Does Compression Imply Generalization? A Minimum Description Length Perspective

Abstract: A central question in modern learning theory is what notion of compression, if any, explains why over-parameterized neural networks generalize. Existing answers measure complexity at different levels, ranging from weights to representations; however, estimating the associated quantities either requires held-out data in principle or relies on the challenging (ultra) high-dimensional mutual information estimation. To overcome these challenges, we approach the question through the minimum description length (MDL) principle, and specifically, we propose the prequential regret as a complementary signal that is computable from the training trajectory alone. We establish that, in the (nearly) interpolating regime, the expected prequential regret upper-bounds the description length of the trained parameter under the algorithm-induced distribution, and that this description length translates into a high-probability generalization bound in the PAC-MDL setting. Along the way, we show that the encoder-complexity correction recently added to the algorithm-level information bottleneck (IB) is upper-bounded by an MDL parameter description length, \new{bringing together} the two views of compression from MDL and IB. Empirically, the prequential regret tracks the generalization error on CIFAR-10 and CIFAR-100, with Pearson correlations exceeding 0.9.