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

TL;DR

This paper develops online algorithms for matrix factorization, private query answering, and discrepancy minimization, achieving error bounds comparable to offline methods while handling data arriving sequentially.

Contribution

It introduces online algorithms that match offline error bounds for matrix factorization and discrepancy minimization, with applications to differential privacy and online data processing.

Findings

Online factorization algorithm matches offline bounds up to logarithmic factors.

Achieves differential privacy for online statistical queries with near-optimal error.

Provides online discrepancy minimization competitive with 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 $\gamma_2$ 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 $\gamma_2$ norm (and also against hereditary discrepancy) up to logarithmic factors.