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For linear models ($L=1$), both $\\ell_\\infty$- and $\\ell_2$-SAM recover the $\\ell_2$ max-margin classifier, matching gradient descent (GD). However, for depth $L = 2$, the behavior changes drastically -- even on a single-example dataset. For $\\ell_\\infty$-SAM, the limit direction depends critically on initialization and can converge to $\\mathbf{0}$ or to any standard basis vector, in stark contrast to GD, whose limit aligns with the basis vector of the dominant data coordinate. For $\\ell_2$-SAM, we show that although its limit direction matches the $\\ell_1$ max-margin solution as in the case of GD, its finite-time dynamics exhibit a phenomenon we call \"sequential feature amplification\", in which the predictor initially relies on minor coordinates and gradually shifts to l","title":"Minor First, Major Last: A Depth-Induced Implicit Bias of Sharpness-Aware Minimization","url":"https://arxiv.org/abs/2603.08290","vendor":"arxiv_cs_ai"},"summary":"arXiv:2603.08290v2 Announce Type: replace-cross \nAbstract: We study the implicit bias of Sharpness-Aware Minimization (SAM) when training $L$-layer linear diagonal networks on linearly separable binary classification. For linear models ($L=1$), both $\\ell_\\infty$- and $\\ell_2$-SAM recover the $\\ell_2$ max-margin classifier, matching gradient descent (GD). However, for depth $L = 2$, the behavior changes drastically -- even on a single-example dataset. For $\\ell_\\infty$-SAM, the limit direction depends critically on initialization and can converge to $\\mathbf{0}$ or to any standard basis vector, in stark contrast to GD, whose limit aligns with the basis vector of the dominant data coordinate. For $\\ell_2$-SAM, we show that although its limit direction matches the $\\ell_1$ max-margin solution as in the case of GD, its finite-time dynamics exhibit a phenomenon we call \"sequential feature amplification\", in which the predictor initially relies on minor coordinates and gradually shifts to l","title":"Minor First, Major Last: A Depth-Induced Implicit Bias of Sharpness-Aware Minimization","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-19T04:43:36Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2603.08290"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:74d16d784fdc8200266397411a6e9eb89c33239e59830ecaa15e52564bce2b7cc53f1eafb43dc8c871d7658b64a8898bb91374392f8330763e8656ec776f2302","signer":"crovia.substrate","subject":{"observed_at":"2026-05-19T04:43:36Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2603.08290"},"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":"bbb21836e7f199b8a187f8acd08b88a213e7ce171298ec651b50096f383abc14","leaf_index":143162,"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":"df7cbd4f5759323f14d83c325ef46cc2f16faa0fd011992bf5f16927fcedf346","side":"right"},{"sibling":"90fc73eb7deb9e1a03a726ae6c7181c8565bcb6dccb35107bdb2f9a2abc37815","side":"left"},{"sibling":"76852e6e1341fa2043ee71e487bb73c7433f3fd977815c85bb02d69b1bc2ed6b","side":"right"},{"sibling":"5e4cec3f83748763828bf99606ecb418101de1b5f7731a4ef40be05d5824cf49","side":"left"},{"sibling":"2c3333ebae2565582fc1c7391957b06c6e0f20656f48ba1449fbf137d96771e4","side":"left"},{"sibling":"64a5ec52eed2a8189bc3dcf5ead2feaebbee89147caaa56d81f5d48755a48d02","side":"left"},{"sibling":"85fd29731e65c36b4f2a5f4f39c039cb540b551881d70cd5789c3e56208c17c8","side":"right"},{"sibling":"4b6dab10c74fb2a96436053a067988cc08e1b2f810f1362c194b7439e788d860","side":"right"},{"sibling":"b986468aca0b7804b8a608705cafc663139e79ff550a69be0b9dd58ec70714f7","side":"left"},{"sibling":"db97141c585f6a1e6bebe92b3ea300ea0f38a2321ca286d85850b11b2dd162a6","side":"left"},{"sibling":"202f1bead178ef3785968d50d3d188264a95192a077654c331612e04a34cbfbe","side":"left"},{"sibling":"72249c8c8b068386e35d16f4bd0bbeb9ba820ca217ef0f0d28396c9fe493f5f0","side":"left"},{"sibling":"ea64599340f7ffdf17ad0cbc1d9401ef8870a347e3847bdc106d06b1673df09c","side":"right"},{"sibling":"8f4c0fbe56b6c010fbb8c782ebcd478079bb3f991d8704e2534209a075d9163c","side":"left"},{"sibling":"4db1f363729507e27a60851cf6ed334d7b9acdef194ed7d419aba4d2bd367a4a","side":"right"},{"sibling":"a86ee18c45e7fcc408b6007eaece05aa75b2d9ae30252e9e878462b4dffbef7b","side":"right"},{"sibling":"1d18e7663d43ccff0122ecc7ee12645bb16afb607b218e81b1ea2408f863cb78","side":"right"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"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":143302,"merkle_root":"999156d40a7c61d9ddd52b7338f3cbda3e68f53bace070c7b616ea194e23b123","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260519T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-19T05:37:30Z","sig_algorithm":"ed25519","signature":"b1a252cc66ff32bed1d10dd88a6b2a200e3856d3dbcfcc4ee55e02e00f3d548e854ed9c544704b222bd5d315492c4a935ba2d90d727c585a67899b0ad602fc05","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_febfd743e0e57590c41c62e03b1378657ae2f2ffefbea4123cb3ea61a98c325e"}}