Computing the Information Content of Trained Neural Networks
This addresses the fundamental question of how neural networks generalize despite having more weights than data, with implications for ML theory and practitioners.
The paper tackles the problem of quantifying the information stored in neural network weights to understand generalization, deriving a consistent estimator and closed-form upper bound for infinitely wide networks that are non-vacuous and correlate with empirical generalization at finite width.
How much information does a learning algorithm extract from the training data and store in a neural network's weights? Too much, and the network would overfit to the training data. Too little, and the network would not fit to anything at all. Naïvely, the amount of information the network stores should scale in proportion to the number of trainable weights. This raises the question: how can neural networks with vastly more weights than training data still generalise? A simple resolution to this conundrum is that the number of weights is usually a bad proxy for the actual amount of information stored. For instance, typical weight vectors may be highly compressible. Then another question occurs: is it possible to compute the actual amount of information stored? This paper derives both a consistent estimator and a closed-form upper bound on the information content of infinitely wide neural networks. The derivation is based on an identification between neural information content and the negative log probability of a Gaussian orthant. This identification yields bounds that analytically control the generalisation behaviour of the entire solution space of infinitely wide networks. The bounds have a simple dependence on both the network architecture and the training data. Corroborating the findings of Valle-Pérez et al. (2019), who conducted a similar analysis using approximate Gaussian integration techniques, the bounds are found to be both non-vacuous and correlated with the empirical generalisation behaviour at finite width.