2020-01-17: Gradient descent with momentum --- to accelerate or to super-accelerate? https://arxiv.org/abs/2001.06472v1In this Section, we provided details and derivations related to the message presented in the Introduction (Figure ??): that super-accelerating momentum-based gradient descent is beneficial for minimization in the one-dimensional parabolic case

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We consider gradient descent with `momentum', a widely used method for loss
function minimization in machine learning. This method is often used with
`Nesterov acceleration', meaning that the gradient is evaluated not at the
current position in parameter space, but at the estimated position after one
step. In this work, we show that the algorithm can be improved by extending
this `acceleration' --- by using the gradient at an estimated position several
steps ahead rather than just one step ahead. How far one looks ahead in this
`super-acceleration' algorithm is determined by a new hyperparameter.
Considering a one-parameter quadratic loss function, the optimal value of the
super-acceleration can be exactly calculated and analytically estimated. We
show explicitly that super-accelerating the momentum algorithm is beneficial,
not only for this idealized problem, but also for several synthetic loss
landscapes and for the MNIST classification task with neural networks.
Super-acceleration is also easy to incorporate into adaptive algorithms like
RMSProp or Adam, and is shown to improve these algorithms.