SGD Through the Lens of Kolmogorov Complexity

TL;DR

This paper proves that stochastic gradient descent (SGD) can achieve near-perfect classification accuracy under mild assumptions, including low model complexity and local progress, providing the first convergence guarantee for general, underparameterized models.

Contribution

It introduces a novel convergence analysis of SGD based on Kolmogorov complexity, applicable to a wide range of models without specific architectural constraints.

Findings

SGD achieves (1-ε) accuracy under mild assumptions.

First convergence guarantee for underparameterized models.

Model-agnostic analysis using entropy compression.

Abstract

We prove that stochastic gradient descent (SGD) finds a solution that achieves $(1-\epsilon)$ classification accuracy on the entire dataset. We do so under two main assumptions: (1. Local progress) The model accuracy improves on average over batches. (2. Models compute simple functions) The function computed by the model is simple (has low Kolmogorov complexity). It is sufficient that these assumptions hold only for a tiny fraction of the epochs. Intuitively, the above means that intermittent local progress of SGD implies global progress. Assumption 2 trivially holds for underparameterized models, hence, our work gives the first convergence guarantee for general, underparameterized models. Furthermore, this is the first result which is completely model agnostic - we do not require the model to have any specific architecture or activation function, it may not even be a neural network. Our analysis makes use of the entropy compression method, which was first introduced by Moser and Tardos in the context of the Lov\'asz local lemma.