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Dual-Objective Reinforcement Learning with Novel Hamilton-Jacobi-Bellman Formulations

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.5Download Report PDF
Hamilton-Jacobi AnalysisDual SatisfactionSafe Reinforcement LearningDecomposition
Abstract:

Hard constraints in reinforcement learning (RL) often degrade policy performance. Lagrangian methods offer a way to blend objectives with constraints, but require intricate reward engineering and parameter tuning. In this work, we extend recent advances that connect Hamilton-Jacobi (HJ) equations with RL to propose two novel value functions for dual-objective satisfaction. Namely, we address: 1) the Reach-Always-Avoid (RAA) problem – of achieving distinct reward and penalty thresholds – and 2) the Reach-Reach (RR) problem – of achieving thresholds of two distinct rewards. In contrast with temporal logic approaches, which typically involve representing an automaton, we derive explicit, tractable Bellman forms in this context via decomposition. Specifically, we prove that the RAA and RR problems may be rewritten as compositions of previously studied HJ-RL problems. We leverage our analysis to propose a variation of Proximal Policy Optimization (DO-HJ-PPO), and demonstrate that it produces distinct behaviors from previous approaches, out-competing a number of baselines in success, safety and speed across a range of tasks for safe-arrival and multi-target achievement.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper proposes novel value functions for two dual-objective problems—Reach-Always-Avoid (RAA) and Reach-Reach (RR)—within the Hamilton-Jacobi reachability framework. It resides in the 'Hamilton-Jacobi Reachability and Value Function Formulations' leaf, which contains four papers total, including the original work. This leaf represents a relatively focused research direction within the broader taxonomy of 50 papers across 13 leaf nodes, suggesting a specialized but not overcrowded area where formal HJ-based methods are actively developed.

The taxonomy reveals that the paper's leaf sits within 'Theoretical Foundations and Formulations,' adjacent to 'Duality Theory and Optimization Guarantees' (3 papers) and 'Multi-Objective Problem Formulations' (3 papers). These neighboring branches address complementary concerns—duality gaps, convergence properties, and Pareto optimality—while the HJ reachability leaf emphasizes explicit value-function derivations and Bellman forms. The paper's focus on decomposing dual-objective problems into compositions of simpler HJ-RL problems distinguishes it from multi-objective scalarization methods and from purely algorithmic approaches in the 'Algorithmic Approaches and Training Frameworks' branch.

Among 30 candidates examined, none were found to clearly refute any of the three contributions: the RAA/RR value functions (10 candidates, 0 refutable), the decomposition theorems (10 candidates, 0 refutable), and the DOHJ-PPO algorithm (10 candidates, 0 refutable). This suggests that within the limited search scope, the specific formulations and decomposition results appear distinct from prior work. The three sibling papers in the same leaf address HJ reachability and value functions but do not appear to cover the exact RAA and RR problem structures or their compositional characterizations.

Based on the top-30 semantic matches and the taxonomy structure, the work appears to occupy a relatively novel position within HJ-based dual-objective RL. The limited search scope means that more exhaustive examination—particularly of temporal logic and automaton-based methods mentioned in the abstract—could reveal additional overlaps. However, the explicit Bellman forms and decomposition approach seem to differentiate this work from both neighboring theoretical formulations and applied constraint-satisfaction methods.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
30
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: Dual-objective reinforcement learning with reach-avoid constraints addresses the challenge of training agents that must simultaneously optimize performance while guaranteeing safety through formal reach-avoid specifications. The field's structure reflects a blend of rigorous theory and practical deployment. Theoretical Foundations and Formulations establish the mathematical underpinnings, particularly through Hamilton-Jacobi reachability and value function methods that provide formal guarantees for constraint satisfaction. Algorithmic Approaches and Training Frameworks translate these foundations into learnable policies, exploring techniques ranging from constrained policy optimization to dual-agent architectures that decouple reward maximization from safety enforcement. Application-Specific Implementations demonstrate the breadth of domains—from autonomous navigation and traffic control to robotics and resource allocation—where dual objectives arise naturally. Game-Theoretic and Adversarial Formulations extend the framework to multi-agent settings, capturing competitive or cooperative dynamics where reach-avoid constraints must be satisfied under strategic interactions. Several active lines reveal key trade-offs between computational tractability and safety guarantees. Works like HJ Reachability[11] and Dual Agent Safety[3] emphasize rigorous value-function approaches that offer strong theoretical assurances but can be computationally intensive, while methods such as Safety Policy Optimization[8] and Safe CoR[9] pursue scalable training frameworks that balance empirical performance with constraint adherence. The original paper, Dual Objective HJB[0], sits squarely within the Hamilton-Jacobi reachability cluster, contributing formal value-function formulations for dual-objective problems. Compared to neighbors like Shared Driving Collision[28], which applies reach-avoid logic to human-robot interaction, Dual Objective HJB[0] focuses on the foundational mathematical characterization rather than domain-specific tuning. This positioning highlights an ongoing tension: how to preserve the elegance of HJ-based guarantees while enabling the flexibility needed for diverse real-world applications, a question that continues to drive research across all branches of the taxonomy.

Claimed Contributions

Novel value functions for Reach-Always-Avoid and Reach-Reach problems

The authors introduce two new value functions for dual-objective satisfaction in reinforcement learning: the Reach-Always-Avoid (RAA) problem, which requires reaching a goal while perennially avoiding hazards, and the Reach-Reach (RR) problem, which requires reaching two distinct goals in either order. These formulations extend existing Hamilton-Jacobi reachability methods to more complex compositional tasks.

10 retrieved papers
Decomposition theorems for RAA and RR value functions

The authors prove that the RAA and RR value functions can be decomposed into combinations of simpler reach, avoid, and reach-avoid value functions. Specifically, Theorem 1 shows RAA decomposes into avoid and reach-avoid problems, while Theorem 2 shows RR decomposes into three reach problems. This decomposition enables tractable solutions using existing methods.

10 retrieved papers
DOHJ-PPO algorithm for dual-objective reinforcement learning

The authors develop DOHJ-PPO, a novel algorithm that extends Proximal Policy Optimization to solve the RAA and RR problems. The algorithm bootstraps concurrently solved decompositions for coupling on-policy rollouts, using stochastic relaxations of the Bellman equations (SRBE and SRABE) to handle stochastic policies and dynamics.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Novel value functions for Reach-Always-Avoid and Reach-Reach problems

The authors introduce two new value functions for dual-objective satisfaction in reinforcement learning: the Reach-Always-Avoid (RAA) problem, which requires reaching a goal while perennially avoiding hazards, and the Reach-Reach (RR) problem, which requires reaching two distinct goals in either order. These formulations extend existing Hamilton-Jacobi reachability methods to more complex compositional tasks.

Contribution

Decomposition theorems for RAA and RR value functions

The authors prove that the RAA and RR value functions can be decomposed into combinations of simpler reach, avoid, and reach-avoid value functions. Specifically, Theorem 1 shows RAA decomposes into avoid and reach-avoid problems, while Theorem 2 shows RR decomposes into three reach problems. This decomposition enables tractable solutions using existing methods.

Contribution

DOHJ-PPO algorithm for dual-objective reinforcement learning

The authors develop DOHJ-PPO, a novel algorithm that extends Proximal Policy Optimization to solve the RAA and RR problems. The algorithm bootstraps concurrently solved decompositions for coupling on-policy rollouts, using stochastic relaxations of the Bellman equations (SRBE and SRABE) to handle stochastic policies and dynamics.

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