Recent studies have proposed interpreting the training process from an ergodic perspective. Building on this foundation, we present a unified framework for understanding and accelerating the training of deep neural networks via stochastic gradient descent (SGD). By analyzing the geometric landscape of the objective function we introduce a practical diagnostic, the running estimate of the largest Lyapunov exponent, which provably distinguishes genuine convergence toward stable minimizers from mere statistical stabilization near saddle points. We then propose a ghost category extension for standard classifiers that adds auxiliary ghost output nodes so the model gains extra descent directions that open a lateral corridor around narrow loss barriers and enable the optimizer to bypass poor basins during the early training phase. We show that this extension strictly reduces the approximation error and that after sufficient convergence the ghost dimensions collapse so that the extended model coincides with the original one and there exists a path in the enlarged parameter space along which the total loss does not increase. Taken together, these results provide a principled architecture level intervention that accelerates early stage trainability while preserving asymptotic behavior and simultaneously serves as an architecture-friendly regularizer.

We consider dynamical and geometrical aspects of deep learning. For many standard choices of layer maps we display semi-invariant metrics which quantify differences between data or decision functions. This allows us, when considering random layer maps and using non-commutative ergodic theorems, to deduce that certain limits exist when letting the number of layers tend to infinity. We also examine the random initialization of standard networks where we observe a surprising cut-off phenomenon in terms of the number of layers, the depth of the network. This could be a relevant parameter when choosing an appropriate number of layers for a given learning task, or for selecting a good initialization procedure. More generally, we hope that the notions and results in this paper can provide a framework, in particular a geometric one, for a part of the theoretical understanding of deep neural networks.