{"paper":{"title":"Avoiding unsafe sets when training with Langevin Dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","stat.ML"],"primary_cat":"cs.LG","authors_text":"Adam M. Oberman","submitted_at":"2026-07-08T15:36:45Z","abstract_excerpt":"Training a model with noisy gradient descent can be idealized as overdamped Langevin dynamics on the loss landscape, and a natural safety question is to bound the probability $\\nu_t(\\mathcal{A}_H) = \\mathbb{P}(Q_t \\in \\mathcal{A}_H)$ that the trajectory lies in a designated failure region $\\mathcal{A}_H$. We study this for a smooth, strongly convex loss in $d$ dimensions and a failure region separated from the minimizer by an energy gap. Three bounds emerge. At the end of training, the equilibrium mass $\\pi(\\mathcal{A}_H)$ is exponentially small in $d$, with a complementary energy-barrier rate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07538","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.07538/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}