Private Rate-Constrained Optimization with Applications to Fair Learning

TL;DR

The paper addresses private learning under rate constraints, where conventional DP-SGD struggles due to non-decomposability of rate constraints. It introduces RaCO-DP, a differentially private SGDA variant that uses generalized rate constraints and privately computed histograms to evaluate constraints at each step, with post-processing ensuring no extra privacy cost. The authors provide convergence guarantees to an approximate stationary point and demonstrate strong empirical performance, notably Pareto-dominance over state-of-the-art private baselines on demographic parity and false negative rate constraints across multiple datasets. The work advances the privacy-utility-fairness landscape by enabling flexible, multiclass rate constraints with direct constraint control and substantial computational efficiency, suggesting broad practical impact for private, fair learning in sensitive domains.

Abstract

Many problems in trustworthy ML can be formulated as minimization of the model error under constraints on the prediction rates of the model for suitably-chosen marginals, including most group fairness constraints (demographic parity, equality of odds, etc.). In this work, we study such constrained minimization problems under differential privacy (DP). Standard DP optimization techniques like DP-SGD rely on the loss function's decomposability into per-sample contributions. However, rate constraints introduce inter-sample dependencies, violating the decomposability requirement. To address this, we develop RaCO-DP, a DP variant of the Stochastic Gradient Descent-Ascent (SGDA) algorithm which solves the Lagrangian formulation of rate constraint problems. We demonstrate that the additional privacy cost of incorporating these constraints reduces to privately estimating a histogram over the mini-batch at each optimization step. We prove the convergence of our algorithm through a novel analysis of SGDA that leverages the linear structure of the dual parameter. Finally, empirical results on learning under group fairness constraints demonstrate that our method Pareto-dominates existing private learning approaches in fairness-utility trade-offs.

Figures (8)

• Figure 1: Each rate constraint of the form \ref{['eq:rate-soft-general']} builds local datasets based on the global partition. A class-1 (class-0) prediction is shown with a blue (red) square. Prediction rates $P_0, P_1$ are shown as fractions. As an example, let $D_1$, $D_2$, and $D_3$ be the set of Hispanic, Black, and Caucasian individuals in the dataset, respectively. Constraint $\Gamma_1$ builds its local datasets as $\{\{D_1\}, \{D_2 \cup D_3\}\}$, i.e. {Hispanic, Non-Hispanic}, from the global partition $\{D_1, D_2, D_3\}$ using the set of index subsets $\mathcal{I}_1 = \{I_1, I_2\}$.
• Figure 2: (Left) Disparity-Error trade-off curves of DP fair training algorithms on Adult under demographic parity constraints. RaCO-DP Pareto dominates the SOTA method (DP-FERMI), closing the optimality gap with non-private (SGDA). (Right) RaCO-DP vs. Non-Private SGDA on ACSEmployment with 18 constraints, showing that RaCO-DP adapts to multiple sensitive groups.
• Figure 3: Satisfiability on Adult. Trade-off between test error and constraint violation for different target values $\gamma$ (dashed lines), averaged over 20 runs. RaCO-DP achieves demographic parity constraint satisfaction.
• Figure 4: Demographic Parity Constraint
• Figure 5: False Negative Rate Constraint

Theorems & Definitions (38)

• Definition 2.1: Differential Privacy
• Definition 2.2: GDA
• Remark 1
• Definition 3.1: Demographic Parity
• Remark 2
• Remark 3
• Remark 4
• Theorem 4.1
• Definition 5.1
• Theorem 5.2: Informal