{"paper":{"title":"Stochastic Mirror Descent under Iterate-Dependent Markov Noise: Analysis in the Asymptotic and Finite Time Regimes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Stochastic mirror descent converges almost surely under iterate-dependent Markov noise for both convex and non-convex problems.","cross_cats":["cs.SY","eess.SY"],"primary_cat":"math.OC","authors_text":"Anik Kumar Paul, Shalabh Bhatnagar","submitted_at":"2026-05-15T02:19:06Z","abstract_excerpt":"We study a stochastic optimization problem in which the sampling distribution depends on the decision variable, and the available samples are generated through an iterate-dependent Markov chain. Such settings arise naturally in problems with decision-dependent uncertainty; however, they introduce bias and temporal dependence, which render standard techniques developed for i.i.d.\\ noise inapplicable. In this work, we analyze the stochastic mirror descent algorithm under iterate-dependent Markov noise. We first establish almost sure convergence for both convex and non-convex problems under the m"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We first establish almost sure convergence for both convex and non-convex problems under the mild assumption of Lipschitz continuity of the objective function, without requiring differentiability. We then derive finite-time concentration bounds for smooth objectives. In the convex setting, the resulting sample complexity matches the classical rate of stochastic mirror descent under i.i.d. noise.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Markov chain generated by the iterate-dependent sampling distribution satisfies sufficient mixing or ergodicity conditions that allow the bias and temporal dependence to be controlled; this property is invoked to justify the almost-sure convergence but is not stated explicitly in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves almost sure convergence and finite-time sample complexity bounds for stochastic mirror descent under iterate-dependent Markov noise for both convex and non-convex objectives.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Stochastic mirror descent converges almost surely under iterate-dependent Markov noise for both convex and non-convex problems.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"42bf05ea629a5ede35084d839557e0418752d48aceb27d78de16c9484e4bb352"},"source":{"id":"2605.15538","kind":"arxiv","version":1},"verdict":{"id":"660d5c2a-3283-4535-8109-ca21d43f604c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T14:31:42.166022Z","strongest_claim":"We first establish almost sure convergence for both convex and non-convex problems under the mild assumption of Lipschitz continuity of the objective function, without requiring differentiability. We then derive finite-time concentration bounds for smooth objectives. In the convex setting, the resulting sample complexity matches the classical rate of stochastic mirror descent under i.i.d. noise.","one_line_summary":"Proves almost sure convergence and finite-time sample complexity bounds for stochastic mirror descent under iterate-dependent Markov noise for both convex and non-convex objectives.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Markov chain generated by the iterate-dependent sampling distribution satisfies sufficient mixing or ergodicity conditions that allow the bias and temporal dependence to be controlled; this property is invoked to justify the almost-sure convergence but is not stated explicitly in the abstract.","pith_extraction_headline":"Stochastic mirror descent converges almost surely under iterate-dependent Markov noise for both convex and non-convex problems."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15538/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T15:01:17.475857Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:37:37.871798Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T14:22:01.391541Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.029625Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"shingle_duplication","ran_at":"2026-05-19T13:49:41.829472Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T13:49:41.366697Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.614737Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"19fa80867ae89a96c8f1a60a902fbbdf034d6957a2a0cef4e007b78c6798066e"},"references":{"count":29,"sample":[{"doi":"","year":2021,"title":"A. AGARWAL, S. M. KAKADE, J. D. LEE,ANDG. MAHAJAN,On the theory of policy gradient methods: Optimality, approximation, and distribution shift, Journal of Machine Learning Research, 22 (2021), pp. 1–76","work_id":"3fa84b97-9ab4-4128-b8c4-3692ea687fa4","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"V. S. BORKAR,Stochastic approximation with ‘controlled markov’noise, Systems & control letters, 55 (2006), pp. 139–145","work_id":"f9e956c2-74b8-4356-b6da-c5f4e78fd707","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"BOUMAL,An Introduction to Optimization on Smooth Manifolds, Cambridge University Press, 2023","work_id":"9ca73bbd-6898-4416-9651-fb7a8be7e08b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"BUBECK,Introduction to Online Optimization, Lecture notes, Princeton University, 2012","work_id":"3479d70e-86ab-4b9e-bd0a-be12dce37d86","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"E. CHE, J. DONG,ANDX. T. TONG,Stochastic gradient descent with adaptive data, Operations Research, (2026)","work_id":"531d1973-a75e-40da-8fec-59951e6f74e0","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":29,"snapshot_sha256":"835bb10b574836119da8a1b13d4e335fdb412a1c99415583127749ce685b1cb3","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"b089a1f3a07d7799f12f8b7c443f5357848fd5113108e376807ed220d9b81148"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}