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For small $B$ and large $\\eta$, SGD corresponds to a stochastic evolution of the parameters, whose noise amplitude is governed by the ''temperature'' $T\\equiv \\eta/B$. Yet this description is observed to break down for sufficiently large batches $B\\geq B^*$, or simplifies to gradient descent (GD) when the temperature is sufficiently small. Understanding where these cross-overs take place remains a central challenge. Here, we resolve these questions for a teacher-student perceptron classification model and show empirically that our key predictions still apply to deep networks. Specifically, we obtain a phase diagram in the $B$-$\\eta$ plane that separates three dynamical phases: (i) a noise-dominated SGD governed by temperature, (ii) a large-first-step-dominated SGD and (iii) GD. These different phases also correspond to different regimes of generalization error. Remarkably, our analysis reveals that the batch size $B^*$ separating regimes (i) and (ii) scale with the size $P$ of the training set, with an exponent that characterizes the hardness of the classification problem.","url_abs":"https://arxiv.org/abs/2309.10688v4","url_pdf":"https://arxiv.org/pdf/2309.10688v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"on-the-different-regimes-of-stochastic","repo_url":"https://github.com/pcsl-epfl/regimes_of_sgd","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"jax","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[{"method_slug":"sgd","method_name":"SGD"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2309.10688","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2309.10688"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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