Online Matrix Factorization, Online Private Query Release, and Online Discrepancy Minimization
It provides the first online algorithms for these problems with guarantees matching offline bounds, benefiting applications requiring real-time responses with privacy or discrepancy constraints.
The paper addresses online matrix factorization, online private query release, and online discrepancy minimization, achieving error bounds competitive with offline algorithms up to logarithmic factors.
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.