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PreprintarXiv licencearXiv

The Last Iterate of the Projected Stochastic Subgradient Method Pays a Logarithm, Even in Dimension One

  • Guglielmo Beretta
  • Tommaso Cesari
  • Roberto Colomboni
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Abstract

We prove that the projected stochastic subgradient method with the standard anytime stepsizes ηt = γ/√t can suffer expected last-iterate optimization error of order (log n)/√n already on the real line. Combined with the classical in-expectation upper bound, this establishes the sharp worst-case order of the projected stochastic subgradient method.

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Cite

BibTeXRIS
BibTeX
@misc{beretta2026last,
  title = {{The Last Iterate of the Projected Stochastic Subgradient Method Pays a Logarithm, Even in Dimension One}},
  author = {Guglielmo Beretta and Tommaso Cesari and Roberto Colomboni},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30143},
  eprint = {2610.30143},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30143},
}
RIS
TY  - UNPB
TI  - The Last Iterate of the Projected Stochastic Subgradient Method Pays a Logarithm, Even in Dimension One
AU  - Guglielmo Beretta
AU  - Tommaso Cesari
AU  - Roberto Colomboni
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.30143
UR  - https://arxiv.org/abs/2610.30143
AB  - We prove that the projected stochastic subgradient method with the standard anytime stepsizes η_t = γ/√t can suffer expected last-iterate optimization error of order (log n)/√n already on the real line. Combined with the classical in-expectation upper bound, this establishes the sharp worst-case order of the projected stochastic subgradient method.
ER  -