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[D] Simplest mathematical example of a function that can only be solved by gradient descent (alien.top)

submitted 1 year ago by EatMeMonster@alien.top to c/machinelearning@academy.garden

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I'm trying to teach a lesson on gradient descent from a more statistical and theoretical perspective, and need a good example to show its usefulness.

What is the simplest possible algebraic function that would be impossible or rather difficult to optimize for, by setting its 1st derivative to 0, but easily doable with gradient descent? I preferably want to demonstrate this in context linear regression or some extremely simple machine learning model.

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[–] lithiumdeuteride@alien.top 1 points 1 year ago (5 children)

How about this function:

f(x) = abs(x) - cos(x)

Setting the first derivative equal to zero leads to trouble, as the derivative doesn't exist at the minimum, and there are infinitely many points which have zero slope. Nevertheless, the function has a clear minimum which gradient descent of any finite step size should find.

[–] yannbouteiller@alien.top 1 points 1 year ago

It does answer OP's question, but is of limited practical relevance for an ML course IMHO.

We typically use GD in approximately pseudoconvex optimization landscapes, not because there are infinitely many or even any single saddle point. To escape local optima and saddle points, we rely on other tricks like SGD.

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