Implicit bias of SGD in $L_{2}$-regularized linear DNNs: One-way jumps from high to low rank
This addresses the challenge of implicit bias in optimization for deep learning, offering theoretical insights into rank selection, though it is incremental as it builds on existing work on minima in linear networks.
The paper tackles the problem of SGD getting stuck at rank-overestimating minima in L2-regularized deep linear networks for tasks like matrix completion, showing that SGD can probabilistically jump from higher to lower rank minima but not back, enabling convergence to lower-rank solutions.
The $L_{2}$-regularized loss of Deep Linear Networks (DLNs) with more than one hidden layers has multiple local minima, corresponding to matrices with different ranks. In tasks such as matrix completion, the goal is to converge to the local minimum with the smallest rank that still fits the training data. While rank-underestimating minima can be avoided since they do not fit the data, GD might get stuck at rank-overestimating minima. We show that with SGD, there is always a probability to jump from a higher rank minimum to a lower rank one, but the probability of 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 absorbing for small enough ridge parameters $λ$ and learning rates $η$: SGD has prob. 0 of leaving $B_{r}$, and from any starting point there is a non-zero prob. for SGD to go in $B_{r}$.