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We manage to train neural nets to approximate complicated data sets via rather simple process: back propagation.

It is indeed a marvel that it works nearly as well as it does.

But then again, evolution is even dumber (in the sense that it only makes random choices that thrive or perish, and can't even take gradients into account), but evolution has still managed to produce intelligent critters.

I guess when you have enough dimensions greedy approaches to optimisation / hill climbing can work well enough, even when you have challenging problems?

Especially if you are allowed to move to some meta levels. Eg evolution doesn't build planes, it built brains that can figure out how to build planes. Similarly with back propagation perhaps.

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