Does Compression Imply Generalization? A Minimum Description Length Perspective

Title: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.

Machine Learning (cs.LG)

arXiv:2610.21376 [cs.LG]

(or arXiv:2610.21376v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.21376 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.