{"paper":{"title":"Complex Stochastic Gradient Descent and Directional Bias in Reproducing Kernel Hilbert Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Complex SGD converges under the same assumptions as real SGD and extends directional bias to kernel regression in complex RKHS.","cross_cats":["cs.NA","math.CV","math.NA"],"primary_cat":"cs.LG","authors_text":"Emeric Battaglia, Natanael Alpay","submitted_at":"2026-04-24T21:08:39Z","abstract_excerpt":"Stochastic Gradient Descent (SGD) is a known stochastic iterative method popular for large-scale convex optimization problems due to its simple implementation and scalability. Some objectives, such as those found in complex-valued neural networks, benefit from updates like in SGD and Gradient Descent (GD) with a newly defined ``gradient'' that allows for complex parameters. This complex variant of the SGD/GD methods has already been proposed, but convergence guarantees without analyticity constraints have not yet been provided. We propose a variant of SGD (complex SGD) that allows for complex "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We propose a variant of SGD (complex SGD) that allows for complex parameters, and we provide convergence guarantees under assumptions that parallel those from the real setting. Notably, these results extend to GD as well, and with the same set of assumptions, we confirm that some directional bias results extend from the real to the complex setting for kernel regression problems.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumptions that parallel those from the real setting (such as convexity, smoothness, or bounded variance) remain valid and sufficient when parameters and gradients are complex-valued, without additional analyticity requirements.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Complex SGD converges without analyticity constraints and extends real-valued directional bias results to complex RKHS, with demonstrations on Fock and Hardy spaces.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Complex SGD converges under the same assumptions as real SGD and extends directional bias to kernel regression in complex RKHS.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"bad28f07d790d43d962be30f050deba51e6ba3b19adbb3945234c55cf072853a"},"source":{"id":"2604.23017","kind":"arxiv","version":2},"verdict":{"id":"a5be9ee8-b7bf-4041-b74d-6b47b74cd1e5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T12:10:09.206864Z","strongest_claim":"We propose a variant of SGD (complex SGD) that allows for complex parameters, and we provide convergence guarantees under assumptions that parallel those from the real setting. Notably, these results extend to GD as well, and with the same set of assumptions, we confirm that some directional bias results extend from the real to the complex setting for kernel regression problems.","one_line_summary":"Complex SGD converges without analyticity constraints and extends real-valued directional bias results to complex RKHS, with demonstrations on Fock and Hardy spaces.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumptions that parallel those from the real setting (such as convexity, smoothness, or bounded variance) remain valid and sufficient when parameters and gradients are complex-valued, without additional analyticity requirements.","pith_extraction_headline":"Complex SGD converges under the same assumptions as real SGD and extends directional bias to kernel regression in complex RKHS."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.23017/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T09:41:03.607611Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:33:18.228910Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b9389f42952c5ad4c5a9fe82a5a86c4b292ac719464092f232b05ae554d180a7"},"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"}