We consider the problem of testing and learning from data in the presence of resource constraints, such as limited memory or weak data access, which place limitations on the efficiency and feasibility of testing or learning. In particular, we ask the following question: Could a resource-constrained learner/tester use interaction with a resource-unconstrained but untrusted party to solve a learning or testing problem more efficiently than they could without such an interaction? In this work, we answer this question both abstractly and for concrete problems, in two complementary ways: For a wide variety of scenarios, we prove that a resource-constrained learner cannot gain any advantage through classical interaction with an untrusted prover. As a special case, we show that for the vast majority of testing and learning problems in which quantum memory is a meaningful resource, a memory-constrained quantum algorithm cannot overcome its limitations via classical communication with a memory-unconstrained quantum prover. In contrast, when quantum communication is allowed, we construct a variety of interactive proof protocols, for specific learning and testing problems, which allow memory-constrained quantum verifiers to gain significant advantages through delegation to untrusted provers. These results highlight both the limitations and potential of delegating learning and testing problems to resource-rich but untrusted third parties.

翻译：我们研究了在存在资源约束（如有限内存或弱数据访问能力）的条件下进行数据测试与学习的问题，这些约束对测试或学习的效率与可行性构成了限制。具体而言，我们提出以下问题：资源受限的学习者/测试者能否通过与资源不受限但不可信方进行交互，以比无交互时更高的效率解决学习或测试问题？在本工作中，我们通过两种互补的方式，从抽象层面和具体问题层面对此问题作出回答：针对多种场景，我们证明资源受限的学习者无法通过与不可信证明者进行经典交互获得任何优势。作为一个特例，我们表明在绝大多数以量子内存为关键资源的测试与学习问题中，内存受限的量子算法无法通过与内存不受限的量子证明者进行经典通信来突破其固有局限。与之相对，当允许量子通信时，我们针对特定学习与测试问题构建了多种交互式证明协议，使得内存受限的量子验证者能够通过委托给不可信证明者获得显著优势。这些结果共同揭示了将学习与测试问题委托给资源丰富但不可信第三方时存在的局限性与潜在可能性。