Adaptive Schauder Stochastic Mirror Descent in Banach Spaces

In this paper, we extend stochastic mirror descent (SMD) to infinite-dimensional Banach spaces for solving a class of risk functional minimization problems, where stochastic gradient information is only available through sampling. We first choose the Bregman distance according to the uniform convexity properties of the Banach space. For the non-uniformly convex space $\mathcal{L}_{ρ_X}^1(Ω)$, we instead construct a Bregman distance induced by the entropy function. Based on a Schauder basis of the Banach space, we introduce a family of finite-dimensional subspaces that adapt to the sample size $n$. At each SMD iteration, we restrict the subproblem to the corresponding finite-dimensional subspace and project the stochastic gradient onto the associated finite-dimensional dual space, thereby introducing a new regularization strategy. This regularization strategy allows us to explicitly solve the SMD subproblem efficiently and to achieve a bias--variance trade-off. The algorithm requires $\mathcal{O}(n^{1+θ})$ time and $\mathcal{O}(n^θ)$ memory, where $θ>0$ can be chosen arbitrarily small when the minimizer has sufficient regularity. By developing a new analytical framework, we prove that the proposed algorithm achieves a convergence rate of $\mathcal{O}\left(n^{-1/p_1}\right)$ up to logarithmic factors, where $p_1\geq2$ is determined by the convexity properties of the underlying space. In the misspecified setting where the minimizer satisfies only weaker regularity conditions, we show that the proposed algorithm still converges to the minimum. We further extend the algorithm to inverse problems and validate its effectiveness in numerical experiments.