{"paper":{"title":"OSDN: Improving Delta Rule with Provable Online Preconditioning in Linear Attention","license":"http://creativecommons.org/licenses/by/4.0/","headline":"OSDN augments the Delta Rule with an online diagonal preconditioner equivalent to per-feature key scaling, delivering super-geometric convergence and 39% lower recall residual at 1.3B parameters.","cross_cats":["cs.CL"],"primary_cat":"cs.LG","authors_text":"Chenyu Zhou, Dongdong Ge, Hongpei Li, Jianghao Lin, Yinyu Ye, Yuerou Liu","submitted_at":"2026-05-13T12:59:26Z","abstract_excerpt":"Linear attention and state-space models offer constant-memory alternatives to softmax attention, but often struggle with in-context associative recall. The Delta Rule mitigates this by writing each token via one step of online gradient descent. However, its step size relies on a single scalar gate that ignores the feature-wise curvature of the inner objective. We propose Online Scaled DeltaNet (OSDN), which augments the scalar gate with a diagonal preconditioner updated online via hypergradient feedback. Crucially, this right-preconditioning is algebraically equivalent to a per-feature scaling"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By exploiting the exact-quadratic structure of the inner regression loss, we establish super-geometric convergence against a right-Newton comparator and prove an algorithm-aligned token-local residual contraction bound; at 1.3B parameters OSDN achieves a 39% reduction in the recall residual ratio.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The inner objective remains exactly quadratic and the online hypergradient update for the diagonal preconditioner can be maintained without breaking the chunkwise parallel pipeline or requiring high-dimensional state.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"OSDN adds online diagonal preconditioning to the Delta Rule, preserving chunkwise parallelism while proving super-geometric convergence and delivering 32-39% recall gains at 340M-1.3B scales.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"OSDN augments the Delta Rule with an online diagonal preconditioner equivalent to per-feature key scaling, delivering super-geometric convergence and 39% lower recall residual at 1.3B parameters.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f76df0f9d6254453bc369ab5be0b1c7dfd479f2e502f474059bd423ac56db383"},"source":{"id":"2605.13473","kind":"arxiv","version":1},"verdict":{"id":"414a0e29-9c84-4451-8f0b-a5440e79128a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:04:37.167670Z","strongest_claim":"By exploiting the exact-quadratic structure of the inner regression loss, we establish super-geometric convergence against a right-Newton comparator and prove an algorithm-aligned token-local residual contraction bound; at 1.3B parameters OSDN achieves a 39% reduction in the recall residual ratio.","one_line_summary":"OSDN adds online diagonal preconditioning to the Delta Rule, preserving chunkwise parallelism while proving super-geometric convergence and delivering 32-39% recall gains at 340M-1.3B scales.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The inner objective remains exactly quadratic and the online hypergradient update for the diagonal preconditioner can be maintained without breaking the chunkwise parallel pipeline or requiring high-dimensional state.","pith_extraction_headline":"OSDN augments the Delta Rule with an online diagonal preconditioner equivalent to per-feature key scaling, delivering super-geometric convergence and 39% lower recall residual at 1.3B parameters."},"references":{"count":73,"sample":[{"doi":"","year":2023,"title":"What learning algorithm is in-context learning? investigations with linear models","work_id":"79950075-a669-4e48-8bc9-28cc29e8d753","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Zoology: Measuring and improving recall in efficient language models","work_id":"a51e0ddf-22e2-4d3c-9759-7201f7d9a699","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Simple linear attention language models balance the recall-throughput tradeoff","work_id":"c542bc4f-de21-42d2-812b-b46f5fa9b434","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Just read twice: closing the recall gap for recurrent language models, 2024 b","work_id":"0700c833-caaf-47d2-b97c-2f4a50eed025","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Hinton, V olodymyr Mnih, Joel Z","work_id":"2acf842b-c183-4bfb-8391-2b456d531aa2","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":73,"snapshot_sha256":"8fbc836eb1ff49d3919ad7ba3273ee946089009a486517d6162d19a767f6e859","internal_anchors":5},"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"}