Gradient descent

In Mist Mountain there was a deep valley, and legend said there was a clear spring at the bottom. When the guide led the apprentices down, the fog was so thick they could not see ten paces ahead, and they could only feel the terrain with their feet. The first apprentice was impatient and picked a direction at random, charging ahead. At first he did descend quickly, but before long he hit a rock wall; turning back, he overshot the low point, and after much back-and-forth it was dark.

The second apprentice was afraid of mistakes and moved only half an inch at each step. The direction was steady, but by afternoon he was still halfway up the mountain. The master said that steps too small can also use up a lifetime on the road.

The third apprentice, wherever he went, first probed the wooden staff around him, found the direction in which the ground sloped most downward, and then walked a short distance. After walking, he probed again, not rushing on forever because of the previous direction.

Sometimes the ground was like a shallow bowl, and after a few steps you had to change direction; sometimes the slope was steep, and your steps had to shorten; sometimes you entered a gentle area where descent slowed, yet you could still keep approaching the sound of the spring through tiny inclines.

In the evening, they finally reached the bottom of the valley. The apprentice realized that the master had not seen a map of the whole mountain, but had repeatedly used the slope at the current position: walk in the direction that lowers elevation, with a step not so large that it overshoots the low point, and not so small that it stalls.

One apprentice recorded the slopes felt through the wooden staff on his sleeve and discovered that the same mountain was not the same everywhere. Steep slopes required holding back, gentle slopes allowed more freedom, and loose rocks meant testing again; with only initial courage, one would neither find the spring nor protect one's legs.

In the evening, they passed a small hollow. The sound of water was close, but it was not the legendary clear spring. The impatient apprentice wanted to stop and celebrate, but the master told him to scout the surroundings again: if there was still a lower path nearby, the small low point in front of them could not be treated as the endpoint. Finding the valley floor depends on continuous probing, and also on knowing when it is already hard to go any lower. Later the master had them draw the slope direction and step length of every step. The winding lines on the map showed exactly that going downhill is not one big gamble, but continuous correction through local signals when the whole picture is unclear.

The master planted his staff by the spring and said that going down the mountain rarely relies on guessing the endpoint correctly once; it relies on correcting each step based on the error right in front of you. When the mountain is unclear, steady small steps are more reliable than a reckless dash.