Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks
This provides theoretical insights into the implicit regularization and convergence behavior of SGD for deep neural networks, addressing a fundamental problem in machine learning optimization.
The paper proves that stochastic gradient descent (SGD) performs variational inference by minimizing an average potential over the posterior distribution of weights with entropic regularization, but not for the original loss function, and shows that SGD trajectories for deep networks do not converge classically, instead forming closed loops due to highly non-isotropic gradient noise with covariance rank as low as 1% of its dimension.
Stochastic gradient descent (SGD) is widely believed to perform implicit regularization when used to train deep neural networks, but the precise manner in which this occurs has thus far been elusive. We prove that SGD minimizes an average potential over the posterior distribution of weights along with an entropic regularization term. This potential is however not the original loss function in general. So SGD does perform variational inference, but for a different loss than the one used to compute the gradients. Even more surprisingly, SGD does not even converge in the classical sense: we show that the most likely trajectories of SGD for deep networks do not behave like Brownian motion around critical points. Instead, they resemble closed loops with deterministic components. We prove that such "out-of-equilibrium" behavior is a consequence of highly non-isotropic gradient noise in SGD; the covariance matrix of mini-batch gradients for deep networks has a rank as small as 1% of its dimension. We provide extensive empirical validation of these claims, proven in the appendix.