SGD Through the Lens of Kolmogorov Complexity
This offers a theoretical foundation for SGD convergence in machine learning, applicable broadly but incremental in extending existing analysis.
The paper proves that stochastic gradient descent (SGD) achieves near-perfect classification accuracy under assumptions of local progress and low Kolmogorov complexity, providing the first convergence guarantee for general underparameterized models without architectural constraints.
We prove that stochastic gradient descent (SGD) finds a solution that achieves $(1-ε)$ 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 Lovász local lemma.