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In differential privacy, $\textit{continual observation}$ refers to problems in which we wish to continuously release a function of a dataset that is revealed one element at a time. The challenge is to maintain a good approximation while keeping the combined output over all time steps differentially private. In the special case of $\textit{continual counting}$ we seek to approximate a sum of binary input elements. This problem has received considerable attention lately, in part due to its relevance in implementations of differentially private stochastic gradient descent. $\textit{Factorization mechanisms}$ are the leading approach to continual counting, but the best such mechanisms do not work well in $\textit{streaming}$ settings since they require space proportional to the size of the input. In this paper, we present a simple approach to approximating factorization mechanisms in low space via $\textit{binning}$, where adjacent matrix entries with similar values are changed to be identical in such a way that a matrix-vector product can be maintained in sublinear space. Our approach has provable sublinear space guarantees for a class of lower triangular matrices whose entries are monotonically decreasing away from the diagonal. We show empirically that even with very low space usage we are able to closely match, and sometimes surpass, the performance of asymptotically optimal factorization mechanisms. Recently, and independently of our work, Dvijotham et al. have also suggested an approach to implementing factorization mechanisms in a streaming setting. Their work differs from ours in several respects: It only addresses factorization into $\textit{Toeplitz}$ matrices, only considers $\textit{maximum}$ error, and uses a different technique based on rational function approximation that seems less versatile than our binning approach. |
