Metric Entropy Duality and the Sample Complexity of Outcome Indistinguishability

• URL: http://arxiv.org/abs/2203.04536v1
• Date: Wed, 9 Mar 2022 06:02:31 GMT
• Title: Metric Entropy Duality and the Sample Complexity of Outcome Indistinguishability
• Authors: Lunjia Hu, Charlotte Peale, Omer Reingold
• Abstract summary: In outcome indistinguishability, the goal is to output a predictor that cannot be distinguished from the target predictor. We show that the sample complexity of outcome indistinguishability is characterized by the metric entropy of $P$ w.r.t. This equivalence makes an intriguing connection to the long-standing metric entropy duality conjecture in convex geometry.
• Score: 7.727052811126007
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We give the first sample complexity characterizations for outcome indistinguishability, a theoretical framework of machine learning recently introduced by Dwork, Kim, Reingold, Rothblum, and Yona (STOC 2021). In outcome indistinguishability, the goal of the learner is to output a predictor that cannot be distinguished from the target predictor by a class $D$ of distinguishers examining the outcomes generated according to the predictors' predictions. In the distribution-specific and realizable setting where the learner is given the data distribution together with a predictor class $P$ containing the target predictor, we show that the sample complexity of outcome indistinguishability is characterized by the metric entropy of $P$ w.r.t. the dual Minkowski norm defined by $D$, and equivalently by the metric entropy of $D$ w.r.t. the dual Minkowski norm defined by $P$. This equivalence makes an intriguing connection to the long-standing metric entropy duality conjecture in convex geometry. Our sample complexity characterization implies a variant of metric entropy duality, which we show is nearly tight. In the distribution-free setting, we focus on the case considered by Dwork et al. where $P$ contains all possible predictors, hence the sample complexity only depends on $D$. In this setting, we show that the sample complexity of outcome indistinguishability is characterized by the fat-shattering dimension of $D$. We also show a strong sample complexity separation between realizable and agnostic outcome indistinguishability in both the distribution-free and the distribution-specific settings. This is in contrast to distribution-free (resp. distribution-specific) PAC learning where the sample complexity in both the realizable and the agnostic settings can be characterized by the VC dimension (resp. metric entropy).

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