Singular Curvature in ReLU Training:Differentiation and the Gradient-Flow Limit Need Not Commute

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.

Constrained Variable Projection for Structured Problems

Variable projection is a classical technique for separable nonlinear least-squares problems, in which variables that enter linearly are eliminated exactly, yielding a reduced nonlinear problem. By expressing this framework as a particular instance of a broader class of bilevel optimization problems, we develop a constrained variable-projection framework for data-science models, where the remaining variables are subject to convex constraints and the eliminated variables arise from a lower-level least-squares problem. In particular, by interpreting variable projection as a collapsed bilevel optimization problem, we derive exact reduced-gradient formulas compatible with automatic differentiation and propose a conditional-gradient algorithm for the resulting constrained reduced problem. We establish convergence guarantees under standard smoothness and compactness assumptions, and discuss extensions to structured lower-level variables. Numerical experiments on sparse autoencoding, dictionary learning, blind deconvolution, and few-shot learning suggest that the method can improve wall-clock efficiency and data efficiency relative to natural joint-optimization baselines.