(Also: I don’t Feel the Idea of “minimal effective parameterisation†truly captures what’s taking place listed here, but for The explanation you point out you.)
It is a little bit of a tangent, however the Levin sure is in fact about Kolmogorov complexity. It truly is a reasonably easy theorem; the evidence is constructive, and basically displays that a specified perform file which corresponds to numerous parameters in the parameter-House can not be also complicated, by setting up a straightforward program which computes f. Quite about, if the parameter-Area is finite and discrete, then we could build a Huffman code for the perform space (exactly where the distribution around the operate-Place may be the distribution that corresponds on the uniform distribution around the parameter-House).
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Due to the fact high probability functions tend to be more most likely, Therefore the greater probably features correspond to shorter plans. Considering that shorter systems are always extra probably inside the prior that simulates all achievable plans, they need to be predicted to get greater plans, and so generalize effectively.
Also: considering that we’re estimating several probabilities by sampling, we in essence must discretise the functionality House. Should you have any concrete solutions for ways to get around this then we’re all ears!
I'm not sure I concur using this type of -- I do think Kolmogorov complexity is usually a relevant Idea of complexity On this context.
I don’t Feel This really is in conflict Along with the declare from the paper, which is that _most_ on the inductive bias originates from the architecture and initialization.
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In essence, lower-complexity capabilities correspond to greater volumes from the parameter-Place of neural networks. If capabilities with large volumes even have massive basins of attraction, and when employing SGD is about such as taking place a random basin (weighted by its dimensions), then This may basically clarify why neural networks perform.
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The paper talks a bunch about such things as SGD remaining (almost) Bayesian and also the neural network prior obtaining very low Kolmogorov complexity; I discovered these to generally be interruptions from the main stage. Over and above that, approximating the random sampling chance that has a Gaussian system is a reasonably sensitive affair and I have concerns with regard to the applicability to authentic neural networks.
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(Even though you are referring to the overparameterized case, the place this argument isn't vacuous in addition to applies largely to neural nets and not other ML types, I do not discover this argument incredibly compelling a priori, while I concur that depending on empirical proof it is probably generally right.)