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~\cite{denisov} 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.
