{"paper":{"title":"Zeroth-Order Optimization at the Edge of Stability","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Zeroth-order methods remain mean-square stable only when their step size satisfies a bound that depends on the full Hessian spectrum rather than its largest eigenvalue alone.","cross_cats":["math.DS","math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Bingcong Li, Liang Zhang, Michael Muehlebach, Minhak Song, Niao He, Sewoong Oh","submitted_at":"2026-04-16T06:23:18Z","abstract_excerpt":"Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square st"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator... mean-square stability of ZO methods depends on the entire Hessian spectrum.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The linear stability analysis assumes the loss can be locally approximated by a quadratic form whose Hessian is constant over the relevant trajectory, and that the two-point estimator is used without additional smoothing or variance reduction.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Zeroth-order methods achieve mean-square stability when the step size satisfies a condition involving the entire Hessian spectrum, with full-batch ZO optimizers operating at the edge of stability and large steps regularizing the Hessian trace.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Zeroth-order methods remain mean-square stable only when their step size satisfies a bound that depends on the full Hessian spectrum rather than its largest eigenvalue alone.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"532643a344f2c4571bf4ea935d01ee819ff3b696b8d96b7c29d30a87f1eb086c"},"source":{"id":"2604.14669","kind":"arxiv","version":2},"verdict":{"id":"64c66983-492d-41d4-bcd9-fbb0eec76e7d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T11:15:27.370755Z","strongest_claim":"we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator... mean-square stability of ZO methods depends on the entire Hessian spectrum.","one_line_summary":"Zeroth-order methods achieve mean-square stability when the step size satisfies a condition involving the entire Hessian spectrum, with full-batch ZO optimizers operating at the edge of stability and large steps regularizing the Hessian trace.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The linear stability analysis assumes the loss can be locally approximated by a quadratic form whose Hessian is constant over the relevant trajectory, and that the two-point estimator is used without additional smoothing or variance reduction.","pith_extraction_headline":"Zeroth-order methods remain mean-square stable only when their step size satisfies a bound that depends on the full Hessian spectrum rather than its largest eigenvalue alone."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.14669/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"}