$ is close to the stationary distribution $\nu$, however, it doesn’t guarantee that $X_{t_{mix}}$ is approximately independent of $X_0$, let alone consecutive states $X_{t_{mix}}$ and $X_{t_{mix}+1}$, those are definitely not independent. A broad vision can be found in the referenced work of Paulin (2018) where they introduce the technique of partitioning the chain (with $t_{mix}$ as partition size) and constructing a Marton coupling for the partition whose mixing matrix satisfies Eq(2.1) in Proposition 2.4 of Paulin (2018). From that we can infer that dependence decays exponentially with the mixing time, therefore, for approximate independence we usually need to wait longer, often several times $t_{mix}$.

Regarding the second part of the question on what makes the theoretical development challenging. While we do not claim that our approach was uniquely difficult, we can nonetheless highlight several challenges we encountered during the development: First, to apply McDiarmid-type inequality for dependent random variables, we need to prove that our chosen function (negative empirical return) satisfies a bounded-differences condition with real and explicit (especially for RL) constants and this is what we've shown in Lemma 3.1 with its proof in Appendix (B.1, B.2, and B.3). Additionally, that derivation must carefully track how this perturbation propagates to future states, and this was discussed and proven in Appendix B.4. Another interesting discussion on the contrast to the literature of PAC-Bayes RL can be found in the paragraph at line 277 and Appendix B.6. Finally, The algorithmic part was as challenging as the theoretical part, the main difficulty lies in maintaining the statistical guarantee provided by the bound without hindering learning, because the KL penalizes straying too far from the prior. Thus, we proposed the prior resetting mechanism that helps controlling the KL, but we doubted it will violate the theoretical guarantees, that's why to make sure we respect them the algorithm collects fresh data at each update using a frozen policy network.

Regarding $\tau_{min}$ estimation:

We direct the reviewer to our response to Reviewer pTac addressing this issue.

Regarding clinical relevance:

We agree that simulation-to-reality gap is a concern, as stated in the title "Towards Safe and Generalizable...", which is why we view our work as providing one component of a comprehensive safety framework for clinical RL deployment, not a complete solution. More discussion can be found in the answer to question 2 of reviewer yN38, as well as the first part of our response to Reviewer pTac.

Common response to all reviewers

We thank the reviewers for their time and effort in reviewing our work. We are glad that our work combining PAC-Bayes + RL + Healthcare setting has received several favourable comments, as expressed in "The application of the PAC-Bayesian framework to reinforcement learning is novel and insightful", "bound has improved constants relative to earlier work on PAC-Bayes bounds in RL", and "The insight of operating the PAC-Bayes value-error bound in Theorem 3.2 for exploration and exploitation is interesting".

We also acknowledge the concerns and criticisms, addressed individually below. While the healthcare setting is part of the novelty in our contribution, we realize that there is a need to better contextualize our work, especially the claims regarding safety, to avoid unintended overclaiming/overselling and generally to improve the clarity. We intend to address this in the revised paper. For instance, to bring the title into a shape that represents the work better, we're considering "PAC-Bayesian Reinforcement Learning Trains Generalizable Treatment Policies" if this sits well. Likewise, we have also streamlined the abstract, and we'll do the corresponding updates in the revised paper to address your feedback.

Should our paper be accepted, we commit to implementing the improvements described in our rebuttal, and to generally shape the paper into a high-quality contribution. We do believe our work contributes an interesting combination of PAC-Bayes, RL, and a healthcare setting demonstration, which deserves the attention of the NeurIPS community; and so we kindly ask the reviewers to reconsider their evaluation to support acceptance.

Response to Reviewer pTac

Thank you for your constructive review and for recognizing the novelty of our approach. We appreciate your thoughtful questions and concerns.

Regarding finite mixing time and robustness:

You raise an excellent point about the mixing time assumption. Our theoretical bound indeed requires $\tau_{min}$ to be finite, and this is a clear limitation if we don't meet this need in practice. Alternative approaches could use spectral methods that can provide tighter bounds in addition, but they require estimating the spectral (or pseudo spectral) gap, which necessitate sufficient state-action coverage: We need enough visitation counts $N(s,a,s')$ to construct a reliable empirical transition matrix $\mathcal{\hat{P}}_n$. However when the state space is continuous, constructing transition matrices is infeasible.

In practice (Algorithm 1, line 17), we use autocorrelation decay of the reward signal to estimate $\tau_{min}$ because it is simple and can be computed from streaming trajectories without storing full visitation counts. If an overestimation happens, our bound remains valid but becomes potentially looser, so it's harmless in practice. If someone wants a tight bound, they can resort to the option of collecting more trajectories, though it can also be costly as discussed in the answer to question 1 of reviewer yN38. However, on the other hand, underestimation can be a problem, because the bound becomes tighter causing overconfidence; one potential (untested) solution is to compute autocorrelation not only from the reward but from other information that flows in the chain (e.g., vital signs extracted from the states over time), this way we can verify from multiple sources.

Regarding KL divergence alternatives:

We completely agree about KL limitations in high dimensions. As mentioned in our limitations (Section 5), Wasserstein-based PAC-Bayes bounds might offer compelling advantage by respecting parameter space geometry. We view this along with avoiding posterior collapse as an exciting direction for future work that could significantly tighten theoretical certificates.

Regarding Figure 2(c) clarity:

We apologize for the confusion. Indeed Figure 2(c) isn't clear, it shows episodic returns for Humanoid, where PB-SAC (pink) closely tracks SAC (green) despite the additional PAC-Bayes regularization. The apparent visual complexity arises from the high variance in Humanoid and from not properly taking the mean over different runs. We will fix this issue as well as the small font size in the legend in the final version.

Regarding Equation (6):

Yes, the event in the indicator should be $\neq$. Thank you for noticing the typo! It's fixed now.

Common response to all reviewers

We thank the reviewers for their time and effort in reviewing our work. We are glad that our work combining PAC-Bayes + RL + Healthcare setting has received several favourable comments, as expressed in "The application of the PAC-Bayesian framework to reinforcement learning is novel and insightful", "bound has improved constants relative to earlier work on PAC-Bayes bounds in RL", and "The insight of operating the PAC-Bayes value-error bound in Theorem 3.2 for exploration and exploitation is interesting".

We also acknowledge the concerns and criticisms, addressed individually below. While the healthcare setting is part of the novelty in our contribution, we realize that there is a need to better contextualize our work, especially the claims regarding safety, to avoid unintended overclaiming/overselling and generally to improve the clarity. We intend to address this in the revised paper. For instance, to bring the title into a shape that represents the work better, we're considering "PAC-Bayesian Reinforcement Learning Trains Generalizable Treatment Policies" if this sits well. Likewise, we have also streamlined the abstract, and we'll do the corresponding updates in the revised paper to address your feedback.

Should our paper be accepted, we commit to implementing the improvements described in our rebuttal, and to generally shape the paper into a high-quality contribution. We do believe our work contributes an interesting combination of PAC-Bayes, RL, and a healthcare setting demonstration, which deserves the attention of the NeurIPS community; and so we kindly ask the reviewers to reconsider their evaluation to support acceptance.

Response to Reviewer yN38

Thank you for your thoughtful review. We appreciate your recognition of the novelty and insight in applying the PAC-Bayesian framework to reinforcement learning.

AQ1, Regarding healthcare-specific significance:

The healthcare setting presents unique challenges beyond other decision-making domains, we try to outline how our work meets some of those challenges:

• Irreversibility of failures. Unlike other domains where the consequences of failures can be damaged equipment (robotics) or wasted compute (LLMs), failures in medical decisions can cause permanent harm. A statistical guarantee can certify an RL algorithm and enable clinicians to verify that $\mathbb{P}(`return` \ge `critical\\_threshold`) > 1-\delta$ (i.e. with high probability, the return won't drop bellow threshold) before deployment.
• Regulatory requirements. Healthcare algorithms face strict approval processes requiring formal safety guarantees. Theoretical certificates of the kind we put forward in this work are a step towards addressing this need.
• Limited exploration. One cannot freely explore treatment options on patients. One may infer that our bound in Eq. 9, with UCB-like structure, helps guide safe exploration (the KL penalizes going too far from the prior), which is beneficial for off-policy learning from historical data.

To address your concern comprehensively, we will add the following paragraph to our revised paper to clarify these domain-specific challenges: "Healthcare presents distinct challenges for reinforcement learning that go beyond those encountered in domains such as robotics or chemistry. First, the heterogeneity of environments and variables is particularly pronounced: medical datasets often contain dozens of interdependent variables (vital signs, lab results, medication history), and even after feature selection, discarding variables risks losing essential clinical information. Second, partial observability is intrinsic to medical data, despite all diagnostic efforts, the true physiological state remains fundamentally uncertain, necessitating algorithms that can generalize robustly. Finally, exploration through online learning is ethically and practically infeasible. As a result, learning must be conducted offline using historical data, making our PAC-Bayes framework with its formal guarantees particularly valuable."

AQ2, Regarding concrete healthcare guidance:

As example to illustrate the advantage of using PAC-Bayes RL in a healthcare setting, consider the ICU-Sepsis experiments (Section 4). This experiment demonstrates practical value and addresses the request in the following sense. The bound certifies with $95$% confidence ($\delta = 0.05$) that the survival probability exceeds a threshold, directly informing deployment decisions. For instance, in adaptive dosing for chronic conditions, our framework can guarantee that expected patient health scores won't fall below certain values, it only remains to verify whether those values align with the critical threshold defined by healthcare professionals, something where traditional RL cannot provide.

As for the second part of the question about the benefit compared to general PAC learning bounds, we can identify two main things: I. General PAC learning bounds are heavily studied and provide rigorous guarantees but they are generally based on assumptions (e.g. independent data) not met in RL, and furthermore can be loose in many problems of interest. II. PAC-Bayes uniquely allows encoding prior clinical knowledge through $\mu$ (a.k.a prior-informed PAC-Bayes). We can for instance initialize the prior using existing clinical guidelines (e.g., surviving sepsis campaign protocols). This makes $\mathrm{KL}(\rho \| \mu)$ small when the learned policy aligns with established practices. Therefore tightening the bound compared to general PAC analysis that treats all policies equally.

Thank you for highlighting these important aspects. We hope this clarifies how our work specifically addresses healthcare challenges while maintaining the theoretical rigor.

We took your suggestion very seriously and have restructured our introduction accordingly. The new structure opens with a concrete clinical scenario, establishes healthcare-specific challenges that motivate our work, and only then introduces the technical framework. Below is an excerpt:

Introduction

Sequential decision-making under uncertainty is particularly critical in healthcare settings. Consider sepsis treatment in the Intensive Care Unit (ICU), where clinicians face a cascade of interconnected decisions: when to initiate antibiotics, how to titrate intravenous fluids while monitoring organ function, whether to start vasopressor agents and at what dose, and if mechanical ventilation becomes necessary [1 ]. Each intervention shapes not only immediate physiological responses—blood pressure, oxygen saturation, urine output—but also influences which treatment options remain viable hours later. A fluid bolus that stabilizes blood pressure now might worsen pulmonary edema later; aggressive vasopressor dosing could restore perfusion but risk peripheral ischemia. These decisions unfold against a backdrop of incomplete information and patient heterogeneity, where two patients with similar presentations may respond differently to identical interventions. This exemplifies a broader challenge in healthcare: optimizing treatment sequences where each action constrains future possibilities, outcomes manifest over extended timescales, and the stakes of suboptimal decisions are measured in permanent organ damage or mortality rather than mere inefficiency.

Reinforcement learning (RL) offers a natural framework for such sequential decision-making, learning policies that adapt based on patient responses over time. Recent work has explored RL for critical care management [2 , 3 ], precision drug dosing [3], infectious disease control [x, x], and identifying treatment paths that avoid adverse outcomes [4 , 2 ]. These applications demonstrate RL’s potential to enhance clinical decision-making by learning from historical treatment data.

The healthcare setting imposes three fundamental constraints that distinguish it from general RL. (1) Irreversibility of failures: Unlike in other domains, where failures may result in damaged equipment (e.g., robotics) or wasted computational resources (e.g., LLMs), failures in medical decision-making can lead to permanent harm. Imprecisely calibrated therapeutic interventions, although potentially beneficial in addressing specific physiological deficits, can inadvertently worsen chronic conditions by disrupting compensatory mechanisms or exacerbating the patient’s overall state [ 5 ]. Thus, we require to accept only policies that highly deliver treatment outcomes above a minimum acceptable level {footnote: For a pre-set (user-chosen and typically problem-dependent) confidence level $\delta$, We only accept policies that satisfy $\mathbb{P}(`return` \ge `critical\\_threshold`) > 1-\delta$ (i.e., with high probability, the return of such policy won't drop bellow threshold)} before being used in real clinical settings. (2) Regulatory requirements: Healthcare algorithms face strict approval processes requiring formal safety guarantees [ 6 ], theoretical certificates of this kind are a step toward addressing this need. (3) Limited exploration: Ethical and practical constraints prohibit exploratory actions on patients. Learning must rely primarily on historical data, necessitating methods that can provide guarantees while guiding any limited exploration safely.

There is a widely unmet demand for rigorous, high-confidence guarantees on the generalization capabilities of machine learning models [ 7]. Providing such guarantees faces a fundamental theoretical challenge: the sequential, dependent nature of RL data violates the independence assumptions underlying most generalization results. Standard bounds in machine learning [8] and concentration inequalities [9 ] assume i.i.d. samples—clearly inappropriate for trajectories where each state depends on previous actions. While there has been developments on martingale-based concentration inequalities [10, 11] by constructing martingale sequences from value function errors or Bellman residuals, these require identifying suitable martingale structures that may not naturally arise in all problems. The PAC-Bayesian framework [12, 13 , 14, 15 ] offers an alternative path: it can provide data-dependent bounds while incorporating prior knowledge (valuable for encoding clinical guidelines). However, extending PAC-Bayes to handle Markovian dependencies remains relatively unexplored.

[Rest of introduction continues with some related works, contributions, and paper organization...]

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Thanks again for this valuable guidance. We believe it has significantly improved the paper's narrative flow. We would also appreciate your thoughts on our proposed revised title in the common response, and any further comments.

Thank you for the valuable comments and pushing us to develop a more meaningful healthcare-specific contribution.

Our Response

Your comment about needing a "non-trivial solution with healthcare implications" resonated deeply. Initially, we had been thinking about clinically reinterpreting standard PAC-Bayes bounds, but as you said these felt too generic.

We therefore carefully rethought the notion of irreversibility. In healthcare, certain states represent permanent, irreversible harm. Which isn't just about negative rewards; but rather about states from which recovery is impossible.

This realization reminded us of a very recent work by Richens et al. (2025) (and references therein) on the necessity of world models. They formalize the problem with controlled Markov processes (cMPs), which elegantly handles goal-directed behavior through temporal logic. Their framework defines goals in terms of reaching desirable states. However, we observed that, in healthcare, it is equally important to avoid undesirable states. This, led us to extend their framework for healthcare to explicitly model states that must never be entered.

Problem Formulation

We extend the standard RL formulation to a healthcare MDP $(S, A, P, R, \gamma, \mathcal{C})$, where $\mathcal{C} \subseteq S$ represents irreversible critical states (e.g., organ failure, death). This directly reflects our first constraint: the irreversibility of failures. We recognized that trajectories entering $\mathcal{C}$ are fundamentally distinct—they represent catastrophic failures with typically bounded, negative future returns.

This motivated a natural decomposition of the value function: $V^\pi = V^\pi\_{\text{safe}} \cdot (1 - \rho\_\mathcal{C}(\pi)) + V^\pi\_{\text{unsafe}} \cdot \rho\_\mathcal{C}(\pi)$

where $\rho\_\mathcal{C}(\pi) = \mathbb{P}(\text{entering } \mathcal{C})$ is the irreversibility risk.

Developing the Bound

The challenge was to apply PAC-Bayes to this decomposition. We could not treat all trajectories uniformly. We therefore:

1. Apply PAC-Bayes only to safe trajectories (those avoiding $\mathcal{C}$)
2. Separately bound the loss from unsafe trajectories
3. Combine them using the empirical frequency of entering $\mathcal{C}$

This yields our main result that has been rigorously proven with the new setup—a bound with three interpretable components:

$V^\pi \geq \underbrace{\hat{V}^{\pi,\text{safe}}\_D}\_{\text{safe trajectory value}} - \underbrace{\sqrt{\frac{R^2\_{\max}\tau\_{\min}(1-\gamma^{2H})}{2T\_{\text{safe}}(1-\gamma^2)}\left(\text{KL}(\rho||\mu) + \ln\frac{2}{\delta}\right)}}\_{\text{PAC-Bayes penalty}} - \underbrace{\rho\_\mathcal{C}(\pi) \cdot \kappa}\_{\text{irreversibility penalty}}$

This bound has a direct clinical interpretability:

• $\hat{V}^{\pi,\text{safe}}\_D$: Performance when treatment succeeds
• $T\_{\text{safe}} = T \cdot (1-\rho\_\mathcal{C}(\pi))$ is the effective number of safe trajectories
• $\kappa = V\_{\max} - V^{\pi}\_{\text{unsafe}}$: value loss from entering critical states (a pessimistic bound)
• $\hat{\rho}\_\mathcal{C}(\pi) = \frac{T\_{\text{unsafe}}}{T} = \frac{T - T\_{\text{safe}}}{T}$: we can use this approximation empirically

Addressing The Three Constraints

Looking back, this bound naturally addresses all three key constraints in healthcare:

Addressing Irreversibility: The decomposition into safe/unsafe trajectories with explicit penalty $\kappa$ quantifies the cost of irreversible failures. Unlike standard RL bounds that treat all trajectories equally, we separately bound the catastrophic loss from entering $\mathcal{C}$.

Regulatory Requirements: This is indeed a mission-critical objective currently hampered by regulatory frameworks lagging behind AI advances and the scarcity of teams combining ML theory, clinical expertise, and regulatory knowledge.

The bound provides a certificate: "With confidence $1-\delta$, the policy achieves a lower value bound while maintaining $\mathbb{P}(\text{adverse events}) \leq \rho\_\mathcal{C}(\pi)$."

Limited Exploration: The dependence on $T\_{\text{safe}}$ shows we need sufficient data from trajectories that successfully avoided critical states.

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The key was recognizing that healthcare isn't just about maximizing expected returns, it's about doing so while providing guarantees against catastrophic, irreversible failures. By extending the cMP framework with explicit modeling of undesirable states, and formalizing their avoidance through Linear Temporal Logic (LTL), we derived this bound quickly in response to your comments, and we will implement experiments based on it in the final version.

We believe this directly addresses both of your requests: I. Reintroducing the problem setup, and II. Providing a non-trivial bound or theoretical insight connected to healthcare implications.

We’re grateful for your thoughtful and constructive feedback and for your openness to revise the score.