Deep Neural Nets as Hamiltonians
This work provides theoretical insights into the structure of neural network landscapes, which is an incremental contribution to deep learning theory for researchers in machine learning and statistical physics.
The paper tackles the problem of analyzing the energy landscape of randomly initialized multi-layer perceptrons (MLPs) viewed as Hamiltonians over inputs, using the replica trick to compute entropy and saddle point equations for overlaps in the infinite width limit. It finds diverse behaviors, such as full replica symmetry breaking for sin activations and replica symmetry for shallow tanh/ReLU or deep shaped networks.
Neural networks are complex functions of both their inputs and parameters. Much prior work in deep learning theory analyzes the distribution of network outputs at a fixed a set of inputs (e.g. a training dataset) over random initializations of the network parameters. The purpose of this article is to consider the opposite situation: we view a randomly initialized Multi-Layer Perceptron (MLP) as a Hamiltonian over its inputs. For typical realizations of the network parameters, we study the properties of the energy landscape induced by this Hamiltonian, focusing on the structure of near-global minimum in the limit of infinite width. Specifically, we use the replica trick to perform an exact analytic calculation giving the entropy (log volume of space) at a given energy. We further derive saddle point equations that describe the overlaps between inputs sampled iid from the Gibbs distribution induced by the random MLP. For linear activations we solve these saddle point equations exactly. But we also solve them numerically for a variety of depths and activation functions, including $\tanh, \sin, \text{ReLU}$, and shaped non-linearities. We find even at infinite width a rich range of behaviors. For some non-linearities, such as $\sin$, for instance, we find that the landscapes of random MLPs exhibit full replica symmetry breaking, while shallow $\tanh$ and ReLU networks or deep shaped MLPs are instead replica symmetric.