TITLE: Why is the optimizing algorithm called "momentum"?
QUESTION [5 upvotes]: The Gradient Descent with Momentum method for optimizing parameters is widely used. It has many variations including famous Adam algorithm. Now I do understand them, but where does the word "momentum" come from? I don't see any relation to the physical momentum: $p=mv$ or statistical momentum of order-$k$: $M_k = E(X^k)$.

REPLY [6 votes]: It is related to the physical momentum. In classic gradient descent the update $\Delta x$ is calculated as $-\alpha \nabla f(x)$ with learning rate $\alpha$ and hence, an update step reads $$ x' = x + \Delta x = x -\alpha \nabla f(x)$$
However, with momentum you remember the update of the last iteration $\Delta x$ and the update consists of a linear combination of the current gradient and the last update, i.e. $\Delta x' = \mu \Delta x - \alpha \nabla f(x')$:
$$ x'' = x' + \Delta x' = x' + \mu \Delta x - \alpha \nabla f(x')$$
The relation to physical momentum comes in by interpreting x as a heavy particle traveling through space and its new direction is a combination of old and new forces acting on it. So, it is more inert to sudden changes in the direction reducing oscillation. In other words, it's travel direction is more robust to sudden changes in the loss landscape. However, it is accelerating faster if the gradients point in the same direction, e.g. "going down hill".