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Local convergence theorems for adaptive stochastic approximation schemes.

T L Lai1, H Robbins

  • 1Department of Mathematical Statistics, Columbia University, New York, New York 10027.

Proceedings of the National Academy of Sciences of the United States of America
|July 1, 1979
PubMed
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This study introduces adaptive stochastic approximation schemes to find the root of regression models. These methods demonstrate convergence, establishing theorems on their rate of convergence.

Area of Science:

  • Statistics
  • Machine Learning

Background:

  • Regression models are fundamental in statistical analysis.
  • Estimating the root of a regression function is a key problem.

Purpose of the Study:

  • To develop and analyze adaptive stochastic approximation schemes for regression models.
  • To establish convergence properties and rates for these schemes.

Main Methods:

  • Utilizing adaptive stochastic approximation algorithms of the form x(n+1) = x(n) - y(n)/(nb(n)).
  • Analyzing the convergence of observed levels x(n) to the regression function root theta.
  • Establishing local convergence theorems.

Main Results:

  • The proposed schemes converge with probability 1 to the unknown root theta.

Related Experiment Videos

  • Convergence rate theorems are established, linking them to the behavior of b(n).

    • Conclusions:

    • Adaptive stochastic approximation provides a robust method for root finding in regression.
    • The theoretical framework offers insights into the efficiency of these estimation techniques.