Paper
Global Convergence of Multiplicative Updates for the Matrix Mechanism: A Collaborative Proof with Gemini 3
Authors
Keith Rush
Abstract
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~\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.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.19465v1</id>\n <title>Global Convergence of Multiplicative Updates for the Matrix Mechanism: A Collaborative Proof with Gemini 3</title>\n <updated>2026-03-19T20:58:02Z</updated>\n <link href='https://arxiv.org/abs/2603.19465v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.19465v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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~\\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.\n 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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='cs.LG'/>\n <category scheme='http://arxiv.org/schemas/atom' term='cs.AI'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math.OC'/>\n <published>2026-03-19T20:58:02Z</published>\n <arxiv:comment>12 pages, 1 figure</arxiv:comment>\n <arxiv:primary_category term='cs.LG'/>\n <author>\n <name>Keith Rush</name>\n </author>\n </entry>"
}