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We interpret the variational inference of the Stochastic Gradient Descent(SGD) as minimizing a new potential function named the\textit{quasi-potential}. We analytically construct the quasi-potentialfunction in the case when the loss function is convex and admits only oneglobal minimum point. We show in this case that the quasi-potential function isrelated to the noise covariance structure of SGD via a partial differentialequation of Hamilton-Jacobi type. This relation helps us to show thatanisotropic noise leads to faster escape than isotropic noise. We then considerthe dynamics of SGD in the case when the loss function is non-convex and admitsseveral different local minima. In this case, we demonstrate an example thatshows how the noise covariance structure plays a role in "implicitregularization", a phenomenon in which SGD favors some particular local minimumpoints. This is done through the relation between the noise covariancestructure and the quasi-potential function. Our analysis is based on LargeDeviations Theory (LDT), and they are validated by numerical experiments.

Quasi-potential as an implicit regularizer for the loss function in the stochastic gradient descent