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u/obvithrowaway34434

r/OpenAI r/SacredGeometry r/math r/ClaudeAI r/accelerate r/OperationsResearch r/MichaelLevinBiology
▲ 249 r/accelerate

"Anthropic is ramping up its efforts very quickly in biology and medicine, and we hope to have incredible results in the coming years and some early glimmers in the coming months" - Interesting quote from Dario's tweet today

Dario made a rare tweet responding to the (imo valid) criticism of Anthropic about regulatory capture and negative messaging regarding the dangers of AI. It was a well-written post, but I found the second thread most compelling.

>I do not agree that my messaging has been disproportionately negative.  In fact it has been about equally balanced between risks and benefits: I’ve written one major essay about each, and even in interviews where I discuss the risks, I make sure to frequently mention the incredible benefits as well as proposing possible solutions to the risks (short clips from my interviews that end up on social media tend to be disproportionately negative, as that gets clicks).  In fact, I wrote Machines of Loving Grace because I didn’t feel the AI industry was painting an inspiring enough picture of how the technology could radically transform the world for the better.  The bulk of the essay is devoted to refuting skepticism of AI’s potential in health and biology, and showing why I think it will actually be possible to cure most human disease in ~5-10 years, as crazy as it may sound to ordinary people and frankly to biologists as well (I used to be one!).  And, if you read my most recent essay (Policy on the AI Exponential), I discuss concrete proposals for how to streamline the FDA process to make sure the deluge of AI-accelerated drugs isn’t slowed down by the regulatory process.  I feel the urgency here: I lost my father to Hepatitis C only a few years before the development of direct-acting antivirals (sofosbuvir), which cure 95% of patients and probably would have cured him.

>I do agree that the public has a negative view of AI (and that this is a big problem), but I don’t think it is primarily caused by me or any other AI leader warning about AI’s risks.  I think it is fundamentally a crisis of trust.  I think that ordinary people don’t trust companies, governments, or the tech industry and always suspect that we are cooking up some new way to screw them over.  The causes of this go back decades and AI is just the latest iteration of it.  I don’t think that a glitzy marketing campaign with a positive spin (which some have advocated that Anthropic do) is the way to win back that trust — at this point, saying that AI will cure cancer is more a cliche than it is inspiring, and most people think it is deceptive.  The thing that will work is *actually curing cancer*.  I think by far the most accurate criticism of AI companies including Anthropic is that we haven’t yet delivered on our big promises to benefit the world.  That is totally on us, and I think it’s the criticism you should be making, instead of all this stuff about messaging and marketing.

>We are however doing our best to fix this: Anthropic is ramping up its efforts very quickly in biology and medicine, and we hope to have incredible results in the coming years and some early glimmers in the coming months.  When we’ve actually accomplished something real, the whole world will hear about it, as loudly as possible, you have my word on that.  But until then I don’t want to make empty promises, and in the meantime I feel compelled to speak honestly about the very real risks of AI and how to address them.  Honesty is the right thing on the merits, and in terms of public credibility and trust it is no worse than, and may in fact be better than, an approach that ignores or distracts from risks which people instinctively understand are real.

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u/obvithrowaway34434 — 4 days ago
▲ 339 r/accelerate

Fable 5 may have disproved the famous Jacobian conjecture

This is huge if true.

From Wikipedia:

In mathematics, the Jacobian conjecture is a famous problem concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself has a Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse. The conjecture was first stated for two variables by Ludwig Kraus in 1884[1] and then stated in full generality in 1939 by Ott-Heinrich Keller.[2] It was subsequently widely publicized by Shreeram Abhyankar,[3] as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.

https://en.wikipedia.org/wiki/Jacobian_conjecture?wprov=sfla1

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u/obvithrowaway34434 — 1 month ago
▲ 227 r/accelerate

GPT-5.6 has already been involved in almost half a dozen mathematical breakthroughs (many of them more than 30 years old) since it was released a little over a week ago

u/obvithrowaway34434 — 1 month ago
▲ 46 r/accelerate

Kimi K3 ranks third overall in the Artificial Analysis Index, after Fable 5 and GPT-5.6 Sol, and costs similar to 5.6 Sol.

The price seems a bit high for this to be a daily driver via the API. It can't compete with Claude/Codex subscription plans, but it could be attractive for enterprises once the weights are released. It feels like another Deepseek R1 moment.

u/obvithrowaway34434 — 1 month ago
▲ 41 r/accelerate

Kimi K3 will be expensive, but still cheaper than GPT-5.6/Opus 4.8

This is more expensive than GLM-5.2. I will be interested to see how token efficient this model is.

u/obvithrowaway34434 — 1 month ago
▲ 901 r/math+2 crossposts

After OpenAI’s CDC proof announcement, GPT-5.6 used a similar prompt to close a 30-year gap in convex optimization, verified in Lean

TL;DR: In a single 148 min session, with a prompt modeled after the one OpenAI used to prove CDC, GPT 5.6 Sol Pro supplied a proof that closed a complexity gap in convex optimization that has existed since 1996. The result was formally verified in Lean. Links to everything and thoughts on AI capabilities are at the bottom of this post.

Disclosure: I am the author of the preprint and Lean repository linked below. I have a PhD in applied mathematics and am a teaching prof in IEOR at UC Berkeley. The result has not yet been peer reviewed.

Following the recent announcement that GPT-5.6 Sol Pro had produced a proof of the Cycle Double Cover Conjecture, I adapted the prompting methodology used in that project to a problem in convex optimization. After 148 minutes of uninterrupted work, GPT-5.6 Sol Pro produced the main argument for a lower bound that I had been unable to prove myself (and a lot of my past work has been proving complexity lower bounds in different settings).

The problem concerns deterministic zeroth-order convex optimization: Let B_d be the Euclidean unit ball in ℝᵈ, and consider all convex, 1-Lipschitz functions f: B_d → ℝ. An algorithm may query any point x ∈ B_d, and receives only the exact real number f(x), no other information (but the algorithm "knows" that f is convex and Lipschitz). The algorithm is otherwise completely unrestricted, and can use unlimited computation and memory. These function-value-only problems arise naturally when an objective is evaluated through a physical experiment or simulator. One can imagine choosing d engineering parameters and observing only the cost returned by the simulation. If evaluations are expensive (think of measuring a physical system), the natural question is how many are fundamentally required. This is formalized as oracle complexity. Specifically, this is the oracle complexity of convex optimization under an exact function value oracle.

Let Q(d, ε) denote the worst-case number of queries required to find an ε-optimal point of f. An algorithm due to Protasov from 1996 shows that order d² function evaluations are sufficient, which gives Q(d, ε) = O(d²), an upper bound on the complexity. Lower bounds were practically nonexistent for this setting, and the strongest previously applicable bound was only Ω(d), inherited from the stronger first-order oracle model (where the algorithm receives both function values and gradients). That means we didn't know for certain whether gradients actually help in optimization, since the function-value only and first-order oracle models have had this same lower bound, and so there was a linear gap in d in the complexity of this fairly fundamental convex optimization setting since 1996. So, can you find an algoritm that is better than Prosatov’s, and only needs d evaluations? Or can you show that no such algorithm can exist, and we can sleep well at night knowing that Protasov’s algorithm using d² evaluations is best possible? What 5.6 Sol proved is the latter.

I had worked on this problem sporadically for about a year (I ran into needing such a bound for a different complexity paper I was working on). I had some ideas that didn't pan out, and also spent long sessions trying to solve it with GPT-5.4 and GPT-5.5 with no luck, after reading of folks like Ernest Ryu having success with these in some work on optimization bounds.

After seeing OpenAI’s CDC result, I wrote a much more elaborate prompt following the same general methodology. My prompt is about ten pages long and attached at the end of the preprint (see collection of links below). There is a lot baked into this prompt, on approaches to try and also on how exactly the model should proceed, but it's built exactly in the style of OpenAI's CDC prompt. One note is that I gave it a relatively small error requirement, to prove the quadratic lower bound under order d⁻⁴ accuracy. After 148 minutes, GPT-5.6 Sol Pro returned a proposed proof resolving the quadratic dimension dependence at accuracy of order d⁻³. After checking things myself, I formally verified the proof in Lean, and it passed the formal verification check. The construction and main invariant used also make genuine sense to me and are closely related to some other results in complexity of convex optimization (for example, Nemirovsky and Yudin's tight bound for first-order convex optimization also uses constructions that are maxes of affine functions).

Lastly, some important comments about the work relating to AI capabilities: In a lot of cases, proving lower bounds like this result relies on finding that right construction that works (in this case, family of difficult functions and a strategy for how an "adversarial" oracle should answer queries from an algorithm to reveal minimal information) and then proving things about it. There are only so many function classes which would be reasonable to look at (here, quadratics for example would have also been reasonable with order d² degrees of freedom, or any variation of maxes of some simpler families of convex functions as well), but the actual proof mechanics once the "correct" function class and correct strategy for adversarial oracle answers is found are often not so complicated, and often employ existing results from convex geometry or similar (this is also the structure of two previous but much more niche, less important results of mine). So I wouldn't really say that this result is using or creating some fundamentally new techniques in convex geometry or optimization theory. What this means from my perspective is that if a result is attainable with existing techniques, modern AI methods will be able to solve those problems. I don't think researchers in math/TCS will be made obsolete, but I think it will instead no longer make sense to work on any low-hanging, or even medium-hanging (you know what I mean) fruit. We'll be needed for problems where actual novel approaches are needed.

Links:

The preprint, Lean code, complete prompts, proof map, and build instructions are available here:

https://github.com/PhillipKerger/zero-order-bounds-lean-verification

ArXiv: Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values

The original uninterrupted 148-minute chat that produced the initial proof:

https://chatgpt.com/share/6a55aa50-b484-83ea-85c0-c7e7b4bda41c

The later chat that led to the d⁻¹ᐟ² refinement:

https://chatgpt.com/share/6a55ad10-7644-83ea-859e-5483d2e0dff0

OpenAI’s CDC prompt, that I structured things after:

https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf

And a more accessible account I wrote on Medium:

https://medium.com/@kerger.p/an-ai-assisted-breakthrough-in-convex-optimization-an-optimization-problem-dating-back-30-years-a-db5c631119de

Edit: This was Sol PRO, not Ultra. I had been working in codex before this, where the level above XHigh is Ultra. But I did this in the web interface, where the highest is Pro, which is in fact not quite the same as Ultra.

u/InstitutionBuilder — 1 month ago