# A Smooth Binary Mechanism for Efficient Private Continual Observation

arXiv: https://arxiv.org/abs/2306.09666 · HTML: https://mlsanity.com/paper/2306.09666/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 16 Jun 2023.

## Builds on (most cited first, 9 of 9)

- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Practical and Private (Deep) Learning without Sampling or Shuffling](https://mlsanity.com/paper/2103.00039/) (2021)
- [Improved Differential Privacy for SGD via Optimal Private Linear Operators on Adaptive Streams](https://mlsanity.com/paper/2202.08312/) (2022)
- [(Amplified) Banded Matrix Factorization: A unified approach to private training](https://mlsanity.com/paper/2306.08153/) (2023)
- [Multi-Epoch Matrix Factorization Mechanisms for Private Machine Learning](https://mlsanity.com/paper/2211.06530/) (2022)
- [Almost Tight Error Bounds on Differentially Private Continual Counting](https://mlsanity.com/paper/2211.05006/) (2022)
- [Constant matters: Fine-grained Complexity of Differentially Private Continual Observation](https://mlsanity.com/paper/2202.11205/) (2022)
- [Differentially Private Histograms under Continual Observation: Streaming Selection into the Unknown](https://mlsanity.com/paper/2103.16787/) (2021)
- [Frequency Estimation Under Multiparty Differential Privacy: One-shot and Streaming](https://mlsanity.com/paper/2104.01808/) (2021)

## Built on by (most cited first, 12 of 12)

- [Efficient and Near-Optimal Noise Generation for Streaming Differential Privacy](https://mlsanity.com/paper/2404.16706/) (2024)
- [A Unifying Framework for Differentially Private Sums under Continual Observation](https://mlsanity.com/paper/2307.08970/) (2023)
- [A Hassle-free Algorithm for Private Learning in Practice: Don't Use Tree Aggregation, Use BLTs](https://mlsanity.com/paper/2408.08868/) (2024)
- [Streaming Private Continual Counting via Binning](https://mlsanity.com/paper/2412.07093/) (2024)
- [Back to Square Roots: An Optimal Bound on the Matrix Factorization Error for Multi-Epoch Differentially Private SGD](https://mlsanity.com/paper/2505.12128/) (2025)
- [Binned Group Algebra Factorization for Differentially Private Continual Counting](https://mlsanity.com/paper/2504.04398/) (2025)
- [Private Continual Counting of Unbounded Streams](https://mlsanity.com/paper/2506.15018/) (2025)
- [Adaptive Data Analysis for Growing Data](https://mlsanity.com/paper/2405.13375/) (2024)
- [Optimal Rates for O(1)-Smooth DP-SCO with a Single Epoch and Large Batches](https://mlsanity.com/paper/2406.02716/) (2024)
- [DPSW-Sketch: A Differentially Private Sketch Framework for Frequency Estimation over Sliding Windows (Technical Report)](https://mlsanity.com/paper/2406.07953/) (2024)
- [Matrix Factorization for Practical Continual Mean Estimation Under User-Level Differential Privacy](https://mlsanity.com/paper/2601.22320/) (2026)
- [Beyond Square Roots: A Memory-Efficient Explicit Factorization for Multi-Epoch Private Learning](https://mlsanity.com/paper/2605.18379/) (2026)

## Abstract (arXiv)

In privacy under continual observation we study how to release differentially private estimates based on a dataset that evolves over time. The problem of releasing private prefix sums of x1,x2,x3,dots ∈0,1 (where the value of each xi is to be private) is particularly well-studied, and a generalized form is used in state-of-the-art methods for private stochastic gradient descent (SGD). The seminal binary mechanism privately releases the first t prefix sums with noise of variance polylogarithmic in t. Recently, Henzinger et al. and Denisov et al. showed that it is possible to improve on the binary mechanism in two ways: The variance of the noise can be reduced by a (large) constant factor, and also made more even across time steps. However, their algorithms for generating the noise distribution are not as efficient as one would like in terms of computation time and (in particular) space. We address the efficiency problem by presenting a simple alternative to the binary mechanism in which 1) generating the noise takes constant average time per value, 2) the variance is reduced by a factor about 4 compared to the binary mechanism, and 3) the noise distribution at each step is identical. Empirically, a simple Python implementation of our approach outperforms the running time of the approach of Henzinger et al., as well as an attempt to improve their algorithm using high-performance algorithms for multiplication with Toeplitz matrices.