PAPER: INTRODUCTIONThe development of special Bellman equations from the Hamilton-Jacobi (HJ) perspective of dynamic programming (DP) has illustrated a novel route to safety and target-achievement in reinforcement learning (RL) Fisac et al. (2019); Hsu et al. (2021). In comparison with the canonical RL discountedsum cost and corresponding additive DP update, these equations, namely the Safety Bellman Equation (SBE) and Reach-Avoid Bellman Equation (RABE), propagate the minimum (worst) penalty and maximum (best) reward, yielding a value function defined by the outlying performance of a trajectory. In mission-critical applications, where avoiding failure is a necessary condition, these equations have proved invaluable in the field of safe control Mitchell et al. (2005); Ames et al. (2016). By focusing on extremal values rather than discounted sums, the HJ-RL equations induce behaviors that act with respect to the best or worst outcomes in time-optimal fashions, performing far more safely than Lagrangian methods Ganai et al. (2023); So et al. (2024). Accordingly, these updates yield policies with significantly improved performance in target-achievement and obstacle-avoidance tasks over long horizons Yu et al. (2022a;b), relevant to fundamental and practical problems in many domains.In this work, we advance the existing HJ-RL formulations by generalizing them to compositional problems. To date, the HJ-RL Bellman equations are limited to three operations: Reach (R), wherein the agent seeks to reach a goal (achieve a reward threshold), Avoid (A), wherein the agent seeks to avoid an obstacle (avoid a penalty threshold), and Reach-Avoid (RA), where the agent avoids obstacles until reaching the goal. In this light, we extend the HJ-RL Bellman equations to two complementary problems concerned with dual-satisfaction, namely the Reach-Reach (RR) problem for reaching two goals and the Reach-Always-Avoid (RAA) problem for continuing to avoid hazards after reaching a goal, demonstrated in Figure 1. We prove that the RAA and RR have a fundamental structure such that their Bellman equations may be decomposed into combinations of SBEs and RABEs. From this theory, we devise DOHJ-PPO, a novel algorithm for learning the RAA and RR values which bootstraps concurrently solved decompositions for coupling on-policy PPO roll-outs. Notably, this allows one to automatically learn to satisfy dual-objective tasks, for example, in the RAA, the F16 learns to fly into the desired airspace without crashing afterward (Figure 1, top middle-left), and in the RR case, the Hopper learns to jump into a target without diving so it may then achieve the second target (Figure 1, bottom left). The RAA and RR problems are distinct from both standard sum-of-reward values and the simpler HJ-RL formulations, providing new perspectives and performant tools for constrained decision-making.Figure 1: Depiction of the Reach-Always-Avoid (RAA) and Reach-Reach (RR) Tasks. In the RAA tasks, the zero-level set of the rewards (goals) and penalties (obstacles) are depicted in green and red respectively, while in the RR problem, the zero-level set of the two rewards (two goals) are depicted in green and blue. The RAA value is defined by the minimum of the minimum penalty and maximum reward, inducing the agents to enter the goals at some time without ever entering the obstacles. The RR value is defined by the minimum of the two maximum rewards, inducing the agents to enter both goals at some time.
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REVIEW
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**Summary Of The Paper**  
This paper advances Hamilton-Jacobi (HJ)-based reinforcement learning (RL) to address compositional safety-critical tasks requiring dual-objective constraints: **Reach-Always-Avoid (RAA)** and **Reach-Reach (RR)**. It introduces novel value functions for these tasks, proves their decomposition into combinations of existing Safety Bellman Equations (SBE) and Reach-Avoid Bellman Equations (RABE), and proposes **DOHJ-PPO**, a modified proximal policy optimization (PPO) algorithm that leverages these decompositions for simultaneous policy learning. Empirically, the method demonstrates improved safety and performance over Lagrangian and HJ-RL baselines in continuous control tasks (e.g., F16 flight, Hopper locomotion) and grid-world simulations.  

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**Strengths**  
1. **Clear Motivation and Theoretical Foundation**: The paper addresses a critical gap in HJ-RL by formalizing dual-objective constraints (RAA/RR) that align with real-world safety-critical scenarios (e.g., perpetual hazard avoidance post-goal attainment). The decomposition of RAA/RR into SBE/RABE (Theorems 1–2) offers a principled pathway for solving these problems, building on established HJ-RL theory.  
2. **Algorithmic Innovation**: DOHJ-PPO integrates concurrent decomposition of subproblems with on-policy PPO rollouts, enabling automatic dual-task satisfaction without manual hyperparameter tuning—a notable practical contribution.  
3. **Empirical Validation**: Results on continuous control tasks (e.g., F16, Hopper) show competitive performance versus baselines, supported by visualizations (Figure 1) and qualitative descriptions of policy behavior.  

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**Weaknesses**  
1. **Incomplete Derivations and Assumptions**: The proofs of Theorems 1–2 lack explicit mathematical conditions (e.g., smoothness of reward/penalty functions, determinism of dynamics) under which the decomposition holds. This raises doubts about generalizability to non-deterministic settings or non-smooth environments.  
2. **Ambiguity in Algorithm Design**: The description of DOHJ-PPO’s "bootstrapping concurrently solved decompositions" (Section 7.2) is opaque. No pseudocode or detailed mechanism is provided, leaving unclear how subproblem solutions are coupled during training.  
3. **Missing Complexity Analysis**: The paper does not analyze computational or memory scaling of DOHJ-PPO, nor does it compare training efficiency to baselines. This limits evaluation of practical applicability to large-scale problems.  
4. **Insufficient Exploration of Trade-offs**: The RAA/RR formulation prioritizes extremal values over discounted sums, yet the paper fails to discuss how this impacts exploration-exploitation balance or sample efficiency in practice.  

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**Questions For The Authors**  
1. **Decomposition Conditions**: What are the precise mathematical assumptions (e.g., continuity, convexity) required for Theorems 1–2 to hold? Are these guarantees valid for non-smooth reward/penalty functions or stochastic dynamics?  
2. **Bellman Equations for RAA/RR**: Please provide the full derivation of the Bellman equations governing RAA and RR value functions, and clarify how they differ structurally from classical Bellman operators.  
3. **Optimality Guarantees**: Under what conditions does the RAA/RR value function ensure strict compliance with dual constraints (e.g., perpetual hazard avoidance)? Is there a risk of suboptimal policies violating constraints due to approximation errors?  
4. **DOHJ-PPO Mechanism**: Can the authors provide pseudocode or a diagram illustrating how "concurrently solved decompositions" are bootstrapped during training? How is the coupling between subproblems implemented in practice?  

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**Limitations Not Addressed By The Authors**  
1. **Reproducibility Concerns**: The paper lacks a dedicated reproducibility section, omitting details on implementation specifics (e.g., hyperparameters, environment versions) needed to replicate experiments.  
2. **Ethical Considerations**: No ethical analysis is provided, particularly regarding deployment risks of policies trained on synthetic tasks (e.g., F16 flight) in real-world safety-critical systems.  
3. **Comparison to Multi-Objective RL**: The paper asserts uniqueness relative to multi-objective RL approaches (e.g., Cai et al., 2023), but does not quantify differences in performance or scalability on shared benchmark tasks.  

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**Soundness**  
**Score: 3 (Good)**  
While the theoretical decomposition of RAA/RR into SBE/RABE is plausible, the lack of rigorous derivation conditions and ambiguity in algorithm design undermine confidence in broader validity. Empirical results are compelling but lack statistical rigor (no confidence intervals or hypothesis tests).  

**Contribution**  
**Score: 3 (Good)**  
The paper makes a meaningful contribution by extending HJ-RL to novel dual-constraint tasks and proposing DOHJ-PPO. However, the novelty of the decomposition and algorithm is partially overshadowed by prior work on reward decomposition (e.g., van Seijen et al., 2017).  

**Confidence**  
**Score: 3 (Moderate)**  
Key technical details remain underspecified, and critical questions about generalizability, optimality, and algorithm mechanics require clarification.  

**Rating**  
**Score: 6 (Accept with Reservations)**  
The paper presents a promising direction for HJ-RL in safety-critical settings but suffers from incomplete theoretical grounding, ambiguous methodology, and missing reproducibility details. With revisions addressing these issues, it warrants acceptance.  

**Brief Justification For Rating**  
The work introduces a novel application of HJ-RL to dual-constraint tasks and proposes a viable algorithm (DOHJ-PPO) with encouraging empirical results. However, critical gaps in theoretical derivation, algorithm transparency, and experimental rigor necessitate substantial revisions before the contribution can be fully evaluated.

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