Metraipolis › archive › dc_7f772e005ed3

Minimax-Optimal Early Stopping for Continuous-Time SGD via the Discrepancy Principle — arXiv:2609.11273

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.11273
License (as archived): CC0-1.0
Archived: 2026-10-16T06:01:14.166Z
SHA-256 of the archived text: 40b40838e846196c35dcff811cc4e29ee23dfcbad5f115ed44fa0ddae935d947
Stable URL of this page: https://metraipolis.com/documento/dc_7f772e005ed3

How to cite

Minimax-Optimal Early Stopping for Continuous-Time SGD via the Discrepancy Principle — arXiv:2609.11273 — arXiv (en), https://arxiv.org/abs/2609.11273 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7f772e005ed3 (Metraipolis archive dc_7f772e005ed3, sha256 40b40838e846196c35dcff811cc4e29ee23dfcbad5f115ed44fa0ddae935d947)

BibTeX

@misc{dc7f772e005ed3,
  title = {Minimax-Optimal Early Stopping for Continuous-Time SGD via the Discrepancy Principle — arXiv:2609.11273},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.11273},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7f772e005ed3; sha256 40b40838e846196c35dcff811cc4e29ee23dfcbad5f115ed44fa0ddae935d947},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.11273. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study early stopping for a continuous-time model of stochastic gradient descent (SGD) in ill-posed linear inverse problems. We consider an a posteriori stopping rule based on the discrepancy principle, which stops the dynamics once the residual reaches the noise level. Unlike deterministic gradient flow, the stochastic dynamics exhibits persistent multiplicative fluctuations whose amplitude scales with the residual itself. Under a source condition on the initial error and a polynomial bound on the spectrum of the empirical covariance operator, we prove that, with high probability, the error at this random stopping time achieves the minimax rate over the corresponding source class, up to a logarithmic factor. Our results therefore establish the discrepancy principle as an adaptive regularization strategy for continuous-time SGD that is minimax-optimal up to a logarithmic factor, despite the persistent fluctuations induced by stochastic sampling. To our knowledge, this is the first convergence-rate result for a continuous-time model of stochastic gradient descent in inverse problems under an a posteriori stopping rule: existing analyses, including those of variance-reduced variants, bound the stopping time but do not quantify the error attained there.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-11 in math.OC).
Categorie arXiv: Numerical Analysis (math.NA) ; Optimization and Control (math.OC); Statistics Theory (math.ST).
Pagina del lavoro: https://arxiv.org/abs/2609.11273 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T05:52:41.434Z dalla pagina https://arxiv.org/catchup/math.OC/2026-09-11?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).