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

TL;DR

This paper extends the stochastic Auxiliary Problem Principle (APP) algorithm to Banach spaces, proving measurability, convergence, and efficiency estimates, thereby broadening its theoretical foundation and practical applicability.

Contribution

It introduces the stochastic APP in Banach spaces, proves measurability of iterates, and extends convergence and efficiency results from Hilbert to Banach spaces.

Findings

Proves measurability of stochastic APP iterates in Banach spaces

Extends convergence results from Hilbert to Banach spaces

Provides efficiency estimates for function values at averaged and last iterates

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 Fej{\'e}r monotonicity to our framework.