Constant matters: Fine-grained Complexity of Differentially Private Continual Observation
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We study fine-grained error bounds for differentially private algorithms for averaging and counting under continual observation. Our main insight is that the factorization mechanism when using lower-triangular matrices, can be used in the continual observation model. We give explicit factorizations for two fundamental matrices, namely the counting matrix M_πΌππππ and the averaging matrix M_πΊππΎππΊππΎ and show fine-grained bounds for the additive error of the resulting mechanism using the completely bounded norm (cb-norm) or factorization norm. Our bound on the cb-norm for M_πΌππππ is tight up an additive error of 1 and the bound for M_πΊππΎππΊππΎ is tight up to β 0.64. This allows us to give the first algorithm for averaging whose additive error has o(log^3/2 T) dependence. Furthermore, we are the first to give concrete error bounds for various problems under continual observation such as binary counting, maintaining a histogram, releasing an approximately cut-preserving synthetic graph, many graph-based statistics, and substring and episode counting. Finally, we present a fine-grained error bound for non-interactive local learning.
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