# Online Learning with an Unknown Fairness Metric

Stephen Gillen, Christopher Jung, Michael Kearns, Aaron Roth

[arXiv]

We consider the problem of online learning in the linear contextual bandits setting,
but in which there are also strong *individual fairness* constraints governed by an unknown
similarity metric. These constraints demand that we select similar actions or individuals
with approximately equal probability, which may be at odds with optimizing reward,
thus modeling settings where profit and social policy are in tension.
We assume we learn about an unknown
Mahalanobis similarity metric
from only weak feedback that identifies fairness violations, but does not quantify their extent. This is intended to represent the interventions of a regulator who "knows unfairness when he sees it"
but nevertheless cannot enunciate a quantitative fairness metric over individuals.
Our main result is an algorithm
in the adversarial context setting that
has a number of
fairness violations that depends only logarithmically on T,
while obtaining an optimal O(sqrt{T}) regret bound to the best fair policy.