From complex to simple : hierarchical free-energy landscape renormalized in deep neural networks

We develop a statistical mechanical approach based on the replica method to study the design space of deep and wide neural networks constrained to meet a large number of training data. Specifically, we analyze the configuration space of the synaptic weights and neurons in the hidden layers in a simple feed-forward perceptron network for two scenarios: a setting with random inputs/outputs and a teacher-student setting. By increasing the strength of constraints,~i.e. increasing the number of training data, successive 2nd order glass transition (random inputs/outputs) or 2nd order crystalline transition (teacher-student setting) take place layer-by-layer starting next to the inputs/outputs boundaries going deeper into the bulk with the thickness of the solid phase growing logarithmically with the data size. This implies the typical storage capacity of the network grows exponentially fast with the depth. In a deep enough network, the central part remains in the liquid phase. We argue that in systems of finite width N, the weak bias field can remain in the center and plays the role of a symmetry-breaking field that connects the opposite sides of the system. The successive glass transitions bring about a hierarchical free-energy landscape with ultrametricity, which evolves in space: it is most complex close to the boundaries but becomes renormalized into progressively simpler ones in deeper layers. These observations provide clues to understand why deep neural networks operate efficiently. Finally, we present some numerical simulations of learning which reveal spatially heterogeneous glassy dynamics truncated by a finite width $N$ effect.