Google DeepMind Says AlphaEvolve Has Helped Lower The Upper Bound For Theoretical Matrix Multiplication

Google DeepMind, with academic collaborators and its AlphaEvolve coding agent, has lowered the theoretical upper bound for matrix multiplication exponent ω to less than 2.371177, improving on the previous best of 2.371339. The result, announced by Pushmeet Kohli, who leads science and strategic initiatives at Google DeepMind, advances a decades-old open problem in computer science with implications for faster and cheaper computing, including AI.

AI continues to help make all kinds of improvements and breakthroughs in math — and this time, in a field that underpins modern AI. Google DeepMind has announced a new record in one of the oldest open problems in computer science — how fast can two matrices theoretically be multiplied. The team, working with academic collaborators and its AlphaEvolve https://officechai.com/ai/how-googles-alphaevolve-ai-wrote-code-that-humans-wouldnt-normally-write/ coding agent, has pushed the upper bound on the matrix multiplication exponent, commonly written as omega ω , down to less than 2.371177, an improvement on the previous best bound of 2.371339. Pushmeet Kohli, who leads science and strategic initiatives at Google DeepMind, announced the result, calling matrix multiplication the basic computational operation that powers modern computing, including AI. The exponent ω describes the theoretical fastest possible speed at which computers can multiply matrices, and figuring out its exact value has been an open challenge in complexity theory for decades. Why matrix multiplication matters so much Matrices are essentially grids of numbers, and multiplying them together is one of those operations that sounds abstract but sits underneath an enormous amount of real computation. Every neural network, including the large language models that power chatbots like Gemini and ChatGPT, is fundamentally built on layers of matrix multiplications. Graphics rendering, scientific simulations, cryptography, and search algorithms all lean on the same operation too. Because it gets used so often and at such massive scale, even a small theoretical improvement in how efficiently matrices can be multiplied has the potential to ripple outward into faster and cheaper computing everywhere. The naive way to multiply two n×n matrices takes roughly n³ operations. Mathematicians have spent more than fifty years trying to figure out how much that can be cut down. In 1969, Volker Strassen showed that matrices could be multiplied using fewer operations than the standard method, kicking off a long line of research into just how low that number could theoretically go. That theoretical floor, if it could ever be reached, is what mathematicians call ω. Researchers have proven that ω is at least 2, since you need at least that many operations just to read through the entries of the matrices being multiplied. What nobody has ever proven is that 2 is achievable. So the field has instead worked backward, trying to find algorithms that push the achievable upper bound as close to 2 as possible. What DeepMind actually did Progress on lowering this bound over the last few years has come from a technique called combination loss analysis, a refinement of an older approach known as the laser method, developed through a series of papers by researchers including Duan and colleagues in 2022, and Virginia Vassilevska Williams, Josh Alman, and others in the years since. Both of those researchers are co-authors on DeepMind’s new paper as well. At the heart of this approach sits a difficult optimization problem, and DeepMind’s contribution was largely about attacking that optimization problem in three ways. First, the team reformulated the problem so it could be solved in a larger, more flexible setting than earlier methods allowed. Second, they built a new optimization algorithm for it using recent advances in machine learning. Third, they handed the resulting algorithm over to AlphaEvolve, DeepMind’s Gemini-powered coding agent, which refined it further. AlphaEvolve is designed to work by generating and testing many variants of a piece of code against a defined objective, gradually evolving toward better solutions, similar in spirit to how it previously found a way to multiply 4×4 matrices using 48 scalar multiplications https://officechai.com/ai/google-alphaevolve-ai-discovers-new-algorithm-for-matrix-multiplication-improves-56-year-old-approach/ instead of the 49 required by Strassen’s original method. This new result operates at a more abstract, theoretical level than that earlier one, since it concerns the asymptotic exponent for arbitrarily large matrices rather than a concrete algorithm for a fixed small size, but it comes from the same underlying idea of letting an AI system search a solution space that has proven extremely difficult for humans to search by hand. The combined approach across all three improvements is what produced the new bound of ω < 2.371177, edging the number closer to the theoretical minimum of 2, even though the actual gap that remains is still substantial in terms of what it would mean for real-world computing. Part of a broader pattern at DeepMind This isn’t the first time AlphaEvolve has been pointed at open mathematical problems. Earlier this year, the system was used to improve lower bounds for five classical Ramsey numbers https://officechai.com/ai/google-deepminds-alphaevolve-cracks-five-ramsey-number-puzzles-that-stumped-mathematicians-for-decades/ , a class of combinatorics problems that had resisted progress for over a decade in some cases. DeepMind has also applied AlphaEvolve across more than fifty open problems spanning analysis, geometry, combinatorics, and number theory, including the kissing number problem, and the system’s results alongside newer tools like the AI co-mathematician https://officechai.com/ai/google-deepmind-releases-ai-co-mathematician-that-creates-new-high-score-on-frontiermath-benchmark/ and AlphaProof Nexus suggest DeepMind is treating pure mathematics as a genuine testing ground for what its agents can do, not just a side project. Demis Hassabis has previously described this kind of work as part of a flywheel, where AI systems that get better at discovering new algorithms and mathematical results can in turn be used to build better AI systems, which then get applied to harder problems. Matrix multiplication sits right at the center of that loop, given how directly it feeds back into the computational cost of training and running the very models doing the discovering. The matrix multiplication exponent problem is unlikely to be solved outright anytime soon, and DeepMind’s paper does not claim otherwise. What it does show is that a fairly narrow, technical optimization problem that has occupied specialists for years can be meaningfully advanced by combining better mathematical formulation, modern machine learning, and an AI coding agent refining the result at the end.