Neural networks (NNs) do not output all functions with equal probability, but seem to be biased towards functions of certain types; heuristically, towards 'simple' functions. In VPCL18, MSVP+19, MVPSL20 evidence is given that functions output by NNs are inclined to have low information-theoretic complexity - nice summaries are given on...
Summary Deep neural networks (DNNs) are generally used with large numbers of parameters relative to the number of given data-points, so that the solutions they output are far from uniquely determined. How do DNNs 'choose' what solution to output? Some fairly recent papers ([1], [2]) seem to suggest that DNNs...
This post is motivated by a recent post of Stuart Armstrong on going from preferences to a utility function. It was originally planned as a comment, but seems to have developed a bit of a life of its own. The ideas here came up in a discussion with Owen Biesel;...
Hmm, so I'm very wary of defending tropical geometry when I know so little about it; if anyone more informed is reading please jump in! But until then, I'll have a go.
tropical geometry might be relevant ML, for the simple reason that the functions coming up in ML with ReLU activation are PL
I'm not sure I agree with this argument.
Hmm, even for a very small value of `might'? I'm not saying that someone who wants to contribute to ML needs to seriously consider learning some tropical geometry, just that if one already knows tropical geometry it's not a crazy idea to poke around a bit and see if there are applications.
... (read more)The use
Hi Dmitry,
To me it seems not unreasonable to think that some ideas from tropical geometry might be relevant ML, for the simple reason that the functions coming up in ML with ReLU activation are PL, and people in tropical geometry have thought seriously about PL functions. Of course this does not guarantee that there is anything useful to be said!
One possible example that comes to mind in the context of your post here is the concept of polyhedral currents. As I understand it, here the notion of "density of polygons' is used as a kind of proxy for the derivative of a PL function? But I think the theory of polyhedral currents... (read more)
Get that agreement in writing.
I'm not sure that would be particularly reassuring to me (writing as one of the contributors). First, how would one check that the agreement had been adhered to (maybe it's possible, I don't know)? Second, people in my experience often don't notice they are training on data (as mentioned in a post above by ozziegooen).
I think this is a key point. Even the best possible curriculum, if it has to work for all students at the same rate, is not going to work well. What I really want (both for my past-self as a student, and my present self as a teacher of university mathematics) is to be able to tailor the learning rate to individual students and individual topics (for student me, this would have meant 'go very fast for geometry and rather slowly for combinatorics'). And while we're at it, can we also customise the learning styles (some students like to read, some like to sit in class, some to work in groups, etc)?
This is technologically more feasible than it was a decade ago, but seems far from common.
Thanks Charlie.
Just to be double-sure, the second process was choosing the weight in a ball (so total L2 norm of weights was <= 1), rather than on a sphere (total norm == 1), right?
Yes, exactly (though for some constant , which may not be , but turn out not to matter).
Is initializing weights that way actually a thing people do?
Not sure (I would like to know). But what I had in mind was initialising a network with small weights, then doing a random walk ('undirected SGD'), and then looking at the resulting distribution. Of course this will be more complicated than the distributions I use above, but I think the shape... (read more)
Neural networks (NNs) do not output all functions with equal probability, but seem to be biased towards functions of certain types; heuristically, towards 'simple' functions. In VPCL18, MSVP+19, MVPSL20 evidence is given that functions output by NNs are inclined to have low information-theoretic complexity - nice summaries are given on lesswrong here and here and elsewhere by the author. However, the converse is not true; some functions with low information-theoretic complexity (such as simple periodic functions) are not so readily output by NNs - this is discussed extensively in the comments to the above posts. Understanding this kind of problem better is widely considered relevant for AI alignment, see e.g. here.
To try... (read 638 more words →)
Maths at my Dutch university also has homework for quite a few of the courses, which often counts for something like 10-20% of final grade. It can usually be submitted online, so you only need to be physically present for exams. However, there are a small number of courses that are exceptions to this, and actually require attendance to some extent (e.g. a course on how to give a scientific presentation, where a large part of the course consists of students giving and commenting on each other's presentations - not so easy to replace the learning experience with a single exam at the end).
But this differs between Dutch universities.
I suspect the arXiv might not be keen on an account that posts papers by a range of people (not including the account-owner as coauthor). This might lead to heavier moderation/whatever. But I could be very wrong!
Some advice for getting papers accepted on arxiv
As some other comments have pointed out, there is a certain amount of moderation on arXiv. This is a little opaque, so below is an attempt to summarise some things that are likely to make it easier to get your paper accepted. I'm sure the list is very incomplete!
In writing this I don't want to give the impression that posting things to arXiv is hard; I have currently 28 papers there, have never had a single problem or delay with moderation, and the submission process generally takes me <15 mins these days.
Endorsement. When you first attempt to submit a paper you may need to be
The arXiv really prefers that you upload in tex. For the author this makes it less likely that your paper will be flagged for moderation etc (I guess). So if it were possible to export to Rex I think that for the purposes of uploading to arXiv this would be substantially better. Of course, I don’t know how much more/less work it is…
Deep neural networks (DNNs) are generally used with large numbers of parameters relative to the number of given data-points, so that the solutions they output are far from uniquely determined. How do DNNs 'choose' what solution to output? Some fairly recent papers ([1], [2]) seem to suggest that DNNs are biased towards outputting solutions which are 'simple', in the sense of Kolmogorov complexity (minimum message length complexity) ([3]). It seems to me that this might be relevant for questions around AI alignment. I've not thought this idea through in detail; rather, I'm writing this post to ask whether people are aware of work in this direction (or see obvious reasons why it... (read 537 more words →)
This post is motivated by a recent post of Stuart Armstrong on going from preferences to a utility function. It was originally planned as a comment, but seems to have developed a bit of a life of its own. The ideas here came up in a discussion with Owen Biesel; all mistakes in this exposition are mine. I'm not very good with the typesetting engine here, so apologies for latex and other problems.
The basic ideas is as follows. Suppose we have a set of objects, and we are given some information on which objects are preferred to which other objects. Then we are interested in whether and in how many ways... (read 1233 more words →)
I love the idea of this! But it worries me a bit that when I look through the ones under "mathematics" the ordering seems pretty erratic. I'm a professional mathematician and managing editor for a good mathematics journal, so this should be the field I know best, and my doubts here make me question the rest.
It's awkward for me to criticise the ranking of specific papers publicly, but to give one example the paper "Progress in the mirror symmetry program?: a criterion for the rationality of cubic fourfolds" seems vastly under-rated on the "big if true" axis relative to many other works (I think the p(generalises) for that paper is fair).
On the other hand, mathematics has a reputation for being hard for outsiders to evaluate; I'm curious of what people think of the rankings in other fields?