How models actually learn, from vanilla gradient descent to Adam
Gradient descent behaves very differently on a round loss surface and a stretched one. Conditioning measures that stretch. Poor conditioning makes the optimizer zig-zag: one direction is steep, another is flat.
For a quadratic loss, conditioning is controlled by the eigenvalues of the Hessian. The condition number κ is the ratio of largest to smallest curvature.
In a pinball machine with tight side bumpers and a long narrow exit lane, a hard hit sends the ball ricocheting side to side while it only slowly moves forward. Poor conditioning does the same to gradient descent: it bounces in the steep direction and crawls in the flat one. The figure below is exactly that machine. Slide κ to stretch the bowl, run the descent, and watch the path ricochet across the narrow direction while inching along the long one. (Leave β at 0 for now; it stars in the Momentum lesson.)