A ball sits on a curvy loss landscape with a shallow local dip on the left and a clearly lower global minimum on the right. The derivative is only the local slope under the ball, drawn as a short amber tangent line, with a teal arrow under the axis showing the pending step of minus learning-rate times slope. Pressing step moves the ball along the curve by that amount; run repeats automatically until the slope is nearly zero, 400 steps pass, or the ball flies out of the valley. A logarithmic slider sets the stride from about 0.01 to 4: tiny strides crawl, moderate strides converge to the nearest bottom, strides around 1 oscillate back and forth across the bowl, and the largest strides diverge. Readouts show w, the loss, the signed slope, and the next step; a sparkline tracks the loss over steps and fading ghost dots trace the ball's recent path.

Downhill is a direction

The ball cannot see the valley — it only feels the slope under its feet (the amber tangent). Each step moves it against that slope: the teal arrow is −lr·L′(w). Set the stride, then step — or run.

The slope is measured only under your feet — no map of the valley — yet following “down” step after step finds a bottom. The stride (learning rate) decides everything: too small crawls, right converges, too big oscillates across the bowl, way too big flies out of the valley entirely. And sometimes you settle in the nearest dip rather than the lowest — which is why where you start and how you stride matter. Every loss curve in the other books is this picture in millions of dimensions.