Entropy and mutual information in models of deep neural networks
This work addresses a theoretical challenge for researchers in machine learning and statistical physics, but it is incremental as it builds on existing methods and does not provide new practical insights.
The paper tackles the problem of computing information-theoretic quantities like entropy and mutual information in deep neural networks, using tractable models and statistical physics methods, but finds that the relationship between compression and generalization remains unclear in their proposed setting.
We examine a class of deep learning models with a tractable method to compute information-theoretic quantities. Our contributions are three-fold: (i) We show how entropies and mutual informations can be derived from heuristic statistical physics methods, under the assumption that weight matrices are independent and orthogonally-invariant. (ii) We extend particular cases in which this result is known to be rigorously exact by providing a proof for two-layers networks with Gaussian random weights, using the recently introduced adaptive interpolation method. (iii) We propose an experiment framework with generative models of synthetic datasets, on which we train deep neural networks with a weight constraint designed so that the assumption in (i) is verified during learning. We study the behavior of entropies and mutual informations throughout learning and conclude that, in the proposed setting, the relationship between compression and generalization remains elusive.