Estimating the empirical distribution of a scalar-valued data set is a basic and fundamental task. In this paper, we tackle the problem of estimating an empirical distribution in a setting with two challenging features. First, the algorithm does not directly observe the data; instead, it only asks a limited number of threshold queries about each sample. Second, the data are not assumed to be independent and identically distributed; instead, we allow for an arbitrary process generating the samples, including an adaptive adversary. These considerations are relevant, for example, when modeling a seller experimenting with posted prices to estimate the distribution of consumers' willingness to pay for a product: offering a price and observing a consumer's purchase decision is equivalent to asking a single threshold query about their value, and the distribution of consumers' values may be non-stationary over time, as early adopters may differ markedly from late adopters.
Our main result quantifies, to within a constant factor, the sample complexity of estimating the empirical CDF of a sequence of elements of $[n]$, up to $\varepsilon$ additive error, using one threshold query per sample. The complexity depends only logarithmically on $n$, and our result can be interpreted as extending the existing logarithmic-complexity results for noisy binary search to the more challenging setting where noise is non-stochastic. Along the way to designing our algorithm, we consider a more general model in which the algorithm is allowed to make a limited number of simultaneous threshold queries on each sample. We solve this problem using Blackwell's Approachability Theorem and the exponential weights method. As a side result of independent interest, we characterize the minimum number of simultaneous threshold queries required by deterministic CDF estimation algorithms.
Private Synthetic Data for Multitask Learning and Marginal Queries
University of Minnesota（明尼苏达大学）；Amazon AWS AI/ML（亚马逊AWS人工智能与机器学习部门）；Columbia University（哥伦比亚大学）；University of Pennsylvania（宾夕法尼亚大学）；Carnegie Mellon University（卡内基梅隆大学）
Private-RAG: Answering Multiple Queries with LLMs while Keeping Your Data Private
Computer Science and Engineering University of California, San Diego（计算机科学与工程大学加州大学圣地亚哥分校）；Halıcıoğlu Data Science Institute University of California, San Diego（Halıcıoğlu数据科学研究所加州大学圣地亚哥分校）；Department of Computer Science University of California, Los Angeles（计算机科学系加州大学洛杉矶分校）
