  Recent empirical advances show that training deep models with large learning
rate often improves generalization performance. However, theoretical
justifications on the benefits of large learning rate are highly limited, due
to challenges in analysis. In this paper, we consider using Gradient Descent
(GD) with a large learning rate on a homogeneous matrix factorization problem,
i.e., $\min_{X, Y} \|A - XY^\top\|_{\sf F}^2$. We prove a convergence theory
for constant large learning rates well beyond $2/L$, where $L$ is the largest
eigenvalue of Hessian at the initialization. Moreover, we rigorously establish
an implicit bias of GD induced by such a large learning rate, termed
'balancing', meaning that magnitudes of $X$ and $Y$ at the limit of GD
iterations will be close even if their initialization is significantly
unbalanced. Numerical experiments are provided to support our theory.
