Limitations of Membership Queries in Testable Learning

• URL: http://arxiv.org/abs/2512.02279v1
• Date: Mon, 01 Dec 2025 23:50:20 GMT
• Title: Limitations of Membership Queries in Testable Learning
• Abstract summary: We show that membership queries cannot decrease the time complexity of testable learning algorithms.<n>In the testable learning model, the learner must output a hypothesis whenever the data satisfies a desired property.<n>We show that TL-Q algorithms in this class imply efficient statistical refutation and learning algorithms.
• Score: 8.837640805363316
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Membership queries (MQ) often yield speedups for learning tasks, particularly in the distribution-specific setting. We show that in the \emph{testable learning} model of Rubinfeld and Vasilyan [RV23], membership queries cannot decrease the time complexity of testable learning algorithms beyond the complexity of sample-only distribution-specific learning. In the testable learning model, the learner must output a hypothesis whenever the data distribution satisfies a desired property, and if it outputs a hypothesis, the hypothesis must be near-optimal. We give a general reduction from sample-based \emph{refutation} of boolean concept classes, as presented in [Vadhan17, KL18], to testable learning with queries (TL-Q). This yields lower bounds for TL-Q via the reduction from learning to refutation given in [KL18]. The result is that, relative to a concept class and a distribution family, no $m$-sample TL-Q algorithm can be super-polynomially more time-efficient than the best $m$-sample PAC learner. Finally, we define a class of ``statistical'' MQ algorithms that encompasses many known distribution-specific MQ learners, such as those based on influence estimation or subcube-conditional statistical queries. We show that TL-Q algorithms in this class imply efficient statistical-query refutation and learning algorithms. Thus, combined with known SQ dimension lower bounds, our results imply that these efficient membership query learners cannot be made testable.

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