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Stochastic Gradient versus Recursive Least Squares Learning

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  • Sergey Slobodyan
  • Anna Bogomolova,
  • Dmitri Kolyuzhnov

Abstract

In this paper, we perform an in—depth investigation of relative merits of two adaptive learning algorithms with constant gain, Recursive Least Squares (RLS) and Stochastic Gradient (SG), using the Phelps model of monetary policy as a testing ground. The behavior of the two learning algorithms is very different. Under the mean (averaged) RLS dynamics, the Self—Confirming Equilibrium (SCE) is stable for initial conditions in a very small region around the SCE. Large distance movements of perceived model parameters from their SCE values, or “escapes”, are observed. On the other hand, the SCE is stable under the SG mean dynamics in a large region. However, actual behavior of the SG learning algorithm is divergent for a wide range of constant gain parameters, including those that could be justified as economically meaningful. We explain the discrepancy by looking into the structure of eigenvalues and eigenvectors of the mean dynamics map under SG learning. Results of our paper hint that caution is needed when constant gain learning algorithms are used. If the mean dynamics map is stable but not contracting in every direction, and most eigenvalues of the map are close to the unit circle, the constant gain learning algorithm might diverge.

Suggested Citation

  • Sergey Slobodyan & Anna Bogomolova, & Dmitri Kolyuzhnov, 2006. "Stochastic Gradient versus Recursive Least Squares Learning," CERGE-EI Working Papers wp309, The Center for Economic Research and Graduate Education - Economics Institute, Prague.
  • Handle: RePEc:cer:papers:wp309
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    References listed on IDEAS

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    1. Chryssi Giannitsarou, 2003. "Heterogeneous Learning," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 6(4), pages 885-906, October.
    2. Seppo Honkapohja & Kaushik Mitra, 2006. "Learning Stability in Economies with Heterogeneous Agents," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 9(2), pages 284-309, April.
    3. Thomas Sargent & Noah Williams & Tao Zha, 2009. "The Conquest of South American Inflation," Journal of Political Economy, University of Chicago Press, vol. 117(2), pages 211-256, April.
    4. William Poole & Robert H. Rasche, 2002. "Flation," Review, Federal Reserve Bank of St. Louis, issue Nov, pages 1-6.
      • William Poole, 2002. "Flation," Speech 49, Federal Reserve Bank of St. Louis.
    5. George W. Evans & Seppo Honkapohja & Noah Williams, 2010. "Generalized Stochastic Gradient Learning," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 51(1), pages 237-262, February.
    6. In-Koo Cho & Noah Williams & Thomas J. Sargent, 2002. "Escaping Nash Inflation," Review of Economic Studies, Oxford University Press, vol. 69(1), pages 1-40.
    7. Kolyuzhnov, Dmitri & Bogomolova, Anna & Slobodyan, Sergey, 2014. "Escape dynamics: A continuous-time approximation," Journal of Economic Dynamics and Control, Elsevier, vol. 38(C), pages 161-183.
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    Cited by:

    1. George W. Evans & Seppo Honkapohja, 2009. "Robust Learning Stability with Operational Monetary Policy Rules," Central Banking, Analysis, and Economic Policies Book Series,in: Klaus Schmidt-Hebbel & Carl E. Walsh & Norman Loayza (Series Editor) & Klaus Schmidt-Hebbel (Series (ed.), Monetary Policy under Uncertainty and Learning, edition 1, volume 13, chapter 5, pages 145-170 Central Bank of Chile.

    More about this item

    Keywords

    Constant gain adaptive learning; E—stability; recursive least squares; stochastic gradient learning.;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness
    • E10 - Macroeconomics and Monetary Economics - - General Aggregative Models - - - General
    • E17 - Macroeconomics and Monetary Economics - - General Aggregative Models - - - Forecasting and Simulation: Models and Applications

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