{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/how-bad-is-worst-case-data-if-you-know-where","title":"Worst-Case Analysis for Randomly Collected Data","arxiv_id":"1911.03605","date":"2019-11-09","proceeding":"NeurIPS 2020 12","authors":["Justin Y. Chen","Gregory Valiant","Paul Valiant"],"abstract":"We introduce a framework for statistical estimation that leverages knowledge of how samples are collected but makes no distributional assumptions on the data values. Specifically, we consider a population of elements $[n]={1,\\ldots,n}$ with corresponding data values $x_1,\\ldots,x_n$. We observe the values for a \"sample\" set $A \\subset [n]$ and wish to estimate some statistic of the values for a \"target\" set $B \\subset [n]$ where $B$ could be the entire set. Crucially, we assume that the sets $A$ and $B$ are drawn according to some known distribution $P$ over pairs of subsets of $[n]$. A given estimation algorithm is evaluated based on its \"worst-case, expected error\" where the expectation is with respect to the distribution $P$ from which the sample $A$ and target sets $B$ are drawn, and the worst-case is with respect to the data values $x_1,\\ldots,x_n$. Within this framework, we give an efficient algorithm for estimating the target mean that returns a weighted combination of the sample values--where the weights are functions of the distribution $P$ and the sample and target sets $A$, $B$--and show that the worst-case expected error achieved by this algorithm is at most a multiplicative $\\pi/2$ factor worse than the optimal of such algorithms. The algorithm and proof leverage a surprising connection to the Grothendieck problem. This framework, which makes no distributional assumptions on the data values but rather relies on knowledge of the data collection process, is a significant departure from typical estimation and introduces a uniform algorithmic analysis for the many natural settings where membership in a sample may be correlated with data values, such as when sampling probabilities vary as in \"importance sampling\", when individuals are recruited into a sample via a social network as in \"snowball sampling\", or when samples have chronological structure as in \"selective prediction\".","url_abs":"https://arxiv.org/abs/1911.03605v2","url_pdf":"https://arxiv.org/pdf/1911.03605v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"how-bad-is-worst-case-data-if-you-know-where","repo_url":"https://github.com/justc2/worst-case-randomly-collected","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}