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.
RAG pipelines return a \emph{ranked list} of passages. We argue this is a mismatch: the downstream language model conditions on a \emph{set}, and the selection problem is fundamentally geometric. We propose \jko, which frames reranking as minimising a free-energy functional $F(p)=\text{relevance}+\text{entropy}+\text{redundancy}$ under Wasserstein-2 gradient flow via the Jordan--Kinderlehrer--Otto proximal scheme. The ground metric $C_{ij}=(1-\cos\langle z_i,z_j\rangle)^2$ encodes the semantic geometry of the embedding manifold. Our central contribution is a \emph{linear-response theory} explaining \emph{why} the Wasserstein geometry helps: the Wasserstein and KL retrieval maps differ only in their proximal Hessian -- dense and geometry-aware for $W^2$, diagonal and geometry-blind for KL -- and this difference damps the mass transport that query paraphrase induces. The theory yields a falsifiable prediction: the stability advantage is monotonically decreasing in step size $h$. We verify this empirically via free-energy descent, frequency-resolved perturbation response, the predicted $h$-dependence, and a certified-radius analysis. Four extensions are introduced: \textbf{\nmjko} (learned ground metric), \textbf{\bwjko} ($W^2$--KL interpolation), \textbf{\samjko} ($2\times$ speedup), and \textbf{\dualrank} (OT dual potentials as confidence signals). Across five BEIR benchmarks, \jko\ outperforms the cross-encoder on all five; the decisive advantage is robustness -- 22--38\% more stable under paraphrase, $2\times$ fewer leaked distractors.
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.
We consider one-hidden layer neural networks trained in the feature-learning regime using gradient descent, and relate the output of the finite-width network $f_{\hatρ_t^m}$ to its infinite-width counterpart $f_{ρ_t^{MF}}$, which evolves in the mean-field dynamics. While constant-time horizon bounds for $\|f_{ρ_t^{MF}} - f_{\hatρ_t^m}\|$ may be obtained via standard Grönwall estimates, the long-time behavior of the fluctuation is a more delicate matter. Uniform-in-time bounds often rely on (local) strong convexity in the landscape or Logarithmic Sobolev inequalities present in noisy gradient dynamics. In this work, we establish non-asymptotic weak propagation-of-chaos that holds uniformly in time, obtained by exploiting instead the convergence rate of the mean-field deterministic Wasserstein-gradient-flow dynamics. Specifically, denoting by $L_t$ the mean-field excess MSE loss at time $t$ and $m$ the number of neurons, under standard regularity assumptions and the condition $\int_0^\infty L_t^{1/2} dt =O(\log d)$, we obtain the uniform in time bound $\|f_{ρ_t^{MF}}- f_{\hatρ_t^m}\|^2 \lesssim \text{poly}(d) m^{-\min(1,c/6)}$ whenever $L_t \lesssim t^{-c}$. Our result holds in a noiseless setting and does not make any assumptions on the geometry of the landscape near the optimum, and extends seamlessly to other forms of discretization, including finite number of samples and time discretization. A key takeaway of our result is that whenever the convergence rate of the mean-field, population-loss dynamics is faster than $t^{-2}$, we can attain a loss of $ε$ with only $\text{poly}(d/ε)$ neurons, training samples, and GD steps.