Here are the slides of a talk Kaarel gave, presenting work with Dmitry establishing that (even arbitrarily overparametrized) neural net bayesian learning has a circuit prior — and thus, when learning a function which is implemented by some small circuit, only requires a small amount of training data to get good test accuracy — for certain scalings of the prior and with various other important caveats. The slides offer a self-contained presentation of the simplest version of the result. See the end of the presentation (slides 36–37) for a bunch of open problems in NN learning theory.
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