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The stochastic Auxiliary Problem Principle in Banach spaces: measurability and convergence

Abstract : The stochastic Auxiliary Problem Principle (APP) algorithm is a general Stochastic Approximation (SA) scheme that turns the resolution of an original optimization problem into the iterative resolution of a sequence of auxiliary problems. This framework has been introduced to design decomposition-coordination schemes but also encompasses many well-known SA algorithms such as stochastic gradient descent or stochastic mirror descent. We study the stochastic APP in the case where the iterates lie in a Banach space and we consider an additive error on the computation of the subgradient of the objective. In order to derive convergence results or efficiency estimates for a SA scheme, the iterates must be random variables. This is why we prove the measurability of the iterates of the stochastic APP algorithm. Then, we extend convergence results from the Hilbert space case to the Banach space case. Finally, we derive efficiency estimates for the function values taken at the averaged sequence of iterates or at the last iterate, the latter being obtained by adapting the concept of modified Fejér monotonicity to our framework.
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https://hal.archives-ouvertes.fr/hal-03115740
Contributor : Thomas Bittar Connect in order to contact the contributor
Submitted on : Tuesday, January 19, 2021 - 8:27:54 PM
Last modification on : Thursday, November 4, 2021 - 12:25:48 PM
Long-term archiving on: : Tuesday, April 20, 2021 - 8:00:20 PM

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  • HAL Id : hal-03115740, version 1
  • ARXIV : 2101.08073

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Thomas Bittar, Pierre Carpentier, Jean-Philippe Chancelier, Jérôme Lonchampt. The stochastic Auxiliary Problem Principle in Banach spaces: measurability and convergence. 2021. ⟨hal-03115740⟩

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