Adaptive Schauder Stochastic Mirror Descent in Banach Spaces
Abstract
In this paper, we introduce an adaptive regularization strategy for stochastic mirror descent (SMD) to solve a class of risk functional minimization problems in infinite-dimensional Banach spaces. Using a Schauder basis, we construct nested finite-dimensional subspaces whose dimensions adapt to the sample size n and balance approximation and optimization errors. We restrict each SMD subproblem to the corresponding subspace and project the stochastic gradient onto its dual space. This yields closed form solutions and coordinate-wise coefficient updates with low computational and storage complexity. For risk minimization in ℒp spaces with 1<p<∞, we construct Bregman distances adapted to the underlying Banach space geometry using max{2,p}-convex functionals induced by uniform convexity. At the ℒ1 endpoint, we instead construct a locally strongly convex functional based on the entropy function. By developing a new analytical framework, we establish a convergence rate of 𝒪(n−min{1/2,1/p}), up to logarithmic factors. In the misspecified setting, we prove convergence of the risk to its minimum under weaker regularity assumptions and derive explicit polynomial rates when the regularity exceeds a specified threshold. Finally, we apply the method to statistical inverse problems and illustrate its empirical performance through numerical experiments in both settings.
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BibTeX
@misc{bai2026adaptive,
title = {{Adaptive Schauder Stochastic Mirror Descent in Banach Spaces}},
author = {Jinhui Bai and Shuai Lu and Lei Shi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.13121},
eprint = {2608.13121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13121},
}
RIS
TY - UNPB
TI - Adaptive Schauder Stochastic Mirror Descent in Banach Spaces
AU - Jinhui Bai
AU - Shuai Lu
AU - Lei Shi
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2608.13121
UR - https://arxiv.org/abs/2608.13121
AB - In this paper, we introduce an adaptive regularization strategy for stochastic mirror descent (SMD) to solve a class of risk functional minimization problems in infinite-dimensional Banach spaces. Using a Schauder basis, we construct nested finite-dimensional subspaces whose dimensions adapt to the sample size n and balance approximation and optimization errors. We restrict each SMD subproblem to the corresponding subspace and project the stochastic gradient onto its dual space. This yields closed form solutions and coordinate-wise coefficient updates with low computational and storage complexity. For risk minimization in ℒ^p spaces with 1<p<∞, we construct Bregman distances adapted to the underlying Banach space geometry using max{2,p}-convex functionals induced by uniform convexity. At the ℒ^1 endpoint, we instead construct a locally strongly convex functional based on the entropy function. By developing a new analytical framework, we establish a convergence rate of 𝒪(n^−min{1/2,1/p}), up to logarithmic factors. In the misspecified setting, we prove convergence of the risk to its minimum under weaker regularity assumptions and derive explicit polynomial rates when the regularity exceeds a specified threshold. Finally, we apply the method to statistical inverse problems and illustrate its empirical performance through numerical experiments in both settings.
ER -