Normalized Square Root: Sharper Matrix Factorization Bounds for Differentially Private Continual Counting

2025-09-17

AI Summary

This work addresses the long-standing problem of establishing tight bounds on the factorization norms $gamma_2$ and $gamma_F$ of the lower-triangular all-ones matrix $M_{ ext{count}}$, which arises in theoretical accuracy analysis of continual counting under differential privacy. Prior bounds exhibited a significant gap—approximately $0.507$ for $gamma_2$—with Mathias’ classical upper bound remaining unimproved for decades. We resolve this by constructing an explicit matrix decomposition, integrating normalized square-root transformation, harmonic analysis, and numerical optimization. Our analysis yields tightened bounds: $gamma_2(M_{ ext{count}}) in left[0.701 + frac{log n}{pi},, 0.846 + frac{log n}{pi} ight]$, reducing the gap to $0.145$; and $gamma_F(M_{ ext{count}}) in left[0.701 + frac{log n}{pi},, 0.748 + frac{log n}{pi} ight]$, with a mere $0.047$ gap. These improvements substantially enhance theoretical accuracy guarantees for private continual counting and private training of deep neural networks.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The factorization norms of the lower-triangular all-ones $n imes n$ matrix, $γ_2(M_{count})$ and $γ_{F}(M_{count})$, play a central role in differential privacy as they are used to give theoretical justification of the accuracy of the only known production-level private training algorithm of deep neural networks by Google. Prior to this work, the best known upper bound on $γ_2(M_{count})$ was $1 + frac{log n}π$ by Mathias (Linear Algebra and Applications, 1993), and the best known lower bound was $frac{1}π(2 + log(frac{2n+1}{3})) approx 0.507 + frac{log n}π$ (Matoušek, Nikolov, Talwar, IMRN 2020), where $log$ denotes the natural logarithm. Recently, Henzinger and Upadhyay (SODA 2025) gave the first explicit factorization that meets the bound of Mathias (1993) and asked whether there exists an explicit factorization that improves on Mathias' bound. We answer this question in the affirmative. Additionally, we improve the lower bound significantly. More specifically, we show that $$ 0.701 + frac{log n}π + o(1) ;leq; γ_2(M_{count}) ;leq; 0.846 + frac{log n}π + o(1). $$ That is, we reduce the gap between the upper and lower bound to $0.14 + o(1)$. We also show that our factors achieve a better upper bound for $γ_{F}(M_{count})$ compared to prior work, and we establish an improved lower bound: $$ 0.701 + frac{log n}π + o(1) ;leq; γ_{F}(M_{count}) ;leq; 0.748 + frac{log n}π + o(1). $$ That is, the gap between the lower and upper bound provided by our explicit factorization is $0.047 + o(1)$.

Problem

• Improving factorization norm bounds for differential privacy
• Narrowing the gap between upper and lower bounds
• Providing explicit factorization for sharper accuracy guarantees

Innovation

• Developed explicit matrix factorization method
• Improved upper and lower bounds
• Reduced gap between bounds significantly