Google AI won the IMO Mathematical Olympiad silver medal, the mathematical reasoning model AlphaProof was launched, and reinforcement learning is so back

For AI, Mathematical Olympiad is no longer a problem.

On Thursday, Google DeepMind’s artificial intelligence completed a feat: using AI to solve the real question of this year’s International Mathematical Olympiad IMO, and it was just one step away from winning the gold medal.

The IMO competition that just ended last week had a total of six questions involving algebra, combinatorics, geometry and number theory. Google’s proposed hybrid AI system got four questions right and scored 28 points, reaching silver medal level.

At the beginning of this month, UCLA tenured professor Terence Tao just promoted the AI Math Olympiad (AIMO Progress Award) with a million-dollar prize. Unexpectedly, the level of AI problem solving has improved to this level before July.

Solve questions simultaneously on IMO and get the hardest questions right

IMO is the oldest, largest and most prestigious competition for young mathematicians, held annually since 1959. Recently, the IMO competition has also been widely recognized as a grand challenge in the field of machine learning, becoming an ideal benchmark for measuring the advanced mathematical reasoning capabilities of artificial intelligence systems.

At this year’s IMO competition, AlphaProof and AlphaGeometry 2 developed by the DeepMind team jointly achieved a milestone breakthrough.

Among them, AlphaProof is a reinforcement learning system for formal mathematical reasoning, while AlphaGeometry 2 is an improved version of DeepMind’s geometry solving system AlphaGeometry.

This breakthrough demonstrates the potential of artificial general intelligence (AGI) with advanced mathematical reasoning capabilities to open new areas of science and technology.

So, how does DeepMind’s AI system participate in the IMO competition?

Simply put, first these mathematical problems are manually translated into formal mathematical language so that the AI ​​system can understand them. In the official competition, human contestants submit answers in two sessions (two days), with a time limit of 4.5 hours per session. The AlphaProof+AlphaGeometry 2 AI system solved one problem in minutes, but took three days to solve other problems. Although if you strictly follow the rules, DeepMind's system has timed out. Some people speculate that this may involve a lot of brute force cracking.

Google said AlphaProof solved two algebra problems and one number theory problem by determining the answers and proving their correctness. These include the hardest problem in the competition, which only five contestants solved at this year's IMO. And AlphaGeometry 2 proves a geometry problem.

The solution given by AI: https://storage.googleapis.com/deepmind-media/DeepMind.com/Blog/imo-2024-solutions/index.html

IMO gold medal winner and Fields Medal winner Timothy Gowers and Dr. Joseph Myers, two-time IMO gold medalist and chairman of the IMO 2024 Problem Selection Committee, scored the solutions given by the combined system according to the IMO scoring rules.

Each of the six questions is worth 7 points, for a maximum total score of 42 points. DeepMind's system received a final score of 28, meaning that all four of the problems it solved received perfect scores - equivalent to the highest score in the silver medal category. This year's gold medal threshold was 29 points, and 58 of the 609 competitors who competed officially earned gold medals.

This graph shows the performance of Google DeepMind’s artificial intelligence system relative to human competitors at IMO 2024. The system scored 28 points out of 42, putting it on par with the competition's silver medalist. Plus, 29 points is enough to get a gold medal this year.

AlphaProof: a formal reasoning method

In the hybrid AI system used by Google, AlphaProof is a self-trained system that uses the formal language Lean to prove mathematical statements. It combines a pre-trained language model with the AlphaZero reinforcement learning algorithm.

Among them, formal languages ​​provide important advantages for formally verifying the correctness of mathematical reasoning proofs. Until now, this has been of limited use in machine learning because the amount of human-written data was very limited.

In contrast, although natural language-based methods have access to larger amounts of data, they produce intermediate reasoning steps and solutions that appear reasonable but incorrect.

Google DeepMind builds a bridge between these two complementary fields by fine-tuning the Gemini model to automatically translate natural language problem statements into formal statements, thereby creating a large library of formal problems of varying difficulty.

Given a mathematical problem, AlphaProof will generate candidate solutions and then prove them by searching for possible proof steps in Lean. Each proof solution found and verified is used to strengthen AlphaProof's language model and enhance its ability to solve subsequent more challenging problems.

To train AlphaProof, Google DeepMind proved or disproved millions of mathematical problems covering a wide range of difficulties and topics in the weeks leading up to the IMO competition. A training loop is also applied during the competition to strengthen the proof of self-generated competition problem variants until a complete solution is found.

AlphaProof reinforcement learning training loop process infographic: About one million informal mathematical problems are translated into formal mathematical language by the formal network. The solver then searches the network for proofs or disproofs of the problem, gradually training itself to solve more challenging problems via the AlphaZero algorithm.

More competitive AlphaGeometry 2

AlphaGeometry 2 is a significantly improved version of the mathematical AI AlphaGeometry that was featured in Nature magazine this year. It is a neuro-symbolic hybrid system in which the language model is based on Gemini and trained from scratch on an order of magnitude more synthetic data than its predecessor. This helps the model solve more challenging geometric problems, including those about object motion and equations of angles, proportions, or distances.

AlphaGeometry 2 uses a symbolic engine that is two orders of magnitude faster than the previous generation. When new problems are encountered, novel knowledge sharing mechanisms enable advanced combinations of different search trees to solve more complex problems.

Prior to this year’s competition, AlphaGeometry 2 could solve 83% of all historical IMO geometry problems from the past 25 years, compared to its predecessor’s 53% solution rate. In IMO 2024, AlphaGeometry 2 solved Problem 4 within 19 seconds of receiving its formalization.

Example of question 4, asking to prove that the sum of ∠KIL and ∠XPY is equal to 180°. AlphaGeometry 2 proposes to construct point E on the line BI such that ∠AEB = 90°. The point E helps to give meaning to the midpoint L of the line segment AB thereby creating many pairs of similar triangles like ABE ~ YBI and ALE ~ IPC to prove the conclusion.

Google DeepMind also reports that as part of the IMO work, researchers are also experimenting with a new natural language reasoning system based on Gemini and a state-of-the-art natural language reasoning system, hoping to achieve advanced problem-solving capabilities. The system does not require translation of questions into formal language and can be combined with other AI systems. In the test of this year's IMO competition questions, it "showed great potential."

Google is continuing to explore AI methods to advance mathematical reasoning and plans to release more technical details about AlphaProof soon.

We’re excited about a future where mathematicians will use AI tools to explore hypotheses, try bold new ways to solve long-standing problems, and quickly complete time-consuming proof elements—and AI systems like Gemini will revolutionize mathematics and broader reasoning aspects become more powerful.

Research team

Google said that the new research was supported by the International Mathematical Olympiad Organization. In addition:

The development of AlphaProof was led by Thomas Hubert, Rishi Mehta and Laurent Sartran; main contributors include Hussain Masoom, Aja Huang, Miklós Z. Horváth, Tom Zahavy, Vivek Veeriah, Eric Wieser, Jessica Yung, Lei Yu, Yannick Schroecker, Julian Schrittwieser, Ottavia Bertolli, Borja Ibarz, Edward Lockhart, Edward Hughes, Mark Rowland and Grace Margand.

Among them, Aja Huang, Julian Schrittwieser, Yannick Schroecker and other members were also core members of the AlphaGo paper 8 years ago (2016). Eight years ago, their AlphaGo, based on reinforcement learning, became famous. Eight years later, reinforcement learning shines again with AlphaProof. Someone lamented in the circle of friends: RL is so back!

AlphaGeometry 2 and natural language inference work is led by Thang Luong. The development of AlphaGeometry 2 was led by Trieu Trinh and Yuri Chervonyi, with important contributions from Mirek Olšák, Xiaomeng Yang, Hoang Nguyen, Junehyuk Jung, Dawsen Hwang and Marcelo Menegali.

Additionally, David Silver, Quoc Le, Hassabis and Pushmeet Kohli are responsible for coordinating and managing the entire project.

Reference content:

https://deepmind.google/discover/blog/ai-solves-imo-problems-at-silver-medal-level/

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