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Finite-Time High-Probability Bounds for Polyak–Ruppert Averaged Iterates of Linear Stochastic Approximation

Author

Listed:
  • Alain Durmus

    (Department of Applied Mathematics, Ecole Polytechnique, 91128 Palaiseau, France)

  • Eric Moulines

    (Department of Applied Mathematics, Ecole Polytechnique, 91128 Palaiseau, France)

  • Alexey Naumov

    (HSE University and Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 101000, Russian Federation)

  • Sergey Samsonov

    (HSE University and Institute for Information Transmission Problems, Moscow 101000, Russian Federation)

Abstract

This paper provides a finite-time analysis of linear stochastic approximation (LSA) algorithms with fixed step size, a core method in statistics and machine learning. LSA is used to compute approximate solutions of a d -dimensional linear system A ¯ θ = b ¯ for which ( A ¯ , b ¯ ) can only be estimated by (asymptotically) unbiased observations { ( A ( Z n ) , b ( Z n ) ) } n ∈ N . We consider here the case where { Z n } n ∈ N is an a sequence of independent and identically distributed random variables sequence or a uniformly geometrically ergodic Markov chain. We derive p th moment and high-probability deviation bounds for the iterates defined by LSA and its Polyak–Ruppert-averaged version. Our finite-time instance-dependent bounds for the averaged LSA iterates are sharp in the sense that the leading term we obtain coincides with the local asymptotic minimax limit. Moreover, the remainder terms of our bounds admit a tight dependence on the mixing time t mix of the underlying chain and the norm of the noise variables. We emphasize that our result requires the LSA step size to scale only with logarithm of the problem dimension d .

Suggested Citation

  • Alain Durmus & Eric Moulines & Alexey Naumov & Sergey Samsonov, 2025. "Finite-Time High-Probability Bounds for Polyak–Ruppert Averaged Iterates of Linear Stochastic Approximation," Mathematics of Operations Research, INFORMS, vol. 50(2), pages 935-964, May.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:2:p:935-964
    DOI: 10.1287/moor.2022.0179
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    References listed on IDEAS

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    1. Jalaj Bhandari & Daniel Russo & Raghav Singal, 2021. "A Finite Time Analysis of Temporal Difference Learning with Linear Function Approximation," Operations Research, INFORMS, vol. 69(3), pages 950-973, May.
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