{"paper":{"title":"A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.CA","math.NA","stat.ML"],"primary_cat":"math.OC","authors_text":"Jonathan W. Siegel, Stephan Wojtowytsch","submitted_at":"2023-10-26T17:33:52Z","abstract_excerpt":"We consider gradient flow/gradient descent and heavy ball/accelerated gradient descent optimization for convex objective functions. In the gradient flow case, we prove the following:\n  1. If $f$ does not have a minimizer, the convergence $f(x_t)\\to \\inf f$ can be arbitrarily slow.\n  2. If $f$ does have a minimizer, the excess energy $f(x_t) - \\inf f$ is integrable/summable in time. In particular, $f(x_t) - \\inf f = o(1/t)$ as $t\\to\\infty$.\n  3. In Hilbert spaces, this is optimal: $f(x_t) - \\inf f$ can decay to $0$ as slowly as any given function which is monotone decreasing and integrable at $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.17610","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/2310.17610/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"}