Fable
The gear in the clock tower that rings the wrong bell when checked backward
The city-center clock tower announced the time every day; if the bell was wrong, the whole city's market opening fell into chaos. The clockmaker initially relied on the last big bell, familiar routes, and rarely erred when the days were small. The clockmaker understood that recording only the final few slowdowns on the clock face couldn't explain which gear the error started from; to improve, the responsibility for the result had to be traced back through every layer.
Later, more and more new gears were installed, and a moment too late or too early could propagate from deep inside to the clock face. As the number of gears increased, an early deviation could be amplified through multiple layers of transmission; adjusting only the end pointer could both hide the root cause and damage parts that were originally fine.
The first repair was rushed: just tighten the big bell, or replace the time-reporting boy. Tightening only the bell face could briefly mask the error, but did not change the parameters of the internal gears. Yet new problems followed, everyone stared only at the most visible place, and small deviations in the distance accumulated ever deeper.
He started from the loss and calculated backward along the transmission chain to determine each gear's contribution to the error, then fine-tuned parameters by gradient; each layer received a signal usable for learning.
He traced back from the moment of the false ring, measuring how much each gear had over-pushed and under-meshed, then made fine adjustments according to severity. This step was troublesome and at first slowed the schedule; but after several rounds, rework decreased, and the master no longer changed the rules on a whim. The apprentice learned to distribute the error back to each layer of gears, so a complex mechanism could gradually become more accurate through local adjustments.
The apprentice finally understood that training is not just about looking at what was answered wrong at the end, but about passing the error signal back through the layers of parameters that produced it.
The apprentice understood for the first time that the wrong sound at the end must be traced back along the transmission path and shared out. From then on, when repairing complex mechanisms, one cannot just curse the place that made the last sound.