U.S. discrimination law can impose liability on firms that fail to adopt a less discriminatory alternative (LDA): a decision policy that achieves the same business objectives while reducing disparate impact on legally protected groups. Recent scholarship argues that this doctrine has direct implications for algorithmic decision-making in high-stakes domains such as employment, lending, and housing, potentially obligating firms to search for "less discriminatory algorithms" (Black et al., 2024). Regulators have at times encouraged proactive LDA searches, reinforcing the expectation of a good-faith effort to identify equally performant models with lower disparate impact. Model multiplicity makes such searches plausible: retraining with different random seeds can yield models with comparable predictive performance but materially different disparate impacts. Yet firms cannot retrain indefinitely, raising a central question: when is the search sufficient to demonstrate good faith? We formalize LDA search under multiplicity as an optimal stopping problem in which a developer seeks to produce evidence that further search is unlikely to yield meaningful improvements. Our main contribution is an adaptive stopping algorithm that provides a high-probability upper bound on the best disparate-impact gains attainable through continued retraining, enabling developers to certify (e.g., to a court) that additional search is unlikely to help. We also show how stronger distributional assumptions over the model space can yield tighter bounds, and we validate the approach on real-world credit and housing datasets.

翻译：美国反歧视法可能对未能采用较少歧视性替代方案（LDA）的企业施加责任：这是一种在实现相同商业目标的同时，减少对受法律保护群体差异性影响的决策策略。近期学术研究认为，这一原则对就业、贷款和住房等高风险领域中的算法决策具有直接影响，可能迫使企业寻找"较少歧视性算法"（Black 等，2024）。监管机构有时鼓励主动进行 LDA 搜索，强化了以善意努力识别性能相当但差异性影响较低的模型的期望。模型多重性使此类搜索成为可能：使用不同随机种子重新训练可在预测性能相当的情况下，产生差异性影响显著不同的模型。然而，企业无法无限期地重新训练，这引出了一个核心问题：何时搜索足以证明善意？我们将多重性下的 LDA 搜索形式化为一个最优停止问题，其中开发者寻求证明进一步搜索不太可能产生有意义的改进。我们的主要贡献在于提出一种自适应停止算法，该算法为通过持续重新训练可获得的最佳差异性影响改进提供了高概率上界，使开发者能够向（例如法院）证明额外搜索不太可能有帮助。我们还展示了模型空间上的更强分布假设如何产生更紧凑的界限，并在真实世界的信贷和住房数据集上验证了该方法。