Momentum on an Ill-Conditioned Loss

A loss with a ravine

Averaging the past gradients

What momentum does and does not fix

Gradient descent on $f(\mathbf{x}) = 0.1x_1^2 + 2x_2^2$ zigzags, bouncing across the steep direction while advancing slowly along the flat one. Which statement describes what momentum changes?

With a momentum coefficient $\beta = 0.9$, a previous velocity of $0$ and a current gradient of $2$, the new velocity $v_t = \beta v_{t-1} + \nabla \mathcal{L}$ equals ____.