arXiv:2608.13121v4 Announce Type: replace-cross 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 $\mathcal{L}^p$ spaces with $1Source：https://arxiv.org/abs/2608.13121
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