Algorithms and Analysis for Optimizing Robust Objectives in Fair Machine Learning

TL;DR

The paper addresses fairness under epistemic uncertainty by modeling a Rawlsian maximin game between a Daemon and an Angel, linking robustness to egalitarian fairness. It derives a hierarchy of robust welfare and malfare objectives, including generalized Gini and Gini–power-mean classes, and shows they arise as solution concepts under various constrained adversary settings. The authors demonstrate that these robust objectives are tractable to optimize via maximin or gradient-based methods under mild convexity assumptions and establish Lipschitz/Hölder continuity and generalization bounds, ensuring reliable performance in fair allocation and learning tasks. By connecting philosophical notions of justice with practical optimization, the work provides principled tools for fair, robust decision-making in ML and allocation problems under demographic uncertainty and distributional shifts.

Abstract

The original position or veil of ignorance argument of John Rawls, perhaps the most famous argument for egalitarianism, states that our concept of fairness, justice, or welfare should be decided from behind a veil of ignorance, and thus must consider everyone impartially (invariant to our identity). This can be posed as a zero-sum game, where a Daemon constructs a world, and an adversarial Angel then places the Daemon into the world. This game incentivizes the Daemon to maximize the minimum utility over all people (i.e., to maximize egalitarian welfare). In some sense, this is the most extreme form of risk aversion or robustness, and we show that by weakening the Angel, milder robust objectives arise, which we argue are effective robust proxies for fair learning or allocation tasks. In particular, the utilitarian, Gini, and power-mean welfare concepts arise from special cases of the adversarial game, which has philosophical implications for the understanding of each of these concepts. We also motivate a new fairness concept that essentially fuses the nonlinearity of the power-mean with the piecewise nature of the Gini class. Then, exploiting the relationship between fairness and robustness, we show that these robust fairness concepts can all be efficiently optimized under mild conditions via standard maximin optimization techniques. Finally, we show that such methods apply in machine learning contexts, and moreover we show generalization bounds for robust fair machine learning tasks.

Figures (6)

• Figure 1: Metaphoric depiction and game-theoretic description of the Rawlsian original position game. A weak Dæ mon (left) plays against an all-seeing Angel (right).
• Figure 2: Metaphoric depiction and game-theoretic description of the modified Rawlsian original position game, with restricted Angel action space. A Dæ mon (left) plays against a comparably powerful Angel (right).
• Figure 3: A simplicial plot over $\triangle_{3}$ of the robustness sets defined by intersection with the $\mathcal{L}_{\infty}$, $\mathcal{L}_{2}$, and $\mathcal{L}_{1}$ norm balls of radius $\frac{1}{5}$ around the point $\bm{w}^{*} = \left\langle \frac{1}{4}, \frac{1}{4}, \frac{1}{2} \right\rangle$. The boundaries of the $\mathcal{L}_{\infty}$, $\mathcal{L}_{2}$, and $\mathcal{L}_{1}$ balls are plotted in solid, dashed, and dotted lines, respectively. Assuming positive radius $r$ such that each $\bm{w}^{*}$-centered norm ball is contained by the unit hypercube, i.e., $\lVert\bm{w}^{*}\rVert_{\infty} \leq \lVert\bm{w}^{*}\rVert_{2} \leq \lVert\bm{w}^{*}\rVert_{1} \leq r$, intersection with the unit simplex yields an equilateral triangular, circular, or hexagonal region, respectively, with $g=3$. In higher dimensions, the regions become simplicial, hyperspherical, or regular-polytopal, respectively.
• Figure 4: Metaphoric depiction and game-theoretic description of the altruistic Dæ mon original position game. A social-planner Dæ mon (left) plays a zero-sum game against an adversarial Angel (right). Both the aggregator-function and the utility-transform formulations of the game are presented.
• Figure 5: Metaphoric depiction and game-theoretic description of the altruistic Angel original position game. A self-interested Dæ mon (left) is coerced into altruistic play by a social-planner Angel (right).

Theorems & Definitions (18)

• Definition 3.1: Weighted Power-Mean Family
• Definition 3.2: Gini Welfare and Malfare
• Lemma 4.1: Maximin Interchangeability
• proof
• Theorem 4.2: Classical Welfare and Malfare Functions as Constrained Angel Solution Concepts
• Theorem 4.3: Robust Welfare and Malfare Functions as Constrained Angel Solution Concepts
• Theorem 4.3: Strategic Gameplay from Nonlinear Objectives
• Theorem 4.3: Power-Means as Utility Transforms
• Definition 4.4: The $\bm{w}^{\uparrow}$-$p$ Gini Power-Mean Class
• Theorem 4.4: Strategic Gameplay in Altruistic Angel Games