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Optimistic Feasible Search for Closed-Loop Fair Threshold Decision-Making

Wenzhang Du

TL;DR

The paper tackles learning a 1D threshold policy under demographic parity and optional service-rate constraints in a non-stationary, closed-loop setting. It introduces Optimistic Feasible Search (OFS), a grid-based, optimism-driven method that screens feasibility via confidence bounds and selects thresholds that maximize optimistic reward, falling back to minimizing optimistic constraint violation when necessary. Across a synthetic MVE-S and two semi-synthetic German Credit and COMPAS benchmarks, OFS consistently achieves higher tail utility with smaller cumulative constraint violations than unconstrained and primal–dual baselines, and approaches oracle performance on feasible fixed-threshold comparisons. The approach is interpretable, reproducible, and demonstrates promising applicability for fairness-aware decision-making in feedback-driven environments, with a clear path for extensions beyond 1D thresholds.

Abstract

Closed-loop decision-making systems (e.g., lending, screening, or recidivism risk assessment) often operate under fairness and service constraints while inducing feedback effects: decisions change who appears in the future, yielding non-stationary data and potentially amplifying disparities. We study online learning of a one-dimensional threshold policy from bandit feedback under demographic parity (DP) and, optionally, service-rate constraints. The learner observes only a scalar score each round and selects a threshold; reward and constraint residuals are revealed only for the chosen threshold. We propose Optimistic Feasible Search (OFS), a simple grid-based method that maintains confidence bounds for reward and constraint residuals for each candidate threshold. At each round, OFS selects a threshold that appears feasible under confidence bounds and, among those, maximizes optimistic reward; if no threshold appears feasible, OFS selects the threshold minimizing optimistic constraint violation. This design directly targets feasible high-utility thresholds and is particularly effective for low-dimensional, interpretable policy classes where discretization is natural. We evaluate OFS on (i) a synthetic closed-loop benchmark with stable contraction dynamics and (ii) two semi-synthetic closed-loop benchmarks grounded in German Credit and COMPAS, constructed by training a score model and feeding group-dependent acceptance decisions back into population composition. Across all environments, OFS achieves higher reward with smaller cumulative constraint violation than unconstrained and primal-dual bandit baselines, and is near-oracle relative to the best feasible fixed threshold under the same sweep procedure. Experiments are reproducible and organized with double-blind-friendly relative outputs.

Optimistic Feasible Search for Closed-Loop Fair Threshold Decision-Making

TL;DR

The paper tackles learning a 1D threshold policy under demographic parity and optional service-rate constraints in a non-stationary, closed-loop setting. It introduces Optimistic Feasible Search (OFS), a grid-based, optimism-driven method that screens feasibility via confidence bounds and selects thresholds that maximize optimistic reward, falling back to minimizing optimistic constraint violation when necessary. Across a synthetic MVE-S and two semi-synthetic German Credit and COMPAS benchmarks, OFS consistently achieves higher tail utility with smaller cumulative constraint violations than unconstrained and primal–dual baselines, and approaches oracle performance on feasible fixed-threshold comparisons. The approach is interpretable, reproducible, and demonstrates promising applicability for fairness-aware decision-making in feedback-driven environments, with a clear path for extensions beyond 1D thresholds.

Abstract

Closed-loop decision-making systems (e.g., lending, screening, or recidivism risk assessment) often operate under fairness and service constraints while inducing feedback effects: decisions change who appears in the future, yielding non-stationary data and potentially amplifying disparities. We study online learning of a one-dimensional threshold policy from bandit feedback under demographic parity (DP) and, optionally, service-rate constraints. The learner observes only a scalar score each round and selects a threshold; reward and constraint residuals are revealed only for the chosen threshold. We propose Optimistic Feasible Search (OFS), a simple grid-based method that maintains confidence bounds for reward and constraint residuals for each candidate threshold. At each round, OFS selects a threshold that appears feasible under confidence bounds and, among those, maximizes optimistic reward; if no threshold appears feasible, OFS selects the threshold minimizing optimistic constraint violation. This design directly targets feasible high-utility thresholds and is particularly effective for low-dimensional, interpretable policy classes where discretization is natural. We evaluate OFS on (i) a synthetic closed-loop benchmark with stable contraction dynamics and (ii) two semi-synthetic closed-loop benchmarks grounded in German Credit and COMPAS, constructed by training a score model and feeding group-dependent acceptance decisions back into population composition. Across all environments, OFS achieves higher reward with smaller cumulative constraint violation than unconstrained and primal-dual bandit baselines, and is near-oracle relative to the best feasible fixed threshold under the same sweep procedure. Experiments are reproducible and organized with double-blind-friendly relative outputs.
Paper Structure (24 sections, 7 equations, 3 figures, 6 tables, 1 algorithm)

This paper contains 24 sections, 7 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Utility
  4. Constraints
  5. Closed-loop dynamics and performance metrics
  6. Optimistic Feasible Search (OFS)
  7. Algorithm
  8. Discussion and intuition
  9. Theoretical perspective (sketch)
  10. Baselines
  11. Closed-loop Benchmarks
  12. Synthetic benchmark (MVE-S)
  13. Semi-synthetic benchmarks (German Credit and COMPAS)
  14. Experiments
  15. Protocol and implementation details
  16. ...and 9 more sections

Figures (3)

  • Figure 1: MVE-S closed-loop learning dynamics (mean$\pm$std over seeds). OFS achieves higher reward with dramatically smaller DP gap and cumulative violation.
  • Figure 2: Semi-synthetic closed-loop learning curves on German Credit and COMPAS. OFS consistently reduces constraint violations while preserving high reward.
  • Figure 3: Oracle tradeoff curves (fixed threshold steady-state evaluation). Each curve is obtained by sweeping thresholds and estimating steady-state reward and DP gap; the oracle best feasible point is the feasible point with maximal reward. OFS operates online yet approaches oracle-quality feasible thresholds.