3 ms·

That was dense but seemed nuanced. Anyone care to summarize for those of us who lack the mathematics nomenclature and context?

by piker 1y ago

qsort 1y ago
I'm not claiming to be an expert, but more or less what the article says is this: - Context: Terence Tao is one of the best mathematician alive. - Context: AlphaEvolve is an optimization tool from Google. It differs from traditional tools because the search is guided by an LLM, whose job is to mutate a program written in a normal programming language (they used Python). Hallucinations are not a problem because the LLM is only a part of the optimization loop. If the LLM fucks up, that branch is cut. - They tested this over a set of 67 problems, including both solved and unsolved ones. - They find that in many cases AlphaEvolve achieves similar results to what an expert human could do with a traditional optimization software package. - The main advantages they find are: ability to work at scale, "robustness", i.e. no need to tune the algorithm to work on different problems, better interpretability of results. - Unsurprisingly, well-known problems likely to be in the training set quickly converged to the best known solution. - Similarly unsurprisingly, the system was good at "exploiting bugs" in the problem specification. Imagine an underspecified unit test that the system would maliciously comply to. They note that it takes significant human effort to construct an objective function that can't be exploited in this way. - They find the system doesn't perform as well on some areas of mathematics like analytic number theory. They conjecture that this is because those problems are less amenable to an evolutionary approach. - In one case they could use the tool to very slightly beat an existing bound. - In another case they took inspiration from an inferior solution produced by the tool to construct a better (entirely human-generated) one. It's not doing the job of a mathematician by any stretch of the imagination, but to my (amateur) eye it's very impressive. Google is cooking.
- nsoonhui 1y ago
  >> If the LLM fucks up, that branch is cut. Can you explain more on this? How on earth are we supposed to know LLM is hallucinating?
  - khafra 1y ago
    Math is a verifiable domain. Translate a proof into Lean and you can check it in a non-hallucination-vulnerable way.
    - griffzhowl 1y ago
      But that's not what they're doing here. They're comparing Alphaevolve's outputs numerically against a scoring function
      - perching_aix 1y ago
        They did also take some of the informal proofs and formalized them using AlphaProof, emitting Lean.
        griffzhowl 1y ago
        Ah ok, I didn't notice that part, thx
  - tux3 1y ago
    In this case AlphaEvolve doesn't write proofs, it uses the LLM to write Python code (or any language, really) that produces some numerical inputs to a problem. They just try out the inputs on the problem they care about. If the code gives better results, they keep it around. They actually keep a few of the previous versions that worked well as inspiration for the LLM. If the LLM is hallucinating nonsense, it will just produce broken code that gives horrible results, and that idea will be thrown away.
  - qsort 1y ago
    We don't, but the point is that it's only one part of the entire system. If you have a (human-supplied) scoring function, then even completely random mutations can serve as a mechanism to optimize: you generate a bunch, keep the better ones according to the scoring function and repeat. That would be a very basic genetic algorithm. The LLM serves to guide the search more "intelligently" so that mutations aren't actually random but can instead draw from what the LLM "knows".
- omnicognate 1y ago
  Important clarification > search is guided by an LLM The LLM generates candidates. The selection of candidates for the next generation is done using a supplied objective function. This matters because the system is constrained to finding solutions that optimise the supplied objective function, i.e. to solving a specific, well-defined optimisation problem. It's not a "go forth and do maths!" instruction to the LLM.
- cubefox 1y ago
  > AlphaEvolve is an optimization tool from Google. It differs from traditional tools because the search is guided by an LLM, whose job is to mutate a program written in a normal programming language (they used Python). To clarify, AlphaEvolve is an evolutionary algorithm which uses a neural network (in this case an LLM), which is based on gradient descent, for mutation. Evolutionary algorithms are generally a less efficient form of optimization compared to gradient descent. But evolutionary algorithms can be applied more widely, e.g. to discrete problems which aren't directly differentiable, like the optimization of Python code. AlphaEvolve combines the two optimization approaches by replacing random mutation with the output of a gradient-based model.
ants_everywhere 1y ago
They put an LLM in a loop that mimics how people do real math, and it did research-level math. Like humans, it wasn't equally capable across all mathematical domains. The experiment was set up to mimic mathematicians who are excellent at proving inequalities, bounds, finding optimal solutions, etc. So more like Ramanujan and Erdős in their focus on a computationally-driven and problem-focused approach.
- j2kun 1y ago
  > that mimics how people do real math Real people do not do math like AlphaEvolve...
  - ants_everywhere 1y ago
    What do you feel is fundamentally different about the feedback loop in AlphaEvolve compared to, say, Einstein and Grossman repeatedly running calculations until they found the right tensor setup for General Relativity? Or Euler filling his notebooks with computations? Or Ramanujan? Or Newton working out infinite series? Or Kepler, etc etc. They are all doing iterative search with feedback from a function that tells them whether they're getting closer or farther from their goal. They try different things, see what works, and keep the stuff that works.
    - youoy 1y ago
      The reward functions in the problems that they proposed alphaevolve are easy. The reward funtions of at least 50% of maths are not. You can say that validating if a proof is correct is a straightforward reward, but the size of interesting theorems over the space of all theorems is very small. And also what does "interesting" could even mean?
    - j2kun 1y ago
      "All human work is just an iterative search with a feedback function" is the kind of trite and ignorant take that belongs only in the private group chats of VC bros.
- 1y ago

That was dense but seemed nuanced. Anyone care to summarize for those of us who lack the mathematics nomenclature and context?

by piker 1y ago

That was dense but seemed nuanced. Anyone care to summarize for those of us who lack the mathematics nomenclature and context?

qsort 1y ago

I'm not claiming to be an expert, but more or less what the article says is this: - Context: Terence Tao is one of the best mathematician alive. - Context: AlphaEvolve is an optimization tool from Google. It differs from traditional tools because the search is guided by an LLM, whose job is to mutate a program written in a normal programming language (they used Python). Hallucinations are not a problem because the LLM is only a part of the optimization loop. If the LLM fucks up, that branch is cut. - They tested this over a set of 67 problems, including both solved and unsolved ones. - They find that in many cases AlphaEvolve achieves similar results to what an expert human could do with a traditional optimization software package. - The main advantages they find are: ability to work at scale, "robustness", i.e. no need to tune the algorithm to work on different problems, better interpretability of results. - Unsurprisingly, well-known problems likely to be in the training set quickly converged to the best known solution. - Similarly unsurprisingly, the system was good at "exploiting bugs" in the problem specification. Imagine an underspecified unit test that the system would maliciously comply to. They note that it takes significant human effort to construct an objective function that can't be exploited in this way. - They find the system doesn't perform as well on some areas of mathematics like analytic number theory. They conjecture that this is because those problems are less amenable to an evolutionary approach. - In one case they could use the tool to very slightly beat an existing bound. - In another case they took inspiration from an inferior solution produced by the tool to construct a better (entirely human-generated) one. It's not doing the job of a mathematician by any stretch of the imagination, but to my (amateur) eye it's very impressive. Google is cooking.

nsoonhui 1y ago
>> If the LLM fucks up, that branch is cut. Can you explain more on this? How on earth are we supposed to know LLM is hallucinating?
- khafra 1y ago
  Math is a verifiable domain. Translate a proof into Lean and you can check it in a non-hallucination-vulnerable way.
  - griffzhowl 1y ago
    But that's not what they're doing here. They're comparing Alphaevolve's outputs numerically against a scoring function
    - perching_aix 1y ago
      They did also take some of the informal proofs and formalized them using AlphaProof, emitting Lean.
      - griffzhowl 1y ago
        Ah ok, I didn't notice that part, thx
- tux3 1y ago
  In this case AlphaEvolve doesn't write proofs, it uses the LLM to write Python code (or any language, really) that produces some numerical inputs to a problem. They just try out the inputs on the problem they care about. If the code gives better results, they keep it around. They actually keep a few of the previous versions that worked well as inspiration for the LLM. If the LLM is hallucinating nonsense, it will just produce broken code that gives horrible results, and that idea will be thrown away.
- qsort 1y ago
  We don't, but the point is that it's only one part of the entire system. If you have a (human-supplied) scoring function, then even completely random mutations can serve as a mechanism to optimize: you generate a bunch, keep the better ones according to the scoring function and repeat. That would be a very basic genetic algorithm. The LLM serves to guide the search more "intelligently" so that mutations aren't actually random but can instead draw from what the LLM "knows".
omnicognate 1y ago
Important clarification > search is guided by an LLM The LLM generates candidates. The selection of candidates for the next generation is done using a supplied objective function. This matters because the system is constrained to finding solutions that optimise the supplied objective function, i.e. to solving a specific, well-defined optimisation problem. It's not a "go forth and do maths!" instruction to the LLM.
cubefox 1y ago
> AlphaEvolve is an optimization tool from Google. It differs from traditional tools because the search is guided by an LLM, whose job is to mutate a program written in a normal programming language (they used Python). To clarify, AlphaEvolve is an evolutionary algorithm which uses a neural network (in this case an LLM), which is based on gradient descent, for mutation. Evolutionary algorithms are generally a less efficient form of optimization compared to gradient descent. But evolutionary algorithms can be applied more widely, e.g. to discrete problems which aren't directly differentiable, like the optimization of Python code. AlphaEvolve combines the two optimization approaches by replacing random mutation with the output of a gradient-based model.

ants_everywhere 1y ago

They put an LLM in a loop that mimics how people do real math, and it did research-level math. Like humans, it wasn't equally capable across all mathematical domains. The experiment was set up to mimic mathematicians who are excellent at proving inequalities, bounds, finding optimal solutions, etc. So more like Ramanujan and Erdős in their focus on a computationally-driven and problem-focused approach.

j2kun 1y ago
> that mimics how people do real math Real people do not do math like AlphaEvolve...
- ants_everywhere 1y ago
  What do you feel is fundamentally different about the feedback loop in AlphaEvolve compared to, say, Einstein and Grossman repeatedly running calculations until they found the right tensor setup for General Relativity? Or Euler filling his notebooks with computations? Or Ramanujan? Or Newton working out infinite series? Or Kepler, etc etc. They are all doing iterative search with feedback from a function that tells them whether they're getting closer or farther from their goal. They try different things, see what works, and keep the stuff that works.
  - youoy 1y ago
    The reward functions in the problems that they proposed alphaevolve are easy. The reward funtions of at least 50% of maths are not. You can say that validating if a proof is correct is a straightforward reward, but the size of interesting theorems over the space of all theorems is very small. And also what does "interesting" could even mean?
  - j2kun 1y ago
    "All human work is just an iterative search with a feedback function" is the kind of trite and ignorant take that belongs only in the private group chats of VC bros.
1y ago