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Register a new userWorst-Case Analysis for Randomly Collected Data
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Justin Chen
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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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.
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One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the best for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its worst performance on any input of a given size, implicitly advocating for the algorithm with the best-possible worst-case performance. Strong worst-case guarantees are the holy grail of algorithm design, providing an application-agnostic certification of an algorithms robustly good performance. However, for many fundamental problems and performance measures, such guarantees are impossible and a more nuanced analysis approach is called for. This chapter surveys several alternatives to worst-case analysis that are discussed in detail later in the book.
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