We introduce a new notion of stability--which we call stable list decoding--and demonstrate its applicability in designing differentially private density estimators. This definition is weaker than global stability [ABLMM22] and is related to the notions of replicability [ILPS22] and list replicability [CMY23]. We show that if a class of distributions is stable list decodable, then it can be learned privately in the agnostic setting. As the main application of our framework, we prove the first upper bound on the sample complexity of private density estimation for Gaussian Mixture Models in the agnostic setting, extending the realizable result of Afzali et al. [AAL24].

翻译：我们引入了一种新的稳定性概念——称为稳定列表解码——并展示了其在设计差分隐私密度估计器中的适用性。该定义弱于全局稳定性[ABLMM22]，且与可复现性[ILPS22]及列表可复现性[CMY23]的概念相关。我们证明，若某分布类具有稳定列表可解码性，则可在不可知设定下实现隐私学习。作为本框架的主要应用，我们首次证明了不可知设定下高斯混合模型私有密度估计的样本复杂度上界，从而扩展了Afzali等人[AAL24]在可实现性设定下的结果。