Saddle-to-Saddle Dynamics in Diagonal Linear Networks
This provides theoretical insights into incremental learning dynamics in neural networks, though it is incremental as it focuses on a specific model.
The paper fully describes the trajectory of gradient flow in diagonal linear networks with vanishing initialization, showing it jumps between saddles to reach the minimum ℓ₁-norm solution, with explicit characterization of saddles and jumping times via a recursive algorithm.
In this paper we fully describe the trajectory of gradient flow over diagonal linear networks in the limit of vanishing initialisation. We show that the limiting flow successively jumps from a saddle of the training loss to another until reaching the minimum $\ell_1$-norm solution. This saddle-to-saddle dynamics translates to an incremental learning process as each saddle corresponds to the minimiser of the loss constrained to an active set outside of which the coordinates must be zero. We explicitly characterise the visited saddles as well as the jumping times through a recursive algorithm reminiscent of the LARS algorithm used for computing the Lasso path. Our proof leverages a convenient arc-length time-reparametrisation which enables to keep track of the heteroclinic transitions between the jumps. Our analysis requires negligible assumptions on the data, applies to both under and overparametrised settings and covers complex cases where there is no monotonicity of the number of active coordinates. We provide numerical experiments to support our findings.