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 $\le T$ for some constant $T$, which may not be $1$, 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 may depend quite a bit on the details of the SGD. For example, I suspect that the result of something like adaptive gradient descent may tend towards more spherical distributions, but I haven't thought about this carefully.

If training large neural networks only moves the parameters a small distance (citation needed), do you still think there's something interesting to say about the effect of training in this lens of looking at the density of nonlinearities?

I hope so! I would want to understand what norm the movements are 'small' in (L2, L$\infty$, ...).

LayerNorm looks interesting, I'll take a look.

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.

1. Endorsement. When you first attempt to submit a paper you may need to be endorsed. JanBrauner kindly offered below to help people with endorsements; I might also be able to do the same, but I've never posted in the CS part of arXiv, so not sure how effective this will be. However, even better to avoid need for moderation. To this end, use an academic email address if you have one; this is quite likely to already be enough. Also, see below on subject classes (endorsement requirements depend on which subject class(es) you want to post in).

2. Choosing subject classes. Each paper gets one or more subject classes, like CS.AI; see [https://arxiv.org/category_taxonomy] for a list. Some subject classes attract more junk than others, and the ones that attract more junk are more heavily moderated. In mathematics, it is math.GM (General Mathematics) that attracts most junk, hence is most heavily moderated. I guess most people here are looking at CS.AI, I don't know what this is like. But one easy thing is to minimise cross-listing (adding additional subject classes for your paper), as then you are moderated by all of them.

3. Write in (la)tex, submit the tex file. You don't have to do this, but it is standard and preferred by the arXiv, and I suspect makes it less likely your paper gets flagged for moderation. It is also an easy way to make sure your paper looks like a serious academic paper.

4. It is possible to submit papers on behalf of third parties. I've never done this, and I suspect such papers will be more heavily moderated.

5. If you have multiple authors, it doesn't really matter who submits. After the submission is posted you are sent a 'paper password' allowing coauthors to 'claim' the paper; it is then associated to their arXiv account, orcid etc (orcid is optional, but a really good idea, and free).

Finally, a request: please be nice to the moderators! They are generally unpaid volunteers doing a valuable service to the community (e.g. making sure I don't have to read nonsense proofs of the Riemann hypothesis every morning). Of course it doesn't feel good if your paper gets held up, but please try not to take it personally.

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…

Hi Charlie, If you can give a short (precise) description for an agent that does the task, then you have written a short programme that solves the task. I think then if you need more space to ‘explain what the agent would do’ then you are saying there also exists a less efficient/compact way to specify the solution. From this perspective I think the latter is then not so relevant. David

P.s. the main thing I have taken so far from the link you posted is that the important part is not exactly about the biases of SGD. Rather, it is about the structure of the DNN itself; the algorithm used to find a (local) optimum plays less of a role than the overall structure. But probably I’m reading too much into your precise phrasing.

Hi Thomas, I agree the proof of the bound is not so interesting. What I found more interesting were the examples and discussion suggesting that, in practise, the upper bound seems often to be somewhat tight.

Concerning differential advancement, I agree this can advance capabilities, but I suspect that advancing alignment is somewhat hopeless unless we can understand better what is going on inside DNNs. On that basis I think it does differentials advance alignment, but of course other people may disagree.

Thanks very much for the link!