Implicit bias of SGD in $L_2$-regularized linear DNNs: One-way jumps from high to low rank

The $L_{2}$-regularized loss of Deep Linear Networks (DLNs) withmore than one hidden layers has multiple local minima, correspondingto matrices with different ranks. In tasks such as matrix completion,the goal is to converge to the local minimum with the smallest rankthat still fits the training data. While rank-underestimating minimacan be avoided since they do not fit the data, GD might getstuck at rank-overestimating minima. We show that with SGD, there is always a probability to jumpfrom a higher rank minimum to a lower rank one, but the probabilityof jumping back is zero. More precisely, we define a sequence of sets$B_{1}\subset B_{2}\subset\cdots\subset B_{R}$ so that $B_{r}$contains all minima of rank $r$ or less (and not more) that are absorbingfor small enough ridge parameters $\lambda$ and learning rates $\eta$:SGD has prob. 0 of leaving $B_{r}$, and from any starting point thereis a non-zero prob. for SGD to go in $B_{r}$.