Continuous Time Analysis of Momentum Methods
This work provides theoretical insights into momentum optimization for neural networks, but it is incremental as it builds on existing methods in a deterministic setting without new empirical results.
The paper tackled the problem of understanding the role of fixed momentum methods like Heavy Ball and Nesterov's accelerated gradients in neural network training by deriving continuous-time approximations of discrete algorithms. The result included proving three approximations, such as showing momentum methods approximate a time-rescaled gradient descent flow asymptotically and that they admit an exponentially attractive invariant manifold reducing dynamics to a modified gradient flow.
Gradient descent-based optimization methods underpin the parameter training of neural networks, and hence comprise a significant component in the impressive test results found in a number of applications. Introducing stochasticity is key to their success in practical problems, and there is some understanding of the role of stochastic gradient descent in this context. Momentum modifications of gradient descent such as Polyak's Heavy Ball method (HB) and Nesterov's method of accelerated gradients (NAG), are also widely adopted. In this work our focus is on understanding the role of momentum in the training of neural networks, concentrating on the common situation in which the momentum contribution is fixed at each step of the algorithm. To expose the ideas simply we work in the deterministic setting. Our approach is to derive continuous time approximations of the discrete algorithms; these continuous time approximations provide insights into the mechanisms at play within the discrete algorithms. We prove three such approximations. Firstly we show that standard implementations of fixed momentum methods approximate a time-rescaled gradient descent flow, asymptotically as the learning rate shrinks to zero; this result does not distinguish momentum methods from pure gradient descent, in the limit of vanishing learning rate. We then proceed to prove two results aimed at understanding the observed practical advantages of fixed momentum methods over gradient descent. We achieve this by proving approximations to continuous time limits in which the small but fixed learning rate appears as a parameter. Furthermore in a third result we show that the momentum methods admit an exponentially attractive invariant manifold on which the dynamics reduces, approximately, to a gradient flow with respect to a modified loss function.