The Stochastic Auxiliary Problem Principle in Banach Spaces: Measurability and Convergence
研究了Banach空间中随机辅助问题原理算法的可测性和收敛性，将Hilbert空间中的收敛结果推广到自反可分Banach空间，并给出了函数值的效率估计。
The stochastic auxiliary problem principle (APP) algorithm is a general stochastic approximation (SA) scheme that turns the resolution of an original convex 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 an 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 reflexive separable 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.