A Theory of Machine Understanding via the Minimum Description Length Principle

Deep neural networks trained through end-to-end learning have achieved remarkable success across various domains in the past decade. However, the end-to-end learning strategy, originally designed to minimize predictive loss in a black-box manner, faces two fundamental limitations: the struggle to form explainable representations in a self-supervised manner, and the inability to compress information rigorously following the Minimum Description Length (MDL) principle. These two limitations point to a deeper issue: an end-to-end learning model is not able to "understand" what it learns. In this paper, we establish a novel theory connecting these two limitations. We design the Spectrum VAE, a novel deep learning architecture whose minimum description length (MDL) can be rigorously evaluated. Then, we introduce the concept of latent dimension combinations, or what we term spiking patterns, and demonstrate that the observed spiking patterns should be as few as possible based on the training data in order for the Spectrum VAE to achieve the MDL. Finally, our theory demonstrates that when the MDL is achieved with respect to the given data distribution, the Spectrum VAE will naturally produce explainable latent representations of the data. In other words, explainable representations--or "understanding"--can emerge in a self-supervised manner simply by making the deep network obey the MDL principle. In our opinion, this also implies a deeper insight: To understand is to compress. At its core, our theory advocates for a shift in the training objective of deep networks: not only to minimize predictive loss, but also to minimize the description length regarding the given data. That is, a deep network should not only learn, but also understand what it learns. This work is entirely theoretical and aims to inspire future research toward self-supervised, explainable AI grounded in the MDL principle.