Adaptive Schauder Stochastic Mirror Descent in Banach Spaces
In this paper, we solve risk functional minimization problems with general losses in Banach spaces within the framework of stochastic approximation. We introduce an adaptive regularization strategy for stochastic mirror descent (SMD). Specifically, using a Schauder basis, we construct nested finite-dimensional subspaces whose dimensions adapt to the sample size $n$ to balance approximation and optimization errors. We solve each SMD subproblem over the corresponding subspace and project the stochastic subgradient 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 $1<p<\infty$, we construct Bregman distances adapted to the uniform convexity of the underlying Banach space using $\max\{2,p\}$-convex functionals. At the $\mathcal{L}^1$ endpoint, we instead construct a locally strongly convex functional based on the entropy function. By developing a new analytical framework, we show that the risk functional values converge to the minimum at a rate of $\mathcal{O}\left(n^{-\min\{\frac12,\frac1p\}}\right)$, up to logarithmic factors. In the misspecified setting, we prove convergence under weaker regularity assumptions and derive explicit polynomial rates under suitable regularity conditions. Finally, we apply the method to statistical inverse problems and illustrate its empirical performance through numerical experiments in both settings.