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Public-Notes/markov_mgf_bounds.pdf at main · sampower88/Public-Notes

and another note, on a rather different topic, revisiting some basic concentration bounds for 'good' Markov chains. "Exponential Inequalities for Reversible Markov chains with Spectral Gap" github.com/sampower88/P...

github.com

miscellaneous maths notes, suitable for public consumption - sampower88/Public-Notes

1102

The basic idea: write down the training dynamics of a simple NN, summarise them with a two-dimensional system of differential inequalities, solve these inequalities, and deduce things as you would like. All pretty elementary, more-or-less teachable, somewhat extensible to other non-convex problems.

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I finally got around to completing a note that's been on the side for a month or two now: "Optimisation of Wide, Shallow Neural Networks by Gradient Flow" github.com/sampower88/P...

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[ ... ] involves something in the realm of 'galactic algorithm' which doesn't appear to be particularly insightful. So, nominally interesting in either case, but still pretty far from demanding attention (esp. with the readability levels!).

040

Anyways, FWIW: the release treated two problems which I'd consider to be of interest to me. One of them is quite KLS-like, and the write-up was not particularly worth reading (but a better proof could credibly come along one day). Another is related to the complexity of sampling, and [ ... ]

160

no value judgment, but this got a laugh from me github.com/openai/math/...

0122

Yes, though it's somehow not an open-and-shut case; I really hold out hope that a 'teachable' proof might be out there, and perhaps not too far away. So it doesn't mean that I'll stop thinking about it!

100

More-or-less trivia, but it's always fun to remind that the methods used to understand KLS and the theoretical methods used to prove things about Diffusion Models have substantial overlap!

000

One hopes that the problem will still inspire people in this direction, since there are some aspects of KLS which suggest that the 'final proof' (as an ideal object) could end up being very short indeed. Naturally, there's a bit of optimism in this opinion, but I do hope that it's true!

This is less in the sense of "it's obvious", but more in the sense that the conjecture says "for this nice class of measures, there is essentially no obstruction to 'good geometry' other than 'the obvious obstruction' of anisotropy", and finding ways to justify this required some ingenuity.

100

The very cool thing about KLS is that the attempts to make progress on it were successful in introducing interesting new constructions (stochastic localisation, centering the heat flow perspective, etc.) in service of proving results which are somehow "what one would expect".

100

Again having a strange feeling about the nominal confirmation that the KLS conjecture is true, though at least I feel more positive than I did around Navier-Stokes. It's a funny one, in as much as from a "theory for algorithms" perspective, the conjecture has already been "as true as necessary".

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Sam Power @spmontecarlo.bsky.social · 05/10/2026

written up in part because discussions on this topic were fairly ubiquitous over the summer (and probably continue to be!)

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(short, but not as short as the screenshot; approx. 1.5 pages)

020

Sam Power @spmontecarlo.bsky.social · 05/10/2026

A short essay: github.com/sampower88/P...

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Sam Power @spmontecarlo.bsky.social · 04/10/2026

And the roses aren't half bad either!

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Sam Power @spmontecarlo.bsky.social · 04/10/2026

A day out at Anglesey Abbey, home of some of the best dahlias around.

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Still true in the ways that matter!

0444

What could they have meant by this?

080

I think Segovia! These are from a day trip which included both Toledo and Segovia, so there's a chance that there's some Toledo as well, but I think that it was largely Segovia.

020

Great town for shape enjoyers:

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Some months late: shapes in Spain

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Part of my 'revelation' here was that in identifying a list of "often discussed, rarely resolved" topics in my area, my instinctive response to several of them was (approximately) "these are a distraction". It was a strange feeling.

030

The positive spin is that there is indeed something there to be found; the complementary spin being that this phenomenon supplies an optimism which might be hard to shake in the face of a genuine 'dead end'. It's a judgment call at some stage, one way or another.

(and of course, an additional wrinkle is that for some ideas which seem reasonably unlikely to be viable a priori, 'machine learning wizardry' occasionally shows them to be capable of seemingly miraculous things)

It does seem to be an essentially difficult thing to identify when and communicate that "this idea is very pretty mathematics, but is not likely to hold water as a practical algorithmic strategy", particularly writing negatively about a method without strong theoretical (e.g.) backing is difficult.

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Sam Power @spmontecarlo.bsky.social · 30/09/2026

A thought which has become clearer to me recently is that many 'open' / 'unsolved' problems in my area (such as it is) essentially concern things which are nice to talk about, but very difficult to "actually do anything" with. (There is a case that this also makes them not great problems)

1100

nice to see it framed so clearly

1454

There's even the analogy of gardening, which over time, I have increasingly realised is largely what I'm looking for in a video game (less the plants, more the calmness and relative reliability / honesty of incremental progress).

030

One analogy which came to mind was playing through e.g. a role-playing game: there's not necessarily 'novelty' in { levelling up my Red Mage, catching Kyogre, paying off my loan to Tom Nook, ... }, but you've still achieved something on your own terms, and it's natural that this is satisfying.

Another aspect of this process which is reasonably positive is that it typically leads to a firmer mental picture of how things fit together in the research landscape, whether microscopically (old paper in my area) or macroscopically (old paper in some quite-different area). This is still exciting.

150

I think that it's a peculiar experience in part because it's often quite a positive one. Usually, you're getting some positive feedback that you at least had the right idea, that you asked the right question, and so on; you've still succeeded 'at maths', if not at contributing in full novelty.

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Sam Power @spmontecarlo.bsky.social · 26/09/2026

One of my favourite tweets of yesteryear is the attached. I think that in (maths) research we don't talk so often about the (pretty ubiquitous) experience of discovering something in one's one time, and then coming to the realisation that it's not as new as one might have hoped.

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The ladder signals to me a level of ambition that we're not quite at yet (though I guess we've chatted about painting at some stage)!

Sam Power @spmontecarlo.bsky.social · 26/09/2026

I guess it's about 3 months now, and we had a lot of help from parents. The TV unit only arrived yesterday.

Sam Power @spmontecarlo.bsky.social · 26/09/2026

Saturday Morning Breakfast Simons

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yep! I thought it was nice that the perspective is able to clarify what the optimal choice of score is, also.

010

Pilea-repotting day is always a joy. From 2 pots at the start of the day, to 7 at the end!

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I don't know so much about other fields. I think people's perceptions of arXiv are a lot more heterogeneous than one might expect. But yeah, it seems to be in flux to some extent; we don't really talk about the difference so much, though I feel it does exist.

000

It's fascinating to wonder about what will happen to the systems of journal publication and conventional peer review as they exist presently, and (if some perspectives are to be taken seriously) which new structures and priorities might arise out of their possible capitulation.

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