[Paper Review] Envy-Free Classification

Maria-Florina Balcan, Travis Dick+2 more|arXiv (Cornell University)|Jan 1, 2019

Game Theory and Voting Systems8 citations

TL;DR

This paper introduces envy-free classification as a fairness criterion in machine learning, where individuals prefer their own classification outcome over others'. It proves that a small sample suffices to ensure generalizability of envy-freeness when using a mixture of deterministic classifiers from a family with low Natarajan dimension, establishing theoretical guarantees for fair and generalizable classification under heterogeneous preferences.

ABSTRACT

In classic fair division problems such as cake cutting and rent division, envy-freeness requires that each individual (weakly) prefer his allocation to anyone else's. On a conceptual level, we argue that envy-freeness also provides a compelling notion of fairness for classification tasks, especially when individuals have heterogeneous preferences. Our technical focus is the generalizability of envy-free classification, i.e., understanding whether a classifier that is envy free on a sample would be almost envy free with respect to the underlying distribution with high probability. Our main result establishes that a small sample is sufficient to achieve such guarantees, when the classifier in question is a mixture of deterministic classifiers that belong to a family of low Natarajan dimension.

Motivation & Objective

• To establish envy-freeness as a meaningful fairness criterion in classification tasks with heterogeneous individual preferences.
• To investigate whether a classifier that is envy-free on a finite sample remains approximately envy-free with respect to the underlying data distribution.
• To provide theoretical guarantees on the generalizability of envy-free classification under mild complexity assumptions on the classifier family.
• To show that a small sample size is sufficient to achieve high-probability generalization to the underlying distribution when using mixtures of low-Natarajan-dimension classifiers.

Proposed method

• Formalizes envy-freeness in classification as each individual weakly preferring their own outcome to any other individual's.
• Models the classifier as a mixture of deterministic classifiers drawn from a family with low Natarajan dimension, enabling complexity control.
• Uses sample-based empirical envy-freeness as a proxy for population-level fairness, analyzing the generalization gap.
• Applies tools from statistical learning theory to bound the probability that a sample-envy-free classifier fails to be approximately envy-free on the full distribution.
• Leverages the low Natarajan dimension to control the complexity of the hypothesis class, ensuring generalization with small samples.
• Derives high-probability generalization bounds that guarantee near-envy-freeness on the underlying distribution with high confidence.

Experimental results

Research questions

• RQ1 Can envy-freeness serve as a viable fairness criterion in classification tasks where individuals have heterogeneous preferences?
• RQ2 How well does empirical envy-freeness on a finite sample generalize to the underlying data distribution?
• RQ3 What complexity assumptions on the classifier family are sufficient to ensure generalizability of envy-freeness with small samples?
• RQ4 How does the Natarajan dimension of the classifier family affect the sample complexity of achieving generalizable envy-freeness?

Key findings

• A classifier that is envy-free on a finite sample is likely to be approximately envy-free with respect to the underlying distribution, with high probability.
• The required sample size for generalization depends on the Natarajan dimension of the classifier family, not on the number of individuals.
• Low Natarajan dimension ensures that the hypothesis class is sufficiently simple to allow generalization from small samples.
• Generalization bounds are derived that guarantee high-probability near-envy-freeness under mild assumptions on the classifier family.
• The results show that envy-freeness can be achieved with small samples when using mixtures of deterministic classifiers from low-complexity families.
• The theoretical framework provides a foundation for designing fair classifiers that are both empirically envy-free and generalizable to unseen data.