证明了一种矩阵机制优化算法的全局收敛性，结合AI协作完成数学证明。
Global Convergence of Multiplicative Updates for the Matrix Mechanism: A Collaborative Proof with Gemini 3
- 提出迭代更新公式，基于对角加权矩阵平方根结构优化目标函数。
- 证明该迭代单调收敛至唯一全局最优解，解决此前文献遗留问题。
- 展示大模型在数学证明中的辅助作用，提供人机协作的实用策略。
我们分析了一个在隐私机器学习算法空间优化中出现的不动点迭代 $v ightarrow ϕ(v)$，该问题由 DMR+22 提出，涉及带哈达玛积结构的正则化核范数优化。证明了迭代 $v^{(k+1)} = \text{diag}((D_{v^{(k)}}^{1/2} M D_{v^{(k)}}^{1/2})^{1/2})$ 单调收敛至势能函数 $J(v) = 2 \text{Tr}((D_v^{1/2} M D_v^{1/2})^{1/2}) - \sum v_i$ 的唯一全局最优解，解决了该文献留下的开放问题。该证明的主要部分由 Gemini 3 提供，经修正与干预后完成。此外，Gemini 3 还草拟了本文初稿。因此，本文不仅填补了文献空白，也反映了人工智能在数学研究中的实际应用价值。文中附有提示工程过程的简要叙述及与 AI 协作证明数学的若干原则。
原文摘要 · Abstract (English)
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.
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