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

Abstract

We analyze a fixed-point iteration $v \leftarrow \phi(v)$ 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 $v^{(k+1)} = \text{diag}((D_{v^{(k)}}^{1/2} M D_{v^{(k)}}^{1/2})^{1/2})$ converges monotonically to the unique global optimizer of the potential function $J(v) = 2 \text{Tr}((D_v^{1/2} M D_v^{1/2})^{1/2}) - \sum v_i$, 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.