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The study on the implicit regularization induced by gradient-basedoptimization is a longstanding pursuit. In the present paper, we characterizethe implicit regularization of momentum gradient descent (MGD) with earlystopping by comparing with the explicit $\ell_2$-regularization (ridge). Indetails, we study MGD in the continuous-time view, so-called momentum gradientflow (MGF), and show that its tendency is closer to ridge than the gradientdescent (GD) [Ali et al., 2019] for least squares regression. Moreover, weprove that, under the calibration $t=\sqrt{2/\lambda}$, where $t$ is the timeparameter in MGF and $\lambda$ is the tuning parameter in ridge regression, therisk of MGF is no more than 1.54 times that of ridge. In particular, therelative Bayes risk of MGF to ridge is between 1 and 1.035 under the optimaltuning. The numerical experiments support our theoretical results strongly.

The Implicit Regularization of Momentum Gradient Descent with Early Stopping