Improving the matrix multiplication exponent with modern optimization and AlphaEvolve

In the vast landscape of computer science, few operations are as fundamental as multiplying two large grids of numbers, a process known as matrix multiplication. This mathematical task underpins everything from training artificial intelligence models to rendering realistic images in video games. For decades, scientists have known that this operation can be performed faster than the standard, straightforward method, but the precise limit of how fast it can possibly go has remained one of the field's most stubborn mysteries. This limit is described by a single number, a mathematical exponent that dictates how the time required for the calculation grows as the size of the grids increases. The smaller this number, the more efficient the computer can be. While the theoretical minimum is known to be at least two, the best proven upper bound has hovered just above 2.37 for years, a barrier that researchers have been chipping away at with increasingly sophisticated mathematical tools.

A team of researchers from Google DeepMind, along with collaborators from several universities, has now pushed this boundary slightly further. By combining modern optimization techniques with a new form of artificial intelligence, they have established a new record, proving that the exponent can be lowered to less than 2.371177. This is a small numerical shift, but in the context of this specific problem, it represents a significant step forward. The previous best result, achieved in 2025, stood at 2.371339. The new finding does not solve the ultimate mystery of the exact limit, nor does it immediately change how computers multiply matrices in practice, but it tightens the theoretical constraints on the problem, showing that the ceiling is lower than previously thought possible.

The path to this new record began with a mathematical framework known as the laser method, a technique developed over forty years ago to design faster matrix multiplication algorithms indirectly. The most recent refinement of this method, called combination loss analysis, relies on solving a massive, complex optimization problem. This problem involves finding the best way to break down a large mathematical structure into smaller pieces. The researchers found that the difficulty of this problem depends on a parameter representing the depth of the breakdown. Previous attempts had stopped at a depth of three, which limited the number of variables they could adjust. The new team realized that by increasing this depth to four, they could explore a much larger space of possibilities, but doing so would require solving a problem with millions of variables, a task far too large for the traditional algorithms used in the past.

To tackle this scale, the researchers turned to techniques borrowed from machine learning. Instead of using standard mathematical solvers, they reformulated the problem so it could be handled by gradient descent, a method commonly used to train neural networks. This approach allowed them to use powerful computer hardware to process the data in parallel, handling the explosion in complexity that came with the deeper breakdown. They treated the mathematical variables as if they were the adjustable weights in a learning model, iteratively refining them to find a better solution. This shift in strategy alone improved the bound by a measurable amount, demonstrating that modern computational tools could unlock potential that older methods had missed.

However, the team did not stop there. They employed a system called AlphaEvolve, an artificial intelligence designed to write and improve its own code. Rather than just running the optimization algorithm, they let the AI modify the algorithm itself. The system would generate a new version of the code, run it to see what bound it produced, and then evolve the code further to minimize that bound. This process of self-improvement allowed the researchers to find subtle refinements in the optimization strategy that a human team might have overlooked. The result of this automated evolution was a further improvement, pushing the bound down to the new record of 2.371177.

To ensure that this result was not an artifact of computer rounding errors or floating-point inaccuracies, the team performed a rigorous verification step. They took the solution found by their algorithms and converted all the numbers into exact fractions, performing the final calculations with perfect precision. They also replaced every logarithm in the equations with a safe, rational bound that guaranteed the constraints were met. This careful certification process confirmed that the new bound is mathematically valid and free from the numerical noise that often plagues such complex computations.

The researchers note that while their approach has yielded a better bound, the improvements are becoming increasingly difficult to achieve. The gains they made are comparable in magnitude to the incremental progress seen over the last forty years. They suggest that while further modest improvements might be possible by continuing to refine these optimization techniques, achieving a much larger leap in understanding the true limit will likely require entirely new mathematical ideas. For now, the work stands as a testament to the power of combining deep theoretical mathematics with the computational muscle of modern machine learning, proving that even in a field with a long history, there is still room for discovery.