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Machine learning models developed from real-world data can inherit potential, preexisting bias in the dataset. When these models are used to inform decisions involving human beings, fairness concerns inevitably arise. Imposing certain fairness constraints in the training of models can be effective only if appropriate criteria are applied. However, a fairness criterion can be defined/assessed only when the interaction between the decisions and the underlying population is well understood. We introduce two feedback models describing how people react when receiving machine-aided decisions and illustrate that some commonly used fairness criteria can end with undesirable consequences while reinforcing discrimination.
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A new deflation criterion for the QZ algorithm
Abstract The QZ algorithm computes the generalized Schur form of a matrix pencil. It is an iterative algorithm and, at some point, it must decide when to deflate, that is when a generalized eigenvalue has converged and to move on to another one. Choosing a deflation criterion that makes this decision is nontrivial. If it is too strict, the algorithm might waste iterations on already converged eigenvalues. If it is not strict enough, the computed eigenvalues might not have full accuracy. Additionally, the criterion should not be computationally expensive to evaluate. There are two commonly used criteria: theelementwisecriterion and thenormwisecriterion. This paper introduces a new deflation criterion based on the size of and the gap between the eigenvalues. We call this new deflation criterion thestrictcriterion. This new criterion for QZ is analogous to the criterion derived by Ahues and Tisseur for the QR algorithm. Theoretical arguments and numerical experiments suggest that the strict criterion outperforms the normwise and elementwise criteria in terms of accuracy. We also provide an example where the accuracy of the generalized eigenvalues using the elementwise or the normwise criteria is less than two digits whereas the strict criterion leads to generalized eigenvalues which are almost accurate to the working precision. Additionally, this paper evaluates some commonly used criteria for infinite eigenvalues.
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- Award ID(s):
- 2004850
- PAR ID:
- 10537787
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Numerical Linear Algebra with Applications
- Volume:
- 31
- Issue:
- 1
- ISSN:
- 1070-5325
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/nla.2524
