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I have come across several articles online stating that mathematicians don't really understand how machine learning works. One example I can come up with right now is this. There are several others too but can't remember enough to mention. Some other article mentioned that they are like black boxes producing elegant solutions without providing any clear insight as to how the solution was arrived at. Could someone explain it in a bit more detail?

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    $\begingroup$ I think you are over-interpreting the article somewhat. There is a good book on this sort of thing by Anthony and Bartlett, "Neural Networks Learning: Theoretical Foundations" (Cambridge University Press), which gives a good overview of the analysis of this sort of model 15 years ago. I suspect most of the analysis these days is on things like kernel models, which generally work better and are more mathematically tractable. A lot of work is done on this (Computational Learning Theory), and to say that we don't really understand seems a bit unfair. $\endgroup$ – Dikran Marsupial Dec 4 '15 at 7:41
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    $\begingroup$ Something worth reading: The two cultures and Useful things to know about ML which is explained on this blog in a lighter version. $\endgroup$ – gwr Dec 4 '15 at 7:44
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    $\begingroup$ Backprop is just gradient descent optimization of a training criterion, there is nothing particularly mysterious about it. The fact that it is hard to understand how a particular neural network represents the "knowledge" that it encodes is a downside, if you need to be able to explain how it encodes it, but in general it isn't usually very relevant for non-parametric models. @gwr, cheers, the Brieman article looks interesting, the other two links however seem to lead to Brieman's article again. $\endgroup$ – Dikran Marsupial Dec 4 '15 at 8:22
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    $\begingroup$ @DikranMarsupial Thanks, it was not meant as some wierd kind of reinforcing loop but is an error in my using an iPad. So here is Useful things to know about ML and the before mentioned lighter version. $\endgroup$ – gwr Dec 4 '15 at 9:04
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    $\begingroup$ @gwr cheers, I'd agree with most of the section headings, so it looks like a useful resource! $\endgroup$ – Dikran Marsupial Dec 4 '15 at 9:14
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E. g. neural networks function exactly like black boxes. We know general grounds on which they work and how to train them. But we don't know what features does neural network compute on hidden layers. Well, we can guess and test our guesses. But there is no guarantee that we well succeed in our guessing or that discovered features will describe some known characteristics of the phenomenon we are using neural network for.

It is near to impossible to convert trained neural network to a bunch of if-conditions. And if you perform such conversion, then god help you with figuring out meaningful names for features neural network learned in it's hidden layers.

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