On the Computational Power of Online Gradient Descent

TL;DR

This paper demonstrates that online gradient descent can simulate complex computations, making it computationally hard to analyze its behavior precisely, which has implications for understanding its limitations.

Contribution

It shows that online gradient descent can encode arbitrary polynomial-space computations, revealing its high computational complexity.

Findings

Online gradient descent can simulate polynomial-space computations.

It is computationally hard to analyze the detailed behavior of online gradient descent.

Implications for the limits of reasoning about online learning algorithms.

Abstract

We prove that the evolution of weight vectors in online gradient descent can encode arbitrary polynomial-space computations, even in very simple learning settings. Our results imply that, under weak complexity-theoretic assumptions, it is impossible to reason efficiently about the fine-grained behavior of online gradient descent.