Clone-Robust Weights in Metric Spaces: Handling Redundancy Bias for Benchmark Aggregation

TL;DR

This paper introduces a theoretical framework for clone-proof weighting functions in metric spaces, ensuring robustness against adversarial redundancy bias in applications like data aggregation and domain adaptation.

Contribution

It proposes a set of axioms for clone-proof weights, extends the maximum uncertainty principle to metric spaces, and provides methods for constructing such weights in Euclidean spaces.

Findings

Axioms for clone-proof weights are formalized.

Existence of such weights in Euclidean spaces is established.

A general construction method for clone-proof weights is proposed.

Abstract

We are given a set of elements in a metric space. The distribution of the elements is arbitrary, possibly adversarial. Can we weigh the elements in a way that is resistant to such (adversarial) manipulations? This problem arises in various contexts. For instance, the elements could represent data points, requiring robust domain adaptation. Alternatively, they might represent tasks to be aggregated into a benchmark; or questions about personal political opinions in voting advice applications. This article introduces a theoretical framework for dealing with such problems. We propose clone-proof weighting functions as a solution concept. These functions distribute importance across elements of a set such that similar objects (``clones'') share (some of) their weights, thus avoiding a potential bias introduced by their multiplicity. Our framework extends the maximum uncertainty principle to accommodate general metric spaces and includes a set of axioms -- symmetry, continuity, and clone-proofness -- that guide the construction of weighting functions. Finally, we address the existence of weighting functions satisfying our axioms in the significant case of Euclidean spaces and propose a general method for their construction.