A Probabilistic Representation of Deep Learning for Improving The Information Theoretic Interpretability
This work addresses interpretability challenges in deep learning for researchers and practitioners, but it appears incremental as it builds on existing information-theoretic frameworks.
The paper tackles the problem of improving information-theoretic interpretability in MultiLayer Perceptrons (MLPs) by proposing a probabilistic representation, demonstrating that existing mutual information estimators are invalid for MLPs and providing new explanations for information flow and generalization, with validation on synthetic and benchmark datasets.
In this paper, we propose a probabilistic representation of MultiLayer Perceptrons (MLPs) to improve the information-theoretic interpretability. Above all, we demonstrate that the activations being i.i.d. is not valid for all the hidden layers of MLPs, thus the existing mutual information estimators based on non-parametric inference methods, e.g., empirical distributions and Kernel Density Estimate (KDE), are invalid for measuring the information flow in MLPs. Moreover, we introduce explicit probabilistic explanations for MLPs: (i) we define the probability space (Omega_F, t, P_F) for a fully connected layer f and demonstrate the great effect of an activation function on the probability measure P_F ; (ii) we prove the entire architecture of MLPs as a Gibbs distribution P; and (iii) the back-propagation aims to optimize the sample space Omega_F of all the fully connected layers of MLPs for learning an optimal Gibbs distribution P* to express the statistical connection between the input and the label. Based on the probabilistic explanations for MLPs, we improve the information-theoretic interpretability of MLPs in three aspects: (i) the random variable of f is discrete and the corresponding entropy is finite; (ii) the information bottleneck theory cannot correctly explain the information flow in MLPs if we take into account the back-propagation; and (iii) we propose novel information-theoretic explanations for the generalization of MLPs. Finally, we demonstrate the proposed probabilistic representation and information-theoretic explanations for MLPs in a synthetic dataset and benchmark datasets.