A Probabilistic Representation of Deep Learning

TL;DR

This paper introduces a probabilistic framework for deep learning, interpreting neural networks as Gibbs distributions and Bayesian models, providing new insights into hierarchy and generalization.

Contribution

It presents a novel probabilistic representation of DNNs, linking neurons to Gibbs energy, and offers a Bayesian perspective on hierarchy and regularization in deep learning.

Findings

Neurons define the energy of a Gibbs distribution.

Hidden layers formulate Gibbs distributions.

DNNs can be interpreted as Bayesian neural networks.

Abstract

In this work, we introduce a novel probabilistic representation of deep learning, which provides an explicit explanation for the Deep Neural Networks (DNNs) in three aspects: (i) neurons define the energy of a Gibbs distribution; (ii) the hidden layers of DNNs formulate Gibbs distributions; and (iii) the whole architecture of DNNs can be interpreted as a Bayesian neural network. Based on the proposed probabilistic representation, we investigate two fundamental properties of deep learning: hierarchy and generalization. First, we explicitly formulate the hierarchy property from the Bayesian perspective, namely that some hidden layers formulate a prior distribution and the remaining layers formulate a likelihood distribution. Second, we demonstrate that DNNs have an explicit regularization by learning a prior distribution and the learning algorithm is one reason for decreasing the generalization ability of DNNs. Moreover, we clarify two empirical phenomena of DNNs that cannot be explained by traditional theories of generalization. Simulation results validate the proposed probabilistic representation and the insights into these properties of deep learning based on a synthetic dataset.