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

TL;DR

This paper proves the global convergence of a multiplicative update iteration in a matrix optimization problem, completing an open problem and illustrating AI's role in mathematical proof development.

Contribution

It provides a rigorous proof of convergence for a specific fixed-point iteration in matrix optimization, aided by AI (Gemini 3), and discusses AI's practical use in mathematical research.

Findings

Proved monotonic convergence to the global optimizer.

Closed an open problem from prior work.

Demonstrated AI's role in mathematical proof completion.

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.