Improving the matrix multiplication exponent with modern optimization and AlphaEvolve

Key points

• The new result is an upper bound showing that the number of operations needed for matrix multiplication is slightly below the 2.371177 power of the input size, and it is not a measurement of actual runtime for practical matrix multiplication.
• The previous combinatorial loss analysis used about 25,000 variables at maximum recursion depth 3, while the new approach handles about 7 million variables at depth 4.
• The probability distribution is represented with a softmax, and the maximum-entropy distribution is computed with Sinkhorn-Knopp; the optimization uses automatic differentiation, Adam, and parallelized JAX tensor computation.
• Gradient-based methods alone lowered the previous upper bound by about 0.000097, and after AlphaEvolve changed the optimization code the total improvement reached about 0.000162, with one candidate taking about 5 hours on a single GPU.
• The final solution is verified by rounding to rational numbers and applying safe rational bounds in the log domain, and the authors say that any major further improvement will require new mathematical ideas.

Paper links

External research summaries. These are not HDATF publications or measured product results.