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

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.

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

Abstract

We analyze a fixed-point iteration 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 converges monotonically to the unique global optimizer of the potential function , 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.
Paper Structure (13 sections, 5 theorems, 25 equations)

This paper contains 13 sections, 5 theorems, 25 equations.

Table of Contents

  1. The Problem and its Brief History
  2. Theoretical Analysis
  3. Basic Properties
  4. The Scalar Case ($N=1$)
  5. The Matrix Case
  6. Step 1: Construction of the Minorizer (Surrogate Function)
  7. Step 2: Monotonic Ascent via the Surrogate
  8. The Surrogate Maximizer vs. The Algorithm Update.
  9. Proof of Ascent.
  10. Conclusion
  11. Acknowledgments
  12. A User's Journey
  13. On the subtleties of checking correctness

Key Result

Lemma 1

For any real $A$, the nuclear norm can be characterized in the following manner: where $\mathcal{U}(N)$ is the set of unitary matrices. The supremum is attained when $U$ is the unitary factor from the polar decomposition of $A$ (see Section 7.4 of handj).

Theorems & Definitions (12)

  • Definition 1: Nuclear Norm
  • Lemma 1
  • proof
  • Lemma 2: Strict Concavity
  • proof
  • Lemma 3: Coercivity
  • proof
  • Lemma 4: Positive Invariance
  • proof
  • Theorem 1
  • ...and 2 more