{"paper":{"title":"Width Provably Matters in Optimization for Deep Linear Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Simon S. Du, Wei Hu","submitted_at":"2019-01-24T18:43:16Z","abstract_excerpt":"We prove that for an $L$-layer fully-connected linear neural network, if the width of every hidden layer is $\\tilde\\Omega (L \\cdot r \\cdot d_{\\mathrm{out}} \\cdot \\kappa^3 )$, where $r$ and $\\kappa$ are the rank and the condition number of the input data, and $d_{\\mathrm{out}}$ is the output dimension, then gradient descent with Gaussian random initialization converges to a global minimum at a linear rate. The number of iterations to find an $\\epsilon$-suboptimal solution is $O(\\kappa \\log(\\frac{1}{\\epsilon}))$. Our polynomial upper bound on the total running time for wide deep linear networks "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.08572","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}