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PreprintCC BY 4.0arXiv

Stochastic Mirror Descent under Dependent Observations

  • Gersende Fort
  • Anatoli Juditsky
  • Jiaming Liang
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Abstract

We study the non-asymptotic convergence of Stochastic Mirror Descent (SMD) algorithm when the noisy first-order oracle is driven by dependent observations. Departing from the classical i.i.d. assumption underlying most existing guarantees, we introduce a flexible ``forgetting" condition that accommodates geometrically or polynomially dependent sequences. Under this condition we derive explicit finite-sample accuracy bounds for the averaged SMD iterates, recovering the classical O(1/T) and O(1/√T) rates in the large- and small-initial-error regimes. We instantiate the general framework on two applications: high-dimensional linear regression with dependent regressors and nonparametric estimation of the drift function of a functional autoregressive model.

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Cite

BibTeXRIS
BibTeX
@misc{fort2026stochastic,
  title = {{Stochastic Mirror Descent under Dependent Observations}},
  author = {Gersende Fort and Anatoli Juditsky and Jiaming Liang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41251},
  eprint = {2610.41251},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41251},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Stochastic Mirror Descent under Dependent Observations
AU  - Gersende Fort
AU  - Anatoli Juditsky
AU  - Jiaming Liang
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41251
UR  - https://arxiv.org/abs/2610.41251
AB  - We study the non-asymptotic convergence of Stochastic Mirror Descent (SMD) algorithm when the noisy first-order oracle is driven by dependent observations. Departing from the classical i.i.d. assumption underlying most existing guarantees, we introduce a flexible ``forgetting" condition that accommodates geometrically or polynomially dependent sequences. Under this condition we derive explicit finite-sample accuracy bounds for the averaged SMD iterates, recovering the classical O(1/T) and O(1/√T) rates in the large- and small-initial-error regimes. We instantiate the general framework on two applications: high-dimensional linear regression with dependent regressors and nonparametric estimation of the drift function of a functional autoregressive model.
ER  -