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Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

The dynamics of Deep Linear Networks (DLNs) is dramatically affected by the variance $σ^2$ of the parameters at initialization $θ_0$. For DLNs of width $w$, we show a phase transition w.r.t. the scaling $γ$ of the variance $σ^2=w^{-γ}$ as $w\to\infty$: for large variance ($γ<1$), $θ_0$ is very close to a global minimum but far from any saddle point, and for small variance ($γ>1$), $θ_0$ is close to a saddle point and far from any global minimum. While the first case corresponds to the well-studied NTK regime, the second case is less understood. This motivates the study of the case $γ\to +\infty$, where we conjecture a Saddle-to-Saddle dynamics: throughout training, gradient descent visits the neighborhoods of a sequence of saddles, each corresponding to linear maps of increasing rank, until reaching a sparse global minimum. We support this conjecture with a theorem for the dynamics between the first two saddles, as well as some numerical experiments.