Active Testing

One of the motivations for property testing of boolean functions is the idea that testing can serve as a preprocessing step before learning. However, in most machine learning applications, the ability to query functions at arbitrary points in the input space is considered highly unrealistic. Instead, the dominant query paradigm in applied machine learning, called active learning, is one where the algorithm may ask for examples to be labeled, but only from among those that exist in nature. That is, the algorithm may make a polynomial number of draws from the underlying distribution D and then query for labels, but only of points in its sample. In this work, we bring this well-studied model in learning to the domain of testing, calling it active testing. We show that for a number of important properties, testing can still yield substantial benefits in this setting. This includes testing unions of intervals, testing linear separators, and testing various assumptions used in semisupervised learning. For example, we show that testing unions of d intervals can be done withO(1) label requests in our setting, whereas it is known to require Ω( √ d) labeled examples for passive testing (where the algorithm must pay for labels on every example drawn from D) and Ω(d) for learning. In fact, our results for testing unions of intervals also yield improvements on prior work in both the membership query model (where any point in the domain can be queried) and the passive testing model [27] as well. In the case of testing linear separators in R, we show that both active and passive testing can be done with O( √ n) queries, substantially less than the Ω(n) needed for learning and also yielding a new upper bound for the passive testing model. We also show a general combination result that any disjoint union of testable properties remains testable in the active testing model, a feature that does not hold for passive testing. In addition to these specific results, we also develop a general notion of the testing dimension of a given property with respect to a given distribution. We show this dimension characterizes (up to constant factors) the intrinsic number of label requests needed to test that property; we do this for both the active and passive testing models. We then use this dimension to prove a number of lower bounds. For instance, interestingly, one case where we show active testing does not help is for dictator functions, where we give Ω(log n) lower bounds that match the upper bounds for learning this class. Our results show that testing can be a powerful tool in realistic models for learning, and further that active testing exhibits an interesting and rich structure. Our work in addition develops new characterizations of common function classes that may be of independent interest. ∗Georgia Institute of Technology, School of Computer Science. Email: ninamf@cc.gatech. edu. Supported in part by NSF grant CCF-0953192, AFOSR grant FA9550-09-1-0538, a Microsoft Faculty Fellowship and a Google Research Award. †Carnegie Mellon University, Computer Science Department. Email: eblais@cs.cmu. edu. ‡Carnegie Mellon University, Computer Science Department, Email: avrim@cs.cmu. edu. Supported in part by the National Science Foundation under grants CCF-0830540, CCF-1116892, and IIS-1065251. §Carnegie Mellon University, Machine Learning Department, Email: liuy@cs.cmu. edu. Supported in part by NSF grant IIS-1065251 and a Google Core AI grant.