{"paper":{"title":"Unified Optimal Analysis of the (Stochastic) Gradient Method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.NA","math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Sebastian U. Stich","submitted_at":"2019-07-09T15:14:25Z","abstract_excerpt":"In this note we give a simple proof for the convergence of stochastic gradient (SGD) methods on $\\mu$-convex functions under a (milder than standard) $L$-smoothness assumption. We show that for carefully chosen stepsizes SGD converges after $T$ iterations as $O\\left( LR^2 \\exp \\bigl[-\\frac{\\mu}{4L}T\\bigr] + \\frac{\\sigma^2}{\\mu T} \\right)$ where $\\sigma^2$ measures the variance in the stochastic noise. For deterministic gradient descent (GD) and SGD in the interpolation setting we have $\\sigma^2 =0$ and we recover the exponential convergence rate. The bound matches with the best known iteration"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04232","kind":"arxiv","version":2},"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/1907.04232/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"}