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Optimal Oracle Inequalities for Projected Fixed-Point Equations, with Applications to Policy Evaluation

Author

Listed:
  • Wenlong Mou

    (University of California, Berkeley, Berkeley, California 94720)

  • Ashwin Pananjady

    (Georgia Institute of Technology, Atlanta, Georgia 30332)

  • Martin J. Wainwright

    (University of California, Berkeley, Berkeley, California 94720; Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

Abstract
Linear fixed-point equations in Hilbert spaces arise in a variety of settings, including reinforcement learning, and computational methods for solving differential and integral equations. We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. First, we prove an instance-dependent upper bound on the mean-squared error for a linear stochastic approximation scheme that exploits Polyak–Ruppert averaging. This bound consists of two terms: an approximation error term with an instance-dependent approximation factor and a statistical error term that captures the instance-specific complexity of the noise when projected onto the low-dimensional subspace. Using information-theoretic methods, we also establish lower bounds showing that both of these terms cannot be improved, again in an instance-dependent sense. A concrete consequence of our characterization is that the optimal approximation factor in this problem can be much larger than a universal constant. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation, establishing their optimality.

Suggested Citation

  • Wenlong Mou & Ashwin Pananjady & Martin J. Wainwright, 2023. "Optimal Oracle Inequalities for Projected Fixed-Point Equations, with Applications to Policy Evaluation," Mathematics of Operations Research, INFORMS, vol. 48(4), pages 2308-2336, November.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:4:p:2308-2336
    DOI: 10.1287/moor.2022.1341
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