The difficulty of computing stable and accurate neural networks | Hacker News Reader

The difficulty of computing stable and accurate neural networks

pnas.org

59 pointsprogramd24 comments

Huh. I can read and understand the abstract and the introduction, but I can't judge the work after a first pass. This is the kind of paper that cannot be easily skimmed, because it consists almost entirely of densely packed pages chock-full of highly abstract mathematical reasoning.

Not surprisingly, the authors are mathematicians. They claim to prove that

* there are well-conditioned problems for which suitable DNNs exist, but no training algorithm can find arbitrarily good approximations of those suitable DNNs;

* it's possible to find approximations of those suitable DNNs only if we sacrifice digits of accuracy -- i.e., the approximations cannot be arbitrarily good; and

* there is a class of DNNs they propose, which they call "fast iterative restated networks" or FIRENETs, that solve undetermined systems of linear equations over the complex numbers, with a good blend of stability (robustness to adversarial samples) and accuracy (within the claimed theoretical limits).

Finally, the authors provide open-source code (A+ for doing that, but... Matlab!!??): https://www.github.com/Comp-Foundations-and-Barriers-of-AI/f...

Does anyone else here understand the work better than me? I would love an informal explanation that appeals to intuition.

This is a follow-up to an earlier paper on instabilities in NN-based solutions to inverse problems: https://www.pnas.org/doi/10.1073/pnas.1907377117

Your statements appear to be a good summary of the paper.

> Your statements appear to be a good summary of the paper.

Thanks. I found the authors' claims relatively easy to grok. What I'd like to understand, intuitively, is how they got there!

Ah, sorry, I can't help too much there.

I skimmed through the proofs in the SI. Many of the tools they use are taken from the compressed sensing literature (robust null space property, sqrt LASSO paths, sample complexity for CS-MRI style problems...).

I can point you to additional reading, if you'd like.

Thank you. I think I can find introductory materials on my own. This is now on my list of things about which I'd like to learn/relearn more. It's been a long time since I've had any superficial exposure to the topic; my recollection of it is vague and in all likelihood outdated. My memory is a bit "situational" -- I tend to remember best the things I've been thinking about and working on, but as time passes everything else sort of becomes blurrier, less precise in my mind.

What’s wrong with Matlab? It works very well for this, as it’s very stable for numerical modeling work.

It's closed-source; it lacks the kind of tooling/infrastructure you need for working with larger, state-of-the-art DNNs; and very little AI research is done with it, limiting the audience that can can use the code. Python -- or even Julia -- would have been a better choice.

The story has changed somewhat recently, but up until ~2020 most ML for Inverse Problems research was done in Matlab, especially for MRI and CT reconstruction. Thinking of FBPNet and ADMM-Net.

I just recently stumbled upon a practical example of the first item in your list: try approximating sin(FREQ_SCALE * input) using a neural network and gradient descent for various FREQ_SCALE and ranges of input. You will find it to be completely powerless even though you can easily construct a single parameter "NN" with sin activation, that reproduces the function exactly when the parameter is initialized with FREQ_SCALE.

The problem extends to any periodic functions. I am working on a blog post about it.

Interesting work. A DNN is after all something that compresses the input data and hopefully generalizes well over the domain. This 'machine' itself will adhere to principles of Kolmogorov complexity etc..

Traditionally, empirical solutions in the NN literature to address instability are regularization and dropout.

Also, adding layers seems to improve things. The famous example is XOR which cannot be trained by a single layer NN.

How do the theoretical limitations in the paper relate to these? if at all...

PNAS is not exactly the venue of first choice for publishing AI research, so heuristically speaking, this article is likely not worth the reading time. Furthermore neither the abstract nor the first few words of the introduction give me any reason to read on. I'm going to assume this is irrelevant, until someone here can convince me it's not.

They're showing that there are classes of problems that neural networks cannot be trained to consistently approximate, regardless of the amount of data provided. It sounds like you're not interested in the limitations of neural networks, so this paper isn't really relevant to you.

You might as well say it's not worth interviewing anyone who didn't graduate from an Ivy League school, "heuristically speaking".

Innovation can come from anywhere. Either we can take time to evaluate something on its merits (if you have the competency and resources), or leave it be (if you don't), or we accept to be told what is true by institutions with their own interests, who will use their position to further leverage them. And in this path, knowledge is no longer free.

I will read further, but this paper is attacking a real issue with neural nets.

Still, the choice of journal is pretty weird. Occam's razor: If you have any results that feel significant and not just to bump your publication count (the title would imply former) you'd run with it to present on NIPS, ICLR, ICML or submit to JMLR.

It's fair point that it's hard to spend time on something if it doesn't carry enough promise. I've seen enough crackpots from good universities that don't get called out enough. They still exist on the likes of NeurIPS/ICML though.

However, to me this appears to be serious work (whether or not it's remarkable). Crackpots often obscure their message, but here they state their premise quite clearly, and skimming through the arguments they seem sane and approachable.

> You might as well say it's not worth interviewing anyone who didn't graduate from an Ivy League school, "heuristically speaking".

I know this isn't the point you were going for, but this _is_ a heuristic that is deployed in some companies/institutions (at least the UK variant certainly is).

> I know this isn't the point you were going for, but this _is_ a heuristic that is deployed in some companies/institutions (at least the UK variant certainly is).

Which companies/institutions are you thinking of?

Innovation can come from anywhere, but the authors of this paper chose to submit to a venue that is outside the focus of ML research. From what I understand, this paper would've been a great fit for COLT (or even a general/broad ML conference like ICML) and since it wasn't submitted (or wasn't accepted), i think it makes sense to assume that it was written by authors who either chose to circumvent the community, or were rejected by it. I don't find your comparison to ivy league fitting, because while there are still a significant number of great candidates from non ivy leagues, the number of great ML papers that have not been published at top tier ML venues is in the single digits -- most researchers I know would rather just upload their paper to arxiv and move on, rather than publishing in 2nd tier venues. Which should tell you how much of a bad signal the wrong venue can be in ML research.

Granted, it could also be that this was authored by people outside of the field who.sont know the rules, but given how much research gets published each day, it's impossible not to rely on heuristics, especially since this paper was.posted without context.

If signaling is what gets you off, it might help that the authors are based at Cambridge

You would take the opinion of a random stranger on HN over the article getting past peer reviewed and published in PNAS.. Why that heuristic?

OP greatly overestimates the knowledge of the average HN user!

I'm a researcher in the field, and the amount of new research coming out each day is just way too big to handle. Which means I have to rely on heuristics like this to judge what is worth my time. Otherwise I'd be spending 25 hours/day reading papers, and waste brain capacity digesting worthless findings. Not everything that passes peer review is read-worthy, far from it. It just means there aren't any gaping holes in it. Don't get me wrong, the heuristic is far from perfect. But a random paper on HN (which is not known for its abundance of researchers) that was posted without any context explaining why I should care is not a good signal.

If you're a researcher in the field is it not a fundamental skill to be able to make these judgement calls yourself? And if wasted time is a concern to you, what makes you think others would be willing to put in the time to try and convince a random academic that they should or shouldn't read a paper where they've not even specified what they'd want to get out of it?

I am a researcher in this field and PNAS is fine medium to communicate. This paper actually first appeared in arxiv which is where everything seems to go first. Having it go from arvix to PNAS is a good indicator that it is a good paper, and it is! It would be a shame to dismiss it.

Hinton's paper in 2006 on reducing dimensionality with NNs appeared in Science and nobody paid attention to that either at the time.

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