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Optimal non-asymptotic analysis of the Ruppert–Polyak averaging stochastic algorithm

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  • Gadat, Sébastien
  • Panloup, Fabien

Abstract

This paper is devoted to the non-asymptotic analysis of the Ruppert–Polyak averaging method introduced in Polyak and Juditsky (1992) and Ruppert (1988)[26] for the minimization of a smooth function f with a stochastic algorithm. We first establish a general non-asymptotic optimal bound: if θˆn is the position of the algorithm at step n, we prove that E|θˆn−argmin(f)|2⩽Tr(Σ⋆)n+Cd,fn−rβ,where Σ⋆ is the limiting covariance matrix of the CLT demonstrated in Polyak and Juditsky (1992) and Cd,fn−rβ is a new state-of-the-art second order term that translates the effect of the dimension. We also identify the optimal gain of the baseline SGD γn=γn−3/4, leading to a second-order term with r3/4=5/4. Second, we show that this result holds under some Kurdyka-Łojiasewicz-type condition (Kurdyka, 1988; Lojasiewicz, 1963) for function f, which is far more general than the standard uniformly strongly convex case. In particular, it makes it possible to handle some pathological examples such as on-line learning for logistic regression and recursive quantile estimation.

Suggested Citation

  • Gadat, Sébastien & Panloup, Fabien, 2023. "Optimal non-asymptotic analysis of the Ruppert–Polyak averaging stochastic algorithm," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 312-348.
    • Handle: RePEc:eee:spapps:v:156:y:2023:i:c:p:312-348
    DOI: 10.1016/j.spa.2022.11.012
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    References listed on IDEAS

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    1. Godichon-Baggioni, Antoine, 2016. "Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: Lp and almost sure rates of convergence," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 209-222.
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