On Active and Passive Testing

Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property? This is a fundamental question in Property Testing, where traditionally the testing algorithm is allowed to pick its queries among the entire set of inputs. Balcan et al. have recently suggested to restrict the tester to take its queries from a smaller, typically random, subset of the inputs. This model is called active testing, in resemblance of active learning. Active testing gets more difficult as the size of the set we can query from decreases, and the extreme case is when it is exactly the number of queries we perform (so the algorithm actually has no choice). This is known as passive testing, or testing from random examples. In their paper, Balcan et al. have shown that active and passive testing of dictator functions is as hard as learning them, and requires Θ(logn) queries (unlike the classic model, in which it can be done in a constant number of queries). We extend this result to k-linear functions, proving that passive and active testing of them requires Θ(k logn) queries, assuming k is not too large. Other classes of functions we consider are juntas, partially symmetric functions, linear functions, and low degree polynomials. For linear functions we provide tight bounds on the query complexity in both active and passive models (which asymptotically differ). The analysis for low degree polynomials is less complete and the exact query complexity is given only for passive testing. In both these cases, the query complexity for passive testing is essentially equivalent to that of learning. For juntas and partially symmetric functions, that is, functions that depend on a small number of variables and potentially also on the Hamming weight of the input, we provide some lower and upper bounds for the different models. When the functions depend on a constant number of variables, our analysis for both families is asymptotically tight. Moreover, the family of partially symmetric functions is the first example for which the query complexities in all these models are asymptotically different. Our methods combine algebraic, combinatorial, and probabilistic techniques, including the Talagrand concentration inequality and the Erdős–Rado results on ∆-systems. ∗Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Institute for Advanced Study, Princeton, New Jersey, 08540. Email: [email protected]. Research supported in part by an ERC Advanced grant, by a USA-Israeli BSF grant, by an ISF grant, by the Israeli I-Core program and by the Simonyi Fund. †School of Mathematics, Georgia Tech, 686 Cherry st, Atlanta, GA 30332. E-mail: [email protected]. ‡Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. Email: [email protected]. Research supported in part by an ERC Advanced grant and by the Israeli Centers of Research Excellence (I-CORE) program.