Gradient descent (GD) is explicit Euler for gradient flow, but a state-accurate continuous-time surrogate need not remain accurate after differentiation. At every fixed nonresonant step size, ordinary automatic differentiation exactly differentiates the executed hard-ReLU GD program. We prove that, over a fixed finite horizon, the GD states converge and these exact discrete derivatives approach an event-free regional propagator, whereas the derivative of the limiting flow also contains speed-normalized activation-event transfers. A prepoint Stieltjes representation separates the absolutely continuous regional Hessian from atomic interface curvature; one nonzero gradient jump produces an exactly rank-one endpoint discrepancy, and global convexity prevents complete multi-event cancellation whenever an event is strict. Nevertheless, a standard family of globally 1-strongly convex residual-ReLU squared-loss risks realizes arbitrarily large reciprocal sensitivity ratios on open initialization sets, with a uniform transversality margin. The same discrete-versus-flow decomposition extends to parameters and reverse-mode adjoints; resolved smoothing in the scalar or autonomous-normal regime and consistent event localization recover the flow sensitivity. The results concern deterministic full-batch, finite-horizon dynamics with a stable finite itinerary of separated same-direction transverse events; they are consistency theorems, not prevalence claims for large-scale training.
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.
Grokking suggests that fitting the training data and learning a simple underlying rule may occur on different time scales. We formalize this phenomenon by separating the fast decay of the classification loss from the slower simplification of the learned representation, and we call the resulting pair of stopping times two training clocks. For deep linear networks, we show that a post-margin gap-growth or one-step tail-contraction condition reduces the cross-entropy loss to level epsilon on a logarithmic time scale. In contrast, when layerwise weight decay is present, the induced regularization on the end-to-end map can be expressed as a Schatten-type penalty; under a sharp late-time Kurdyka-Lojasiewicz tail, this structural energy closes on a polynomial time scale. The two clocks, therefore, separate fitting from representation simplification. We then explain how the same mechanism can appear in ReLU MLPs. In regions where the activation patterns on the training set remain fixed, the network reduces to a linear model in the active coordinates. In a two-layer ReLU embedding model, chain-rule estimates further show that the classifier head can receive larger effective gradients than the embedding block under controlled downstream norms. This supports a two-stage mechanism in which the classifier fits first, while the representation continues to simplify later. We use modular addition as the main experimental setting. The deep linear theory provides the rigorous core of the analysis. But the ReLU results are formulated as conditional reductions that account for empirical behavior without claiming a global proof for nonlinear training dynamics.