{"paper":{"title":"Toward a Unified Theory of Gradient Descent under Generalized Smoothness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Alexander Tyurin","submitted_at":"2024-12-16T13:46:02Z","abstract_excerpt":"We study the classical optimization problem $\\min_{x \\in \\mathbb{R}^d} f(x)$ and analyze the gradient descent (GD) method in both nonconvex and convex settings. It is well-known that, under the $L$-smoothness assumption ($\\|\\nabla^2 f(x)\\| \\leq L$), the optimal point minimizing the quadratic upper bound $f(x_k) + \\langle\\nabla f(x_k), x_{k+1} - x_k\\rangle + \\frac{L}{2} \\|x_{k+1} - x_k\\|^2$ is $x_{k+1} = x_k - \\gamma_k \\nabla f(x_k)$ with step size $\\gamma_k = \\frac{1}{L}$. Surprisingly, a similar result can be derived under the $\\ell$-generalized smoothness assumption ($\\|\\nabla^2 f(x)\\| \\leq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.11773","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/2412.11773/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"}