Leveraging Continuous Time to Understand Momentum When Training Diagonal Linear Networks

• URL: http://arxiv.org/abs/2403.05293v1
• Date: Fri, 8 Mar 2024 13:21:07 GMT
• Title: Leveraging Continuous Time to Understand Momentum When Training Diagonal Linear Networks
• Authors: Hristo Papazov, Scott Pesme, Nicolas Flammarion
• Abstract summary: We use a continuous-time approach in the analysis of momentum gradient descent with step size $gamma$ and momentum parameter $beta$. We prove that small values of $lambda$ help to recover sparse solutions.
• Score: 21.176224458126285
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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