Online Matrix Factorization, Online Private Query Release, and Online Discrepancy Minimization

2026-05-08

AI Summary

This work addresses the problem of answering online, non-adaptive statistical queries under differential privacy when the total number of queries is unknown a priori. It proposes a novel online matrix factorization algorithm that dynamically maintains a factorized structure and extends, for the first time, the γ₂-norm-based error bounds from offline matrix factorization to the online setting. By integrating differential privacy mechanisms with online optimization techniques, the algorithm achieves a competitive ratio within a logarithmic factor of the hereditary discrepancy. Theoretical analysis demonstrates that the proposed method attains near-optimal error performance in the online regime, providing competitive guarantees—within logarithmic factors—for both the γ₂ norm and hereditary discrepancy.

Abstract

In this paper we consider several related online computation problems. First, we study answering sequences of statistical queries arriving online, and being answered immediately when they arrive with differential privacy. Known matrix factorization mechanisms can answer a set of statistical queries with error bounded by the $γ_2$ norm of their query matrix, but require that all queries are known in advance. We show that nearly the same error bounds can be achieved in the online setting for non-adaptively chosen queries. To do so, we give an online factorization algorithm that competitively matches the best offline factorization up to logarithmic factors. In the online matrix factorization problem, a new row $q_t$ of a matrix arrives at each time step $t$, and the algorithm needs to maintain a factorization $L_tR_t=Q_t$ such that at each time it appends some rows to $R_t$, and outputs a new row $\ell_t$ s.t. $\ell_tR_t=q_t$. Our algorithm maintains the competitiveness over this online process, even if the number of rows to arrive is unknown. As another application, we give an online discrepancy minimization algorithm that achieves discrepancy competitive against the $γ_2$ norm (and also against hereditary discrepancy) up to logarithmic factors.

Problem

• Online Matrix Factorization
• Differential Privacy
• Statistical Queries
• Discrepancy Minimization
• Online Algorithms

Innovation

• Online Matrix Factorization
• Differential Privacy
• Discrepancy Minimization
• γ₂ Norm
• Competitive Algorithm