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

arXiv: https://arxiv.org/abs/2603.19465 · HTML: https://mlsanity.com/paper/2603.19465/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 14 references that are in the map.
- First posted on arXiv 19 Mar 2026.

## Builds on (most cited first, 2 of 2)

- [Improved Differential Privacy for SGD via Optimal Private Linear Operators on Adaptive Streams](https://mlsanity.com/paper/2202.08312/) (2022)
- [Efficient and Near-Optimal Noise Generation for Streaming Differential Privacy](https://mlsanity.com/paper/2404.16706/) (2024)

## Abstract (arXiv)

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.