Global Convergence of Multiplicative Updates for the Matrix Mechanism: A Collaborative Proof with Gemini 3

We analyze a fixed-point iteration v leftarrow φ(v) arising in the optimization of a regularized nuclear norm objective involving the Hadamard product structure, posed in DMR+22 in the context of an optimization problem over the space of algorithms in private machine learning. We prove that the iteration v(k+1) = diag((D_v(k)1/2 M D_v(k)1/2)1/2) converges monotonically to the unique global optimizer of the potential function J(v) = 2 Tr((Dv1/2 M Dv1/2)1/2) - sum vi, closing a problem left open there. The bulk of this proof was provided by Gemini 3, subject to some corrections and interventions. Gemini 3 also sketched the initial version of this note. Thus, it represents as much a commentary on the practical use of AI in mathematics as it represents the closure of a small gap in the literature. As such, we include a small narrative description of the prompting process, and some resulting principles for working with AI to prove mathematics.