{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"b32e8c80a54b6807a50dbd8a16eb8c3fb79a9d19fafa95bc6bf36beff28834aa","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["91fdc52a2137848f1217c031310bc0a08f0dfe8fe5bf8dd82fb22aec4ec71ab9","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"c23d5e3fb22f9f0ccd2af852833ea47d75b0a71ccb5d931c7a4e68a2472c92f1","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"stat.learning-inference.sgd-multi-index-complexity","kind":"well-posed","literature_status":"partially-resolved","n":"2","parents":[],"plain":"A network trained by stochastic gradient descent (repeated small weight updates using random batches of data) must find a few hidden directions in a high-dimensional input. The question is how many training examples it needs as the input dimension grows, compared with the minimum any fast method needs.","posed_since":"","precise":"Inputs $x \\sim N(0, I_d)$ and labels $y = g(W x)$ with $W$ an $r$ x d matrix of rank r fixed as $d \\to \\infty$ and $g$ a fixed link function. For a standard two-layer network trained by online or multi-pass SGD on the square loss without data preprocessing, determine the exponent $\\gamma(g)$ in $n \\sim d^\\gamma$ required to reach test error $o(1)$, and determine whether it equals the benchmark for statistical-query and low-degree algorithms, $n \\sim d^{\\max(1, k_*/2)}$, with $k_*$ the generative exponent for single-index targets and the generative leap exponent for multi-index targets (Damian, Lee and Bruna, arXiv:2506.05500). The answer is $\\gamma$ as a function of the Hermite structure of $g$, with proof.","problem_ref":"91fdc52a2137848f1217c031310bc0a08f0dfe8fe5bf8dd82fb22aec4ec71ab9","references":"","settled_by":"Matching upper and lower bounds on the sample complexity of standard gradient training for general multi-index targets.","status_note":"For single-index polynomial targets, SGD with minibatch reuse on two-layer networks reaches $n \\sim d\\,\\operatorname{polylog} d$ (Lee, Oko, Suzuki and Wu, arXiv:2406.01581, 2024); correlational SGD on multi-index targets is governed by the leap complexity (Abbe, Boix-Adsera and Misiakiewicz 2023); the low-degree benchmark for multi-index targets is set by the generative leap (Damian, Lee and Bruna 2025); whether standard SGD reaches that benchmark for general multi-index $g$ is open (2026).","title":"Sample complexity of gradient training for multi-index 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