This paper develops a neural operator framework for approximating mappings defined on constrained Wasserstein spaces $\mathcal{M}_\lambda$, consisting of probability measures on $I \times \mathbb{R}^d$ with prescribed marginal $\lambda$ on the label space $I$. The core contribution is the DeepONetCyl architecture, which combines cylindrical moment approximations $\Phi_J(\mu) = (\langle \varphi_1, \mu \rangle, \ldots, \langle \varphi_J, \mu \rangle)$ with a DeepONet-type branch–trunk structure to preserve the marginal constraint. This enables learning of heterogeneous (non-exchangeable) mean-field control problems where agent interactions depend on labels, extending prior neural methods beyond the exchangeable case.

This paper extends stochastic approximation (SA) theory to non-Markovian driving noise that is also non-ergodic, establishing that the ergodic decomposition of the original process corresponds to a Doeblin decomposition of an equivalent Markov chain. The core insight is that iterates retain memory of the distant past through the tail $\sigma$-field at $-\infty$, offering a theoretical lens on how learning algorithms might encode long-term dependencies. The author proposes this framework as a paradigm for understanding transformer attention mechanisms and continual learning, where the entire history influences current updates.