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For nonlinear ICL, prior work has related softmax and kernelized attention to functional-gradient-type dynamics, but it remains unclear whether a standard transformer with softmax attention can implement a convergent solver with an end-to-end prediction-error guarantee. In this paper, we study in-context kernel ridge regression (KRR) with Gaussian kernels and show that a standard softmax-attention transformer can approximate the KRR predictor during its forward pass by implementing preconditioned Richardson iteration on the associated kernel linear system. Under bounded-data assumptions, we construct a single-head transformer with $O(\\log(1/\\epsilon))$ blocks and MLP width $O(\\sqrt{N/\\epsilon})$ that achieves $\\epsilon$-accurate prediction for pr","title":"Transformers Can Implement Preconditioned Richardson Iteration for In-Context Gaussian Kernel Regression","url":"https://arxiv.org/abs/2605.08475","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.08475v1 Announce Type: cross \nAbstract: Mechanistic accounts of in-context learning (ICL) have identified iterative algorithms for linear regression and related linear prediction tasks, often using linear or ReLU attention variants. For nonlinear ICL, prior work has related softmax and kernelized attention to functional-gradient-type dynamics, but it remains unclear whether a standard transformer with softmax attention can implement a convergent solver with an end-to-end prediction-error guarantee. In this paper, we study in-context kernel ridge regression (KRR) with Gaussian kernels and show that a standard softmax-attention transformer can approximate the KRR predictor during its forward pass by implementing preconditioned Richardson iteration on the associated kernel linear system. Under bounded-data assumptions, we construct a single-head transformer with $O(\\log(1/\\epsilon))$ blocks and MLP width $O(\\sqrt{N/\\epsilon})$ that achieves $\\epsilon$-accurate prediction for pr","title":"Transformers Can Implement Preconditioned Richardson Iteration for In-Context Gaussian Kernel Regression","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-12T04:43:42Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.08475"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:0e9a9b8ccebaf6556d0a6aae0f50fbf3b5e84a7706b3cafe21d338f820839acf7cfe1ff6a90b9f10e2291e383e002dfb71cb29656e6218299959008cf09f690e","signer":"crovia.substrate","subject":{"observed_at":"2026-05-12T04:43:42Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.08475"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"13370398b24bb5b71f5d154397f00e8ebf0de37918f19e3327b86df1c2d6fa02","leaf_index":128581,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"3f66439e16eb7122f24c28f9649cd6d1d55b8e1e3c8dca4b647fc66c28ee9451","side":"left"},{"sibling":"d4d3dbafeddc7fb65b99315d4c346784dd15bd1813778993881f5b9bd8fd3a24","side":"right"},{"sibling":"8d5069b8091b1aa9a4f512478175d1a6df0939a483bbc1e07bd5b4083587c9d3","side":"left"},{"sibling":"2865b327427305875a30ae6644cea239d2a75b976f62377874621a62db418652","side":"right"},{"sibling":"16789e966e0cf4544982b59006793a26843ccba25fa4aa6ed5b3e341ce969671","side":"right"},{"sibling":"dfd38390185abfa793e2aa7dcb95cb915b04addc87e066e6ca0990d3618463e4","side":"right"},{"sibling":"ba60cf3cfbedfa4be8e552df69fe9f00d4748fabd7f2f1d5098bfd63cd558f19","side":"left"},{"sibling":"2be4781bec47f25a197c8a3d5b3f30a3d2f2d1517add8590753a933d122c2b40","side":"right"},{"sibling":"831b04b4dcc29bcff4577c406291bc4f644057b650a55f064d46d1cab5185326","side":"right"},{"sibling":"1d74b0fb79eace68949b4d82b0e430b1d9eb122c125f67f5be7d50c074c228e7","side":"left"},{"sibling":"c6eaf7a4fcab2db96e9e9423acb6922c80f64882d0f3f50d09e53a4807d23084","side":"left"},{"sibling":"df12eaabc0a370aff5d0488478f48d2b3f90d025c903643f98ea804b017688ec","side":"right"},{"sibling":"ffc4d51379293bc3e1910c7d612f409dc610fd9acf8241793fb89f82e1bad4ef","side":"left"},{"sibling":"62ac6554017807bd83187f5a3e5f4f72d6c482616429c2780e9fff1f4845fa04","side":"left"},{"sibling":"3a5e69cf0803f4c91f3895ed7c9a95748fef240bec4422e167c05300f79f06c0","side":"left"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_c1dcc98094f7b5b7d65db4ca791fc24b7a5783d4fdd8cffa71dbd9001d2238a1"}}