  It is well-known that stochastic gradient noise (SGN) acts as implicit
regularization for deep learning and is essentially important for both
optimization and generalization of deep networks. Some works attempted to
artificially simulate SGN by injecting random noise to improve deep learning.
However, it turned out that the injected simple random noise cannot work as
well as SGN, which is anisotropic and parameter-dependent. For simulating SGN
at low computational costs and without changing the learning rate or batch
size, we propose the Positive-Negative Momentum (PNM) approach that is a
powerful alternative to conventional Momentum in classic optimizers. The
introduced PNM method maintains two approximate independent momentum terms.
Then, we can control the magnitude of SGN explicitly by adjusting the momentum
difference. We theoretically prove the convergence guarantee and the
generalization advantage of PNM over Stochastic Gradient Descent (SGD). By
incorporating PNM into the two conventional optimizers, SGD with Momentum and
Adam, our extensive experiments empirically verified the significant advantage
of the PNM-based variants over the corresponding conventional Momentum-based
optimizers. Code: \url{https://github.com/zeke-xie/Positive-Negative-Momentum}.
