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.

翻译：本文研究多个相关的在线计算问题。首先，我们探讨如何回答在线到达的统计查询序列，并即时以差分隐私方式给出答案。已知的矩阵分解机制能通过查询矩阵的$γ_2$范数界回答一组统计查询，但要求所有查询预先已知。我们证明，对于非适应性选择的查询，在线场景中几乎可实现相同的误差界。为此，我们提出一种在线分解算法，其竞争性能与最佳离线分解相匹敌（至多对数因子差异）。在线矩阵分解问题中，矩阵的新行$q_t$在每个时间步$t$到达，算法需维护分解$L_tR_t=Q_t$，使得每次向$R_t$追加若干行，并输出新行$\ell_t$满足$\ell_tR_t=q_t$。即使未来到达的行数未知，我们的算法仍能在此在线过程中保持竞争性。作为另一应用，我们给出一种在线差异最小化算法，其差异值与$γ_2$范数（以及遗传差异）的竞争比达到对数因子级别。