Understanding the unstable convergence of gradient descent
This work addresses a fundamental issue in optimization for machine learning practitioners, providing insights into why gradient descent often converges despite violating standard assumptions.
The paper investigates the unstable convergence of gradient descent when step sizes exceed the theoretical limit of 2/L for L-smooth costs, a common scenario in machine learning, and identifies key causes and characteristics through theory and experiments.
Most existing analyses of (stochastic) gradient descent rely on the condition that for $L$-smooth costs, the step size is less than $2/L$. However, many works have observed that in machine learning applications step sizes often do not fulfill this condition, yet (stochastic) gradient descent still converges, albeit in an unstable manner. We investigate this unstable convergence phenomenon from first principles, and discuss key causes behind it. We also identify its main characteristics, and how they interrelate based on both theory and experiments, offering a principled view toward understanding the phenomenon.