Deep-layer limit and stability analysis of the basic forward-backward-splitting induced network (II): learning problems

Backpropagation is the computational engine of deep learning, yet its mathematical structure is typically treated as a procedural traversal of computational graphs. We present a global operator theory of the \emph{F-adjoint} framework, which reformulates the layerwise backward recursion of an $L$-depth feedforward network into a single linear system $(I-\cB)\Xs=\bG$, where $\bG$ is a source vector. We prove that the global backward operator $\cB$ is strictly block upper-triangular and nilpotent of index at most $L$. This nilpotency guarantees the exact termination of the Neumann series solution after at most $L$ terms, revealing classical backpropagation to be mathematically equivalent to block back-substitution on an upper bidiagonal system. We formalise \emph{F-symmetry} -- the condition in which the backward pass perfectly mirrors the forward pass -- identifying orthogonal weight matrices as canonical examples. Through worked numerical examples, we demonstrate how this operator perspective exposes the single-path collapse of strictly feedforward networks and its breakdown in residual architectures. Finally, we leverage this compositional structure to rigorously derive the mechanics of residual networks (gradient highways) and transfer learning (gradient truncation). This framework elevates backpropagation from an algorithmic recipe to a global nilpotent-operator formulation.

We study the dynamics of gradient descent in the Edge of Stability regime, where the learning rate is large enough to induce persistent oscillations in the loss and the sharpness. We propose a continuous-time effective model that tracks the evolution of the average trajectory coupled with the time-averaged covariance of its fast oscillations. Our analysis reveals that the natural quantity to monitor in such unstable regimes is an effective free energy, which combines the original risk functional with a curvature-related "entropic" term. Our model allows us to track the envelope of the oscillations even in situations where its dynamics evolve on similar timescales as the averaged weights. Otherwise stated, we can track the spikes that occur during the training of some neural network architectures. For wide two-layer neural networks optimized under stable non-vanishing oscillations, we derive a mean-field limit that results in a novel kinetic equation describing the joint distribution of weights and their fluctuations. We show that this equation can be interpreted as a Wasserstein-2 gradient flow of a macroscopic free energy. Finally, we provide numerical evidence on matrix factorization and deep learning tasks (CIFAR-10) to demonstrate the model's accuracy in capturing the envelope of the oscillations and the predictive power of the effective free energy.

The Edge of Stability (EoS) phenomenon, where gradient descent operates with sharpness exceeding the classical convergence threshold yet the loss decreases over long timescales, is ubiquitous in modern deep learning but remains poorly understood in realistic settings. Prior rigorous analyses have been largely confined to scalar or low-dimensional losses with specific structural forms. In this work, we develop a bifurcation theory framework for gradient descent on the edge of stability that applies directly to overparameterized neural networks. By decomposing the training dynamics into components normal and tangent to the manifold of minimizers, we show that stable EoS training arises from a flip bifurcation in the normal direction, governed by the sign of the first Lyapunov coefficient, while the tangent dynamics drift toward regions of decreasing sharpness. Under mild spectral and geometric assumptions on the loss landscape, we prove convergence to the minimizing manifold when training at the EoS threshold. As a corollary, we recover and unify prior results: we show that the product-stability condition of Gan (2026) is an instance of our framework.