  The momentum acceleration technique is widely adopted in many optimization
algorithms. However, there is no theoretical answer on how the momentum affects
the generalization performance of the optimization algorithms. This paper
studies this problem by analyzing the implicit regularization of momentum-based
optimization. We prove that on the linear classification problem with separable
data and exponential-tailed loss, gradient descent with momentum (GDM)
converges to the L2 max-margin solution, which is the same as vanilla gradient
descent. That means gradient descent with momentum acceleration still converges
to a low-complexity model, which guarantees their generalization. We then
analyze the stochastic and adaptive variants of GDM (i.e., SGDM and
deterministic Adam) and show they also converge to the L2 max-margin solution.
Technically, to overcome the difficulty of the error accumulation in analyzing
the momentum, we construct new potential functions to analyze the gap between
the model parameter and the max-margin solution. Numerical experiments are
conducted and support our theoretical results.
