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Syst. Technol."],"published-print":{"date-parts":[[2026,2,28]]},"abstract":"<jats:p>\n                    Ensuring safety in reinforcement learning (RL) is critical for deploying agents in real-world applications. During training, current safe RL approaches often rely on indicator cost functions that provide sparse feedback, resulting in two key limitations: (i) poor sample efficiency due to the lack of safety information in neighboring states, and (ii) dependence on cost-value functions, leading to brittle convergence and suboptimal performance. After training, safety is guaranteed via formal verification (FV) methods for deep neural networks, whose computational complexity hinders their application during training. We address the limitations of using cost functions via verification by proposing a safe RL method based on a\n                    <jats:italic toggle=\"yes\">violation<\/jats:italic>\n                    value\u2014the risk associated with policy decisions in a portion of the state space. Our approach verifies\n                    <jats:italic toggle=\"yes\">safety properties<\/jats:italic>\n                    (i.e., state-action pairs) that may lead to unsafe behavior, and quantifies the size of the state space where properties are violated. This violation value is then used to penalize the agent during training to encourage safer policy behavior. Given the NP-hard nature of FV, we propose an efficient, sample-based approximation with probabilistic guarantees to compute the violation value. Extensive experiments on standard benchmarks and real-world robotic navigation tasks show that violation-augmented approaches significantly improve safety by reducing the number of unsafe states encountered while achieving superior performance compared to existing methods.\n                  <\/jats:p>","DOI":"10.1145\/3770068","type":"journal-article","created":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T15:05:46Z","timestamp":1759244746000},"page":"1-27","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Verifying Online Safety Properties for Safe Deep Reinforcement Learning"],"prefix":"10.1145","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0069-0182","authenticated-orcid":false,"given":"Luca","family":"Marzari","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Verona, Verona, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1652-0599","authenticated-orcid":false,"given":"Ferdinando","family":"Cicalese","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Verona, Verona, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2592-5814","authenticated-orcid":false,"given":"Alessandro","family":"Farinelli","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Verona, Verona, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6786-7384","authenticated-orcid":false,"given":"Christopher","family":"Amato","sequence":"additional","affiliation":[{"name":"Khoury College of Computer Sciences, Northeastern University, Boston, Massachusetts, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1858-7279","authenticated-orcid":false,"given":"Enrico","family":"Marchesini","sequence":"additional","affiliation":[{"name":"Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA"}]}],"member":"320","published-online":{"date-parts":[[2025,11,18]]},"reference":[{"key":"e_1_3_2_2_2","volume-title":"Proceedings of the Conference on Neural Information Processing Systems (NeurIPS)","author":"Abel David","year":"2021","unstructured":"David Abel, Will Dabney, Anna Harutyunyan, Mark K. 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