Momentum

Authors: Thomas Lee, Greta Gasswint, Elizabeth Henning (SYSEN5800 Fall 2021)

Introduction

Momentum is an extension to the gradient descent optimization algorithm that builds inertia in a search direction to overcome local minima and oscillation of noisy gradients (1). It is based on the same concept of momentum in physics. A classic example is a ball rolling down a hill that gathers enough momentum to overcome a plateau region and make it to a global minima (2). Momentum adds history to the parameter updates which significantly accelerates the optimization process. Momentum controls the amount of history to include in the update equation via a hyperparameter (1). This hyperparameter is a value ranging from 0 to 1. A momentum of 0 is equivalent to gradient descent without momentum (1). A higher momentum value means more gradients from the past (history) are considered (2).

(1) https://machinelearningmastery.com/gradient-descent-with-momentum-from-scratch/

(2) https://towardsdatascience.com/gradient-descent-with-momentum-59420f626c8f

Theory, methodology, and/or algorithmic discussions

Definition

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Algorithm

The main idea behind momentum is to compute an exponential moving average of the gradients and use that to update the weights.

In gradient descent (stochastic) without momentum, the update rule at each iteration is given by:

W = W - αdW

Where:

• W denotes the parameters to the cost function
• dW is the gradient indicating which direction to decrease the cost function by
• α is the learning rate

In gradient descent (stochastic) with momentum, the update rule at each iteration is given by:

VdW = βVdW + (1-β)dW

W = W - αVdW

Where:

• β is a new hyperparameter that denotes the momentum constant

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Graphical Explanation

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Numerical Example

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