An Information Geometric Approach to Fairness With Equalized Odds Constraint

TL;DR

The paper tackles fair representation learning under equalized odds when the sensitive attribute S and task T are not directly observed. It introduces a local χ^2 constraint on P_{S|Y} and uses information-geometric, second-order approximations to replace mutual information terms, reducing the design to a quadratic program with a compression constraint. By analyzing the resulting matrices W^{T;Y} and W^{X;Y}, it derives singular-value–based lower bounds that yield a simple, low-complexity design (P2) and demonstrates near-optimality in certain regimes via a numerical example. The work offers a practical framework for balancing utility, fairness, and compression in privacy-aware settings, with potential for generalization to other fairness criteria and risk models.

Abstract

We study the statistical design of a fair mechanism that attains equalized odds, where an agent uses some useful data (database) $X$ to solve a task $T$. Since both $X$ and $T$ are correlated with some latent sensitive attribute $S$, the agent designs a representation $Y$ that satisfies an equalized odds, that is, such that $I(Y;S|T) =0$. In contrast to our previous work, we assume here that the agent has no direct access to $S$ and $T$; hence, the Markov chains $S - X - Y$ and $T - X - Y$ hold. Furthermore, we impose a geometric structure on the conditional distribution $P_{S|Y}$, allowing $Y$ and $S$ to have a small correlation, bounded by a threshold. When the threshold is small, concepts from information geometry allow us to approximate mutual information and reformulate the fair mechanism design problem as a quadratic program with closed-form solutions under certain constraints. For other cases, we derive simple, low-complexity lower bounds based on the maximum singular value and vector of a matrix. Finally, we compare our designs with the optimal solution in a numerical example.

Figures (2)

• Figure 1: Data representation with an equalized odds constraint. The goal is to design a representation $Y$ of the data $X$ that is useful for the task $T$, is compressed, and leaks within a controlled threshold of the sensitive attribute $S$. Here, the agent has no direct access to $S$ and $T$.
• Figure 2: Comparing our method with the optimal solutions of \ref{['problem']} and \ref{['prob2']}. In high-privacy regimes, $P_2$ is close to \ref{['problem']} (via exhaustive search), and $g^{r}_{\epsilon}(P_{S,X,T})$ dominates $g^{r}_{\epsilon,\chi^2}(P_{S,X,T})$.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Lemma 1
• Theorem 1
• Remark 3
• Proposition 1
• Theorem 2
• Remark 4