Leveraging Continuous Time to Understand Momentum When Training Diagonal
  Linear Networks
Hristo Papazov Scott Pesme Nicolas Flammarion
In this work, we investigate the effect of momentum on the optimisation
trajectory of gradient descent. We leverage a continuous-time approach in the
analysis of momentum gradient descent with step size $\gamma$ and momentum
parameter $\beta$ that allows us to identify an intrinsic quantity $\lambda =
\frac{ \gamma }{ (1 - \beta)^2 }$ which uniquely defines the optimisation path
and provides a simple acceleration rule. When training a $2$-layer diagonal
linear network in an overparametrised regression setting, we characterise the
recovered solution through an implicit regularisation problem. We then prove
that small values of $\lambda$ help to recover sparse solutions. Finally, we
give similar but weaker results for stochastic momentum gradient descent. We
provide numerical experiments which support our claims.