We study the implicit bias of noisy stochastic gradient descent in training wide two-layer ReLU networks for multivariate regression. In a mean-field regime, the training dynamics are approximated by a Wasserstein gradient flow that converges to a unique stationary measure. We characterize the structure of this stationary measure and the predictor it represents. We show that, despite the network being infinitely overparameterized, the learned predictor admits an effectively finite representation: the input weights and biases align along finitely many directions, leading to an effective width collapse. In particular, the solution function is continuous piecewise affine, with affine regions determined by the cells of a finite hyperplane arrangement. The number of learned directions, and hence hyperplanes, is bounded above by $2\mathcal{P}-1$, where $\mathcal{P}$ denotes the number of linear dichotomies realizable on the training inputs. We further establish a non-redundancy property of the learned representation by proving that each learned direction induces a unique ternary activation pattern on the training data. Consequently, the complexity of the learned predictor is governed by the combinatorial geometry of the training data.
This paper develops a mean-field theory for a simplified single-layer causal multi-head self-attention model trained by cross-entropy minimization. Each attention head is treated as a particle in parameter space, and the empirical law of the heads is used as the large-head state variable. In the infinite-head limit, the averaged attention logits define a risk functional on probability measures, whose first variation generates a nonlinear Wasserstein gradient-flow equation. Unlike classical mean-field analyses of shallow networks that often focus on square-loss regression, the present model contains the softmax residual from the cross-entropy objective and the query-key-value structure of masked self-attention. We prove a static finite-head approximation bound for the optimal risk, characterize global minimizers through a variational support condition, and establish a quantitative finite-time propagation-of-chaos estimate comparing finite-head stochastic gradient descent with the limiting PDE. We then study the long-time behavior of the PDE: energy dissipation, convergence to the stationary set under compactness, convergence to a single stationary measure under topological or Kurdyka--Łojasiewicz assumptions, and explicit convergence rates under gradient-domination conditions. Finally, we prove local exponential stability under a Wasserstein strong-monotonicity condition and give verifiable stability and instability criteria for Dirac stationary measures. The results provide a rigorous baseline mean-field framework for attention-head training and clarify the additional compactness, landscape, and curvature assumptions needed to pass from stationarity to convergence and stability.