{"paper":{"title":"Directional convergence and alignment in deep learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE","math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Matus Telgarsky, Ziwei Ji","submitted_at":"2020-06-11T17:50:11Z","abstract_excerpt":"In this paper, we show that although the minimizers of cross-entropy and related classification losses are off at infinity, network weights learned by gradient flow converge in direction, with an immediate corollary that network predictions, training errors, and the margin distribution also converge. This proof holds for deep homogeneous networks -- a broad class of networks allowing for ReLU, max-pooling, linear, and convolutional layers -- and we additionally provide empirical support not just close to the theory (e.g., the AlexNet), but also on non-homogeneous networks (e.g., the DenseNet)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.06657","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/2006.06657/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"}