Neural Networks as Spin Models: From Glass to Hidden Order Through Training

We explore a one-to-one correspondence between a neural network (NN) and a statistical mechanical spin model where neurons are mapped to Ising spins and weights to spin-spin couplings. The process of training an NN produces a family of spin Hamiltonians parameterized by training time. We study the magnetic phases and the melting transition temperature as training progresses. First, we prove analytically that the common initial state before training--an NN with independent random weights--maps to a layered version of the classical Sherrington-Kirkpatrick spin glass exhibiting a replica symmetry breaking. The spin-glass-to-paramagnet transition temperature is calculated. Further, we use the Thouless-Anderson-Palmer (TAP) equations--a theoretical technique to analyze the landscape of energy minima of random systems--to determine the evolution of the magnetic phases on two types of NNs (one with continuous and one with binarized activations) trained on the MNIST dataset. The two NN types give rise to similar results, showing a quick destruction of the spin glass and the appearance of a phase with a hidden order, whose melting transition temperature $T_c$ grows as a power law in training time. We also discuss the properties of the spectrum of the spin system's bond matrix in the context of rich vs. lazy learning. We suggest that this statistical mechanical view of NNs provides a useful unifying perspective on the training process, which can be viewed as selecting and strengthening a symmetry-broken state associated with the training task.