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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.19465 |
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| _version_ | 1866911540393803776 |
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| author | Rush, Keith |
| author_facet | Rush, Keith |
| contents | 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 $v^{(k+1)} = \text{diag}((D_{v^{(k)}}^{1/2} M D_{v^{(k)}}^{1/2})^{1/2})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19465 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global Convergence of Multiplicative Updates for the Matrix Mechanism: A Collaborative Proof with Gemini 3 Rush, Keith Machine Learning Artificial Intelligence Optimization and Control 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 $v^{(k+1)} = \text{diag}((D_{v^{(k)}}^{1/2} M D_{v^{(k)}}^{1/2})^{1/2})$ 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. |
| title | Global Convergence of Multiplicative Updates for the Matrix Mechanism: A Collaborative Proof with Gemini 3 |
| topic | Machine Learning Artificial Intelligence Optimization and Control |
| url | https://arxiv.org/abs/2603.19465 |