GPT-5.6 used a prompt to close a 30-year gap in convex optimization • Hacker News Board

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#171

Post by seiferteric » Sat, Jul 18, 2026, 5:19 PM UTC

qsera wrote: post #162

seiferteric wrote: post #22

Earlier quoted context omitted.

They also have ability to go deep and wide in a way that humans just can't. We have limits, get tired, distracted and biased where AI does not. I think there a lot of problem where all the information needed to solve them is there, but we just can't put the pieces together. Like no matter how many people you throw at some problems, you hit human limits and more people won't help, but AI will because it is just relent…

>biased where AI does not. AI can be totally biased... The fact that it can spout bullshit all day long to a human who can be tired and would actually act on the said bullshit, is not very comforting... For example, an LLM could confidently declare something a tired human would take as a fact, but would backfire in a real world.

Not really the kind of biased I meant though. There was a recent article about a AI disproving I think an Erdos conjecture by doing similar things humans have tried, but it was much messier and less "beautiful". I think it is a common bias in science and math that things should be "beautiful" but there is no real reason to think that.

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#172

Post by WarmWash » Sat, Jul 18, 2026, 5:20 PM UTC

jltsiren wrote: post #161

semiquaver wrote: post #15

Earlier quoted context omitted.

You’re at least 18 months out of date claiming that prompting will be the new hot skill. Turns out LLMs are also good at prompting other LLMs.

One of the key skills of a professor is asking the right questions. Figuring out something worth working on, and then framing it in an appropriate way and asking questions that allow someone with specific tools and skills to make progress in the topic. Usually the tools and skills are those available to a new student, but working with an LLM is similar. That skill comes with experience. Most people don't have it imme…

>That skill comes with experience.

Well it seems more and more that 3 months of 500k GPUs churning through data 24/7 to build high dimensional landscapes also counts as experience.

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#173

Post by lucianbr » Sat, Jul 18, 2026, 5:22 PM UTC

dwohnitmok wrote: post #87

skybrian wrote: post #48

Earlier quoted context omitted.

This apparently required a 10-page prompt. It seems like someone needs to know enough to write it?

The author also used GPT-5.6 to write the prompt. This did involve giving GPT-5.6 access to his previous work and a back and forth process (so definitely still used the author's expertise to some degree), but the prompt itself is also largely AI generated.

What's the difference between using GPT to write the prompt to GPT, and "thinking"? The LLM uses the first tokens to predict more tokens, and then uses those tokens to predict even more tokens.

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#174

Post by alexpotato » Sat, Jul 18, 2026, 5:25 PM UTC

xeromal wrote: post #165

_alternator_ wrote: post #76

I know a bit about this field. This conjecture reads as somewhat more niche than the cyclic double cover conjecture recently proved by OpenAI, but nevertheless represents a real contribution. You want to know how long it takes to solve an optimization problem, in this case over convex, lipschitz functions. (The restriction to a spherical domain is not really a restriction, you can just change variables for any bounde…

Sometimes I read a comment on HN that is so advanced that it's just as readable to me as Greek. Love reading it just to see someone work though!

> so advanced that it's just as readable to me as Greek

I used to feel this way about statistics.

The language and terms are hard to understand and many of the formulas are taught as "just memorize this" instead of building up from first principles.

But then I started using statistics to analyze something I cared a lot about (paintball) and I quickly realized it's like learning anything new:

- there is jargon

- and core concepts

- when you learn the above, it suddenly makes a lot more sense.

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#175

Post by WarmWash » Sat, Jul 18, 2026, 5:25 PM UTC

qsera wrote: post #156

peddling-brink wrote: post #13

Earlier quoted context omitted.

Today maybe. I disagree in the long term. While they’ll never have the same subjective experience as humans, what stops an LLM from applying similar lines of thought* in a manner that results in a novel conjecture? They are prediction machines, and so are we in a way. We can give them nearly limitless resources to scale their predictive capabilities. We have billions of years of training baked in. They distill direct…

>what stops an LLM from applying similar lines of thought* in a manner that results in a novel conjecture? One thing is that an LLM can never assume, or find out, an inconsistency in its training data. Novel ideas often require correction of existing assumptions. As far as I understand, it is impossible, by design, for LLMs to contradict what is in its training data. For example, an LLM trained on the data from an in…

They can, but it's limited to that specific chat context.

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#176

Post by adw » Sat, Jul 18, 2026, 5:25 PM UTC

_alternator_ wrote: post #115

hodgehog11 wrote: post #92

Earlier quoted context omitted.

Very confused by this comment. The older (poorer) parts of the ML literature focus on models with convex and (gradient-)Lipschitz objectives, but that's not representative of reality, not even close. Modern objectives for AI models are famously nonconvex (catastrophically, from the point of view of classical optimisation theory), and that's where the interesting research is.

I'd push back on this. Most of the core optimization techniques (eg, ADAM, stochastic gradient descent) are straight out of the convex optimization literature. Generally you need to use optimizers that work well on convex objectives because near minimizers, functions tend to be convex. (Proof by contradiction: a non-convex point has a strict descent direction.) The fact that neural networks are highly nonconvex has e…

Another intuition is that near a minimum you can Taylor expand the function and show that the higher order coefficients (past the square) are negligible.

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#177

Post by p-e-w » Sat, Jul 18, 2026, 5:25 PM UTC

xorcist wrote: post #145

p-e-w wrote: post #139

Earlier quoted context omitted.

Unless you’re claiming that AIs will suddenly (and very soon) stop improving, they are obviously a threat to everyone’s job. Calling notable conjectures that have been open for decades “low-hanging fruit” is an act of desperation. Most professional mathematicians couldn’t have proved those conjectures if their lives depended on it.

The thought that anything could improve without bounds would be absurd. We are living in the physical world after all. The (open, interesting) question is how close we are to the limit.

It’s safe to assume that after less than a decade of LLM development, we’re nowhere close to the limit yet. In fact, progress still seems to be accelerating at the moment.

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#178

Post by tctcd6 » Sat, Jul 18, 2026, 5:27 PM UTC

barnacs wrote: post #118

threethirtytwo wrote: post #108

Genuine question: If you still or did think LLMs are just stochastic parrots that just summarize everything and have no form of creativity, what do you think after seeing results like this? I'm very curious how people reconcile their fear/hatred of AI with actual objective reality. This is actually what interests me most about the whole AI thing. How we tell ourselves what we tell ourselves.

I hold my stance that LLMs are stochastic parrots. Making the parrots ever more complex and training on ever more data produced by intelligent, creative beings may make them more useful or convincing but does at no point give rise to intelligence or creativity.

Comical human arrogance...

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#179

Post by ck2 » Sat, Jul 18, 2026, 5:29 PM UTC

could machine-learning even handle a TEN PAGE PROMPT just a year ago?

this is changing my mind, at least about experts using advanced tools like any profession where it's like the magic of watching a lifetime of hard-earned skill at work

> 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.

Re: GPT-5.6 used a prompt to close a 30-year gap in convex optimization

#180

Post by xeromal » Sat, Jul 18, 2026, 5:31 PM UTC

alexpotato wrote: post #174

xeromal wrote: post #165

Earlier quoted context omitted.

Sometimes I read a comment on HN that is so advanced that it's just as readable to me as Greek. Love reading it just to see someone work though!

> so advanced that it's just as readable to me as Greek I used to feel this way about statistics. The language and terms are hard to understand and many of the formulas are taught as "just memorize this" instead of building up from first principles. But then I started using statistics to analyze something I cared a lot about (paintball) and I quickly realized it's like learning anything new: - there is jargon - and c…

I gotta know what you use stats for regarding paintball. I haven't played in years but I loved playing back in the tipman 98 custom era (not sure if that's still a popular marker).