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FrontierMath: A benchmark for evaluating advanced mathematical reasoning in AI
- agucova 2y agoFor some context on why this is important: this benchmark was designed to be extremely challenging for LLMs, with problems requiring several hours or days of work by expert mathematicians. Currently, LLMs solve 2% of problems in the set (which is kept private to prevent contamination). They even provide a quote from Terence Tao, which helped create the benchmark (alongside other Field medalists and IMO question writers): > “These are extremely challenging. I think that in the near term basically the only way to solve them, short of having a real domain expert in the area, is by a combination of a semi-expert like a graduate student in a related field, maybe paired with some combination of a modern AI and lots of other algebra packages…” Surprisingly, prediction markets [1] are putting 62% on AI achieving > 85% performance on the benchmark before 2028. [1]: https://manifold.markets/MatthewBarnett/will-an-ai-achieve-85-performance-o?play=true https://manifold.markets/MatthewBarnett/will-an-ai-achieve-8...
- llm_trw 2y agoThese benchmarks are entirely pointless. The people making them are specialists attempting to apply their skills to areas unrelated to LLM performance, a bit like a sprinter making a training regimen for a fighter jet. What matters is the data structures that underlie the problem space - graph traversal. First, finding a path between two nodes; second, identifying the most efficient path; and third, deriving implicit nodes and edges based on a set of rules. Currently, all LLMs are so limited that they struggle with journeys longer than four edges, even when given a full itinerary of all edges in the graph. Until they can consistently manage a number of steps greater than what is contained in any math proof in the validation data, they aren’t genuinely solving these problems; they’re merely regurgitating memorized information.
- dr_dshiv 2y ago> they’re merely regurgitating memorized information Source?
- exe34 2y agohe just explained it to you.
- firebaze 2y agoI'm not sure if it is feasible to provide all relevant sources to someone who doesn't follow a field. It is quite common knowledge that LLMs in their current form have no ability to recurse directly over a prompt, which inherently limits their reasoning ability.
- light_hue_1 2y agoThis is just totally false. That's exactly what countless techniques related to chain of thought do.
- dr_dshiv 2y agoIt’s sometimes like, are these critics using the tools? It’s a strange schism at the moment.
- benchmarkist 2y agoIt will be a useful benchmark to validate claims by people like Sam Altman about having achieved AGI.
- mkl 2y agoMost humans can't solve these problems, so it's certainly possible to imagine a legitimate AGI that can't either.
- aurareturn 2y agoBut humans can solve these problems given enough time and domain knowledge. An LLM would never be able to solve them unless they get smarter. Thats the point. It’s not about whether a random human can solve them. It’s whether AI, in general, can. Humans, in general, have proven to be able to solve them already.
- benchmarkist 2y agoThat's correct. Thanks for clarifying for me because I have gotten tired with the comparison to "99% of humans can't do this" as a counter-argument to AI hype criticism.
- llm_trw 2y agoIt is very much an open question just what an llm can solve when allowed to generate an indefinite number of intermediate tokens and allowed to sample an arbitrary amount of text to ground itself. There are currently no tools that let llms do this and no one is building the tools for answering open ended questions.
- mkl 2y agoI'm responding to this: > It will be a useful benchmark to validate claims by people like Sam Altman about having achieved AGI. I think it is possible to achieve AGI without creating an AGI that is an expert mathematician, and that it is possible to create a system that can do FrontierMath without achieving AGI. I.e. I think failure or success at FrontierMath is orthogonal to achieving AGI (though success at it may be a step on the way). Some humans can do it, and some AGIs could do it, but people and AI systems can have human-level intelligence without being able to do it. OTOH I think it would be hard to claim you have ASI if it can't do FrontierMath.
- nopinsight 2y ago> Currently, all LLMs are so limited that they struggle with journeys longer than four edges, even when given a full itinerary of all edges in the graph. This is probably not the case for LLMs in the o1 series and possibly Claude 3.5 Sonnet. Have you tested them on this claim?
- llm_trw 2y agoYes, they also fail. I've found the original gpt4 to be the most consistent. One of these days I'll spend the couple of thousands needed to benchmark all the top models and see how they actually perform on a task which can't be gamed.
- nopinsight 2y agoWhat kinds of problems in what domains did you test o1 models with? I found that they are good at logic and math problems but still hallucinate. I didn’t try to stretch test them with hard problems though.
- llm_trw 2y agoFinding a path between two vertices when given an itinerary of all the edges in a general graph, exactly what I said in the OP.
- youoy 2y agoNot to mention that math proofs are more than graph trasversals... (Although maybe simple math problems are not) There is the problem of extracting the semantics of math formalisms. This is easier in day to day language, I don't know to what extent LLMs can also extract the semantics and relations of different mathematical abstractions.
- sebzim4500 2y ago>Surprisingly, prediction markets [1] are putting 62% on AI achieving > 85% performance on the benchmark before 2028. Or they know the ancient technique of training on the test set. I know most of the questions are kept secret, but they are being regularly sent over the API to every LLM provider.
- tux3 2y agoAlthough the answer isn't sent, so it would have to be a very deliberate effort to fish those out of the API chatter and find the right domain expert with 4-10 hours to spend on cracking it Just letting the AI train on its own wrong output wouldn't help. The benchmark already gives them lots of time for trial and error.
- youoy 2y agoWhy do people still insist that this is unlikely? Like assuming that the company that payed 15M for chat.com does not have some spare change to pay some graduate students/postdocs to solve some math problems. The publicity of solving such benchmark would definitely raise the valuation so it would 100% be worth it for them...
- llm_trw 2y agoAny benchmark which isn't dynamically generated is useless for that very reason.
- rl3 2y agoSimple: I highly doubt they're willing to risk a scandal that would further tarnish their brand. It's still reeling from last year's drama, in addition to a spate of high-profile departures this year. Not to mention a few articles with insider sources that aren't exactly flattering.
- aiono 2y agoI doubt it would be seen as scandal. They can simply generate training data for these questions just like how they generate for other problems. Only difference is probably pay rate is much higher for this kind training data than most other areas.
- light_hue_1 2y agoIf I was going to bet, I would bet yes, they will reach above 85% performance. The problem with all benchmarks, one that we just don't how to solve, is leakage. Systematically, LLMs are much better at benchmarks created before they were trained than after. There are countless papers that show significant leakage between training and test sets for models. This is in part why so many LLMs are so strong according to benchmarks, particularly older popular benchmarks, but then prove to be so weak in practice when you try them out. In addition to leakage, people also over-tune their LLMs to specific datasets. They also go out and collect more data that looks like the dataset they want to perform well on. There's a lot of behind the scenes talk about unethical teams that collect data which doesn't technically overlap test sets, but is extremely close. You can detect this if you look at the pattern of errors these models make. But no one wants to go out and accuse specific teams, at least not for now.
- nerdponx 2y agoCould you run the benchmark by bootstrapping (average of repeated subsampling), instead of a straight-across performance score, and regain some leakage resistance that way? As well as a better simulation of "out of sample" data, at least for a little while.
- agucova 2y agoThis benchmark’s questions and answers will be kept fully private, and the benchmark will only be run by Epoch. Short of the companies fishing out the questions from API logs (which seems quite unlikely), this shouldn’t be a problem.
- benchmarkist 2y agoI looked at the sample questions and even if they get the questions there is no way they will figure out the answers without making significant breakthroughs in understanding mathematics and logic.
- mewpmewp2 2y agoIdeally they would have batches of those exercises, where the only use the next batch when someone has solved a suspicious amount of those exercises. If it performs much worse on the next batch, that is a tell of leakage.
- equestria 2y agoMarket size matters. There's a whopping total of 71 bidders on that.
- TeMPOraL 2y ago> Surprisingly, prediction markets [1] are putting 62% on AI achieving > 85% performance on the benchmark before 2028. Why surprisingly? 2028 is twice as long as capable LLMs existed to date. By "capable" here I mean capable enough to even remotely consider the idea of LLMs solving such tasks in the first place. ChatGPT/GPT-3.5 isn't even 2 years old! 4 years is a lot of time. It's kind of silly to assume LLM capabilities have already bottomed out.
- ekianjo 2y agoSure but it is also reasonable to consider that the pace of progress is not always exponential or even linear at best. Diminishing returns are a thing and we already know that a 405b model is not 5 times better than a 70b model.
- TeMPOraL 2y agoYes, but! Exponential pace of progress isn't usually just one thing; if you zoom in, any particular thing may plateau, but its impact compounds in enabling growth of successors, variations, and related inventions. Nor is it a smooth curve, if you look closely. I feel statements like "a 405b model is not 5 times better than a 70b model" are zooming in on a specific class of models so much you can see the pixels of the pixel grid. There's plenty of open and promising research in tweaking the current architecture in training or inference (see e.g. other thread from yesterday[0]), on top of changes to architecture, methodology, methods of controlling or running inference on exiting models by lobotomizing them or grafting networks to networks, etc. The field is burning hot right now, we're counting space between incremental improvements and interesting research directions in weeks. The overall exponent of "language models" power may just well continue when you zoom out a little bit further. -- [0] - https://news.ycombinator.com/item?id=42093112 https://news.ycombinator.com/item?id=42093112
- mewpmewp2 2y agoHow do you determine the multiplier. Because e.g. there are many problems that GPT4 can solve while GPT3.5 can't. In this case it is infinitely better.
- ak_111 2y agoWould be interesting to know which model solved the 2% and what is the nature of the problems it solved.
- benchmarkist 2y agoVery cool. It'll be nice to have a benchmark that can be used to validate abstract reasoning capabilities because the hype is really starting to get out of hand.
- sebzim4500 2y agoI mean, this benchmark is really hard. I don't think it's a requirement that a system claiming to be AGI should be able to solve these problems, 99.99% of humans can't either.
- benchmarkist 2y agoAn AGI is often claimed to be a general purpose problem solver and these are exactly the types of problems that a general purpose problem solver would be able to solve if given access to a mathematical library. All existing LLMs have been trained on abstract mathematics and logic but it is obvious that they are incapable of abstract logical reasoning, e.g. solving sudoku puzzles.
- skinner_ 2y agoHere is my prediction, FWIW: the hard part of the problem has already been solved, in the following technical sense: there is a few 1000 lines program that has not been invented yet, but it will be invented soon, that loads a current LLM model, runs fast on current hardware, and you will deem it to be an AGI. In other words, the conditional Kolmogorov complexity of undisputable AGI given the Llama weights is only a few 1000 bytes. We are at the pre-AlphaGo, post Clark-Storkey stage of reasoning. That's my guess, anyway.
- benchmarkist 2y ago[dead]
- sebzim4500 2y agoI think you are likely right but coming up with that final inference strategy is still "the hard part" IMO. Not in terms of computation, but in terms of algorith development.
- westurner 2y agoScholarlyArticle: "FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI" (2024) https://arxiv.org/abs/2411.04872 https://arxiv.org/abs/2411.04872 .. https://epochai.org/frontiermath/the-benchmark https://epochai.org/frontiermath/the-benchmark : > [Not even 2%] > Abstract: We introduce FrontierMath, a benchmark of hundreds of original, exceptionally challenging mathematics problems crafted and vetted by expert mathematicians. The questions cover most major branches of modern mathematics -- from computationally intensive problems in number theory and real analysis to abstract questions in algebraic geometry and category theory. Solving a typical problem requires multiple hours of effort from a researcher in the relevant branch of mathematics, and for the upper end questions, multiple days. FrontierMath uses new, unpublished problems and automated verification to reliably evaluate models while minimizing risk of data contamination. Current state-of-the-art AI models solve under 2% of problems, revealing a vast gap between AI capabilities and the prowess of the mathematical community. As AI systems advance toward expert-level mathematical abilities, FrontierMath offers a rigorous testbed that quantifies their progress.
- westurner 2y agoAdditional AI math benchmarks: - "TheoremQA: A Theorem-driven [STEM] Question Answering dataset" (2023) https://github.com/TIGER-AI-Lab/TheoremQA https://github.com/TIGER-AI-Lab/TheoremQA
- deleted 2y ago[deleted]
- bravura 2y agoRegarding keeping the test set private to avoid contamination, the comments about leakage are spot on. The real test set should always be the future. We should evaluate LLMs on text from beyond their knowledge cutoff date, by computing their per-byte perplexity or per-byte compression ratio. There's a deep theoretical connection between compression and learning. The intuition here is that being able to predict the future of science (or any topic, really) is indicative of true understanding. Slightly more formally: When ICLR 2025 announces and publishes the accepted papers, Yoshua Bengio is less surprised/perplexed by what's new than a fresh PhD student. And Terence Tao is less surprised/perplexed by what will be proven in math in the next 10 years than a graduate student in a related field. This work has it right: https://ar5iv.labs.arxiv.org/html//2402.00861 https://ar5iv.labs.arxiv.org/html//2402.00861
- 3abiton 2y agoInteresting take sounds like MDL (Minimum description length) for LLMs!
- Davidzheng 2y agoNot very impressed by the problems they displayed but I guess there should be some good problems in the set given the comments (not in the sense that I find them super easy but they seems random and not super well-posed, and extremely artificial problems--in the sense that they seem to not be of particular mathematical interest[or at least the mathematical content of the problem is being deliberately hidden for testing purposes] but constructed according to some weird criteria). Would be happy to hear an elaboration on the comments by the well-known mathematicians
- vessenes 2y agoHmm. I’m a hard disagree. The problems they show have a number of really nice properties for LLM assessment: They require broad, often integrated knowledge of diverse areas of mathematics, the answers reduce to a number, often a very large number, and thus extremely difficult to guess, and they require a significant amount of symbolic parsing and (I would say) reasoning skills. If we think about what makes a quality mathematician, I’d propose it’s the ability to come at a problem both from the top —- conceptually — and from the bottom — applying various tools and transformations — with a sort of direction in mind that gets to a result. I’d say these problems strongly encourage that sort of behavior. I’m also someone who thinks building in abilities like this to LLMs would broadly benefit the LLMs and the world, because I think this stuff generalizes. But, even if not, It would be hard to say that an LLM that could test 80% on this benchmark would be not useful to a research mathematician. Terence Tao’s dream is something like this that can hook up to LEAN, leaving research mathematicians as editors, advisors, and occasionally working on the really hard parts while the rest is automated and provably correct. There’s no doubt in my mind that a high scoring LLM for this benchmark would be helpful in that concept.
- Jianghong94 2y agoI guess the primary reason is that the answers must be numbers that can be verified easily. Otherwise, you just flood the validator with long LLM reasoning that's hard to verify. People have been proposing using LEAN as a medium for answers but AFAIK even LEAN is not mainstream in the general math community, so there's always trade-offs. Also, coming up with good problems is an art in its own right; the Soviets was famous for institutionalizing anti-Semitism via special math puzzles for Jews in Moscow Univerisity entrance exams. The questions are constructed as such that are hard to solve but have some elementary solutions to divert criticism.
- MichaelRazum 2y agoHow do they solve the 2%? This is the question. If those problems were unseen, that might be already very impressive.