{"paper":{"title":"Sharp description of local minima in the loss landscape of high-dimensional two-layer ReLU neural networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Local minima in two-layer ReLU networks admit an exact low-dimensional representation via summary statistics.","cross_cats":["cond-mat.dis-nn","cs.LG"],"primary_cat":"stat.ML","authors_text":"Bruno Loureiro, Jie Huang, Stefano Sarao Mannelli","submitted_at":"2026-04-10T15:26:00Z","abstract_excerpt":"We study the population loss landscape of two-layer ReLU networks of the form $\\sum_{k=1}^K \\mathrm{ReLU}(w_k^\\top x)$ in a realisable teacher-student setting with Gaussian covariates. We show that local minima admit an exact low-dimensional representation in terms of summary statistics, yielding a sharp and interpretable characterisation of the landscape. We further establish a direct link with one-pass SGD: local minima correspond to attractive fixed points of the dynamics in summary statistics space. This perspective reveals a hierarchical organisation of minima into discrete families and s"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that local minima admit an exact low-dimensional representation in terms of summary statistics, yielding a sharp and interpretable characterisation of the landscape. We further establish a direct link with one-pass SGD: local minima correspond to attractive fixed points of the dynamics in summary statistics space.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis assumes a realizable teacher-student setting with Gaussian covariates and the specific network form sum ReLU(w_k^T x), which may not hold for more general data distributions or deeper networks.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Local minima of two-layer ReLU networks have an exact representation in terms of a few summary statistics that correspond to attractive fixed points of one-pass SGD, with overparameterization connecting them via flat directions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Local minima in two-layer ReLU networks admit an exact low-dimensional representation via summary statistics.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"31ccac97a9566736aafa280aed4efca3f48892e16d0f876f7b6b90cb8bc34c6b"},"source":{"id":"2604.09412","kind":"arxiv","version":2},"verdict":{"id":"8ab2454f-330d-4b3e-8713-83315da1a67c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T16:40:16.865754Z","strongest_claim":"We show that local minima admit an exact low-dimensional representation in terms of summary statistics, yielding a sharp and interpretable characterisation of the landscape. We further establish a direct link with one-pass SGD: local minima correspond to attractive fixed points of the dynamics in summary statistics space.","one_line_summary":"Local minima of two-layer ReLU networks have an exact representation in terms of a few summary statistics that correspond to attractive fixed points of one-pass SGD, with overparameterization connecting them via flat directions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis assumes a realizable teacher-student setting with Gaussian covariates and the specific network form sum ReLU(w_k^T x), which may not hold for more general data distributions or deeper networks.","pith_extraction_headline":"Local minima in two-layer ReLU networks admit an exact low-dimensional representation via summary statistics."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.09412/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"}