Dan Roy

Over 400 academics in 30+ countries signed our open letter calling for immediate changes in AI development and deployment practices to ensure that cybersecurity incidents do not continue to occur, as future incidents will be far more damaging to our critical infrastructure.

A counterexample to Vaught’s conjecture for infinitary logic has been verified in Lean! In joint work with Nate Ackerman and Robin Knight, we developed and formalized a construction based on Knight’s unpublished manuscript.

NON COMPETES BANNED Today, the PM has announced that his Government will legislate to functionally ban non-compete agreements for startups and scaleups in the UK or as he called it - the Bosman ruling for ruling for the innovation sector' 😂 At the @StartupCltn been working on this for months and we think it’s a huge potential unlock for our tech ecosystem. Here’s what you need to know… - Measures to functionally ban non-competes for startups and scaleups in the UK are going to be set out. - There will also be consultation on a wider set of restrictive practices including gardening leave. - More detail on these plans at the budget in a few weeks (Wednesday 28th October) including a timetable for consultation with a view to legislation in the coming months. We think this is a huge move and so do lots of friends including so many of the great founders that signed the letter with Inherent just the other week. Why? Because talent matters and mobile talent means startups can build. The Valley was built on free-flowing talent. In the UK, we’re now seeing our own versions of this story with companies being built from alumni of both homegrown firms like Revolut and beyond. But we need that to move faster. Hiring and building faster means British startups and scaleups can grow faster. Barriers to that slow our ecosystem down. We won’t pretend it will be perfect, and people (including some of our friends but also lots of fairly vested interests too) will have a right to disagree. But we think that startups will thrive and win in a faster paced Britain. So we are delighted with the progress today - as part of a strong speech by the PM outlining his ambition for building the world’s leading innovation nation including more on tax, growing companies outside London and more. But this is the start, not the end of the process. The announcement launches a consultation. It doesn't mean the issue is settled. How far reform goes, and what it looks like on the ground, is still to be decided. The Prime Minister today has done something that can materially impact the trajectory of the UK tech ecosystem and with it our economy - but only if we make sure it’s delivered. So, the real work starts now.

We’re releasing a broad range of new mathematical results produced by an internal frontier model. We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study, and we have drawn on their advice and public recommendations to inform how we release these results. t.co/7N6TPlft1P

Since my previous post got some attention, let me use the opportunity for something more positive: to explain why Antoine Maillard’s work on ellipsoid fitting matters so much to me personally. Cargèse, 2023. A conference I had organised with @zdeborova A blackboard, and d²/4. That is where I first learned about this work from Antoine. I still have these photos from the lecture! You can see the threshold on the board. What you cannot see is everything those ideas would help set in motion. Looking back, a surprising amount of what I now understand about neural networks, the relation between spectra and generalization, feature learning, emergence, power-law scaling, traces back to this line of work. Because its importance was never just the number 1/4. The important thing was the IDEA. The statistical-physics analysis of Antoine Maillard and Dmitriy Kunisky predicted the threshold and the geometry of the solutions. The work of Afonso Bandeira and Antoine Maillard made a crucial part of the programme rigorous, establishing the sharp threshold for approximate fitting through Gaussian equivalence. t.co/F9CLHANv6G t.co/V15hMsu3kv For me, the powerful lesson was not simply “here is the answer to this problem.” It was “here is a way to attack other problems.” And we did. The next step in our own story came in 2024, when I invited Simon Martin to EPFL to tell us about his work with Francis @BachFrancis . That visit helped bring these ideas together around a different question: what can we actually understand about learning in a two-layer neural network? Together with Antoine Maillard, Emanuele Troiani, Simon and Lenka Zdeborová, we asked what happens when one learns a two-layer neural network with quadratic activation, whose width grows proportionally to the input dimension, from quadratically many samples. And the key technical observation was precisely a connection to extensive-rank matrix denoising AND TO THE ELLIPSOID-FITTING PROBLEM. We derived the asymptotic Bayes-optimal learning curve and introduced GAMP-RIE, an algorithm combining approximate message passing with rotationally invariant matrix denoising to achieve that performance. This is the kind of thing we like here in Lausanne :-) Maillard, Troiani, Martin, Krzakala & Zdeborová: “Bayes-optimal learning of an extensive-width neural network from quadratically many samples” NeurIPS 2024 t.co/aqn9lY3Vli Then Yizhou Xu, Antoine, Lenka and I pushed this further, putting the statistical-physics predictions on a rigorous footing. We developed rigorous asymptotics and universality results for a broad class of structured matrix-sensing problems, including matrices whose rank grows proportionally to their dimension. Among the applications: establishing Bayes-optimal learning predictions for extensive-width quadratic neural networks. Xu, Maillard, Zdeborová & Krzakala: “Fundamental Limits of Matrix Sensing: Exact Asymptotics, Universality, and Applications” COLT 2025 t.co/XTaWqRP6If And then came one of the most satisfying consequences of this whole programme. With Vittorio Erba, Emanuele Troiani and Lenka, we studied empirical risk minimization in overparameterized quadratic networks. L2 regularization on the network weights becomes NUCLEAR-NORM regularization in an equivalent convex matrix-sensing problem. The mapping itself was not new. What excited us was bringing statistical-physics ideas and AMP to bear on it to derive sharp predictions. Suddenly, we could connect global minima, generalization, memorization, weight spectra and the low-rank bias induced by weight decay. This was exciting!!! And d²/4 CAME BACK—as the interpolation threshold in the noise-dominated limit! More generally, the threshold depends on the target structure and the noise. Erba, Troiani, Zdeborová & Krzakala: “The Nuclear Route: Sharp Asymptotics of ERM in Overparameterized Quadratic Networks” NeurIPS 2025 t.co/AucCs0bYdO But this was not the end of the story. In a complementary direction, Simon Martin, Giulio Biroli and Francis Bach have analyzed the actual gradient-flow dynamics of extensive-width quadratic networks. They developed a high-dimensional dynamical description and characterized the resulting spectra, generalization and recovery thresholds. Martin, Biroli & Bach: “High-Dimensional Analysis of Gradient Flow for Extensive-Width Quadratic Neural Networks” JMLR 2026 t.co/hkz9WXM5oy Meanwhile, for us, the matrix/spectral viewpoint opened another door. What if the target itself has a nontrivial spectrum? What if that spectrum follows a power law? How are different spectral modes learned as the amount of data increases? Can this explain neural scaling laws? Can it give us a concrete mechanism for the emergence of newly learned features? This led us to: Defilippis, Xu, Girardin, Troiani, Erba, Zdeborová, Loureiro & Krzakala: “Scaling Laws and Spectra of Shallow Neural Networks in the Feature Learning Regime” ICLR 2026 t.co/fHOSWnIThT There, connections with matrix compressed sensing, LASSO and sparsity allowed us to derive phase diagrams for scaling exponents and connect generalization directly to the spectrum of the learned weights. The spectral viewpoint makes the question concrete: which signal components are detectable, which remain unlearned, and how does that balance change as we get more data? And remarkably, essentially the same viewpoint extends beyond quadratic neural networks. In our work on a simplified single-head attention model, we again characterize learning through the spectrum of a learned matrix. We obtain training and test errors, interpolation and recovery thresholds, and the full singular-value distribution of the learned query–key map, including low-rank structure and spectral outliers. For power-law targets, learning proceeds through sequential spectral recovery: progressively weaker signal directions become learnable as the amount of data increases. A concrete mechanism for emergence in this model—and scaling laws come out of the same analysis! Boncoraglio, Erba, Troiani, Xu, Krzakala & Zdeborová: “Single-Head Attention in High Dimensions: A Theory of Generalization, Weights Spectra, and Scaling Laws” ICML 2026 t.co/jKvUU9ewIn And this brings me back to why I care so much about the original story. A good scientific paper does not merely produce an answer to one isolated problem. Sometimes it gives a community a new connection, a new mapping, a new proof strategy. A PATH. For us, one path ran through: Ellipsoid fitting → Gaussian equivalence → extensive-rank matrix estimation → quadratic neural networks → feature learning → spectra → scaling laws → attention. Gaussian equivalence itself has a long, crazy history (don’t get me started on that one...) Of course, this is not a single linear chain. Many researchers and ideas contributed at every stage, and several of these connections long predate the papers I have mentioned. But having watched part of this story develop from very close by, I wanted to explain how Antoine’s work, with his collaborators, helped set this particular programme in motion—and how much we owe to it. Looking at that blackboard again, I do not just see a threshold. I see ideas that helped us do research we did not know how to do before. To me THAT is what “empowering researchers” actually looks like.

$GOOGL EMPLOYEES RAISE CONCERNS OVER GEMINI 4 CODING PERFORMANCE Google is preparing to launch Gemini 4, but some employees with access to the model say its strong benchmark results aren’t fully translating into real-world performance, according to Bloomberg. The main concerns are around coding. Some internal testers say Gemini 4 struggles with certain coding tasks and front-end design, while the model is also said to be very large, potentially making it more expensive to run. Google disputes that Gemini 4 is underperforming and says there is broad internal consensus that the model is at the frontier. Bloomberg also reports Google abandoned Gemini 3.5 Pro after previously planning to release it in June. Internally, views appear mixed: some employees believe Gemini 4 still trails the latest models from OpenAI and Anthropic in certain areas, while others believe it has caught up. Gemini 4 is said to perform particularly well on multimodal tasks, including understanding video, along with safety, cybersecurity and natural communication. Source: Bloomberg

arXiv has updated our policy on rate limiting for all submitters. This update was made to fairly distribute moderator time & support the arXiv community of staff, volunteers, readers & authors. Please read our announcement to learn more: blog.arxiv.org/2026/10/01/upd…

Gemini 4 Argon is our next era of frontier intelligence. It shows significant improvements across benchmarks, setting a new state of the art for real-world long-horizon software engineering tasks.

‼️The Bitter Lesson for context management: Giving LMs unrestricted control over their context beats human-designed SOTA! Introducing 🩵Context Language Models (CLMs)🩵 - Natively manage their own context - Treat context as a file - Learn policies in CLM weights, no harness

@NandoDF 1 year in AI is …

Hands down a cargo e-bike has been my best SF purchase since covid. The freedom to ride by something and think, "oh lets stop there and get ice cream" or "oh hey that matcha place doesn't have a line right now" and be able to park for free right in front of it opens the door to 10,000 adventures with my kids that I will never forget. Get an angle-grinder resistant lock and bike insurance and you are golden.

I work at OpenAI. In my personal capacity, I also think we need to slow down. x.com/hilbertspaess/…

With AlphaFold we mapped the protein universe - now with AlphaGenome Atlas we’re charting the human genome. It can predict the impact of all 9 billion possible single-letter DNA variants, helping scientists better understand disease. Freely available for academic research: t.co/Gsy6lW3z6O

I was selected for Time Magazine's Top 100 in AI. This is for research done with many fantastic collaborators and esp. with @BetleyJan & @jameschua_sg . I’d also like to thank @ajeya_cotra and @MaxNadeau_ (@coeff_giving) MATS, Constellation, and Rethink Priorities for supporting my research and team over the years. I’ve also benefited from discussions of our papers here on Twitter!

Today's my last day at @UWCheritonCS, before I go on leave. I applied to 30+ faculty positions in Fall 2017. 3 interviewed me, only 1 offer. I'm deeply grateful that Waterloo CS alone saw potential here. I'm sad to be leaving and I will remain its loudest advocate.

It was really interesting to go back and re-read this article from February which, according to my computations, was only six months ago. nytimes.com/2026/02/07/sci…

🚢 Marin 535B-A23B started training this week! As usual, the whole process is open. Voyage plan: pretraining (80%) + midtraining (20%) on 18.75T tokens on 11 x GB200 NVL72 for ~3 months (2.7e24 FLOPs). Post-training will follow. Before kicking off the run, we trained a 4-rung scaling ladder from 1.6B-A61M (48B tokens) to 27.7B-A1.2B (926B tokens) to debug issues, and to make a forecast of our hero run. This is by far our biggest run, so definitely expecting the unexpected.

I have decided to leave math academia. Not because of burnout, the academic job market, or a loss of love for mathematics. Surprisingly, it is because of how much mathematics I could do with LLMs over the last few months. For most of my life, pursuing mathematics felt like the obvious choice. It gave me a sense of purpose, rooted in the search for hidden truths in God’s Book. Over the past few months, that conviction has been shaken. With LLMs, I could get several breakthroughs on problems I care deeply about and had worked on for years during my PhD. This progress might have taken years of my academic life if not for LLMs. This could have strengthened my conviction that mathematics was my calling. Instead, it undermined it. With each passing week, my role in the discovery loop seemed to grow smaller. What made mathematics meaningful to me was never just the final answer. It was the struggle: months or years of exploration, failed approaches, and deep thought before a hidden structure finally became visible. That struggle gave me a sense that the result was truly mine. Prompting my way to answers, without the struggle that once defined the process, no longer felt like the same vocation. The answer might be just as beautiful, but it did not feel earned in the same way. I now believe that we are rapidly approaching a world in which most answers from God’s Book will be only a prompt away. Once I truly internalized that possibility, dedicating much of my life to finding those answers slightly earlier no longer felt as meaningful as it once had. But one problem kept bothering me: How do we know the oracle is right? Intelligence (and hence the amount of math papers) is becoming abundant and cheaper by the day. Trust is not. The proofs produced by these systems can be highly sophisticated, and their errors can be extremely subtle. Determining whether an apparent breakthrough was actually correct sometimes took me days. Until it is verified, a beautiful proof is not different from slop. That gap drew me toward formal verification and eventually toward formalizing math proofs in Lean, including proofs of several longstanding conjectures and Erdős problems discovered with the help of LLMs. I began to see verification as a central intellectual bottleneck. Intelligence is useful only when its outputs can be trusted. So I am leaving math academia. I am shifting my attention from discovering answers to building systems that can certify them. I am very excited to be joining @PramaanaLabs to work on this challenge. Over the next few years, I hope to help build a future in which increasingly powerful systems are not merely intelligent, but provably correct and reliable across problems far beyond mathematics.

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