Testing (Subclasses of) Halfspaces

The purpose of this document is to summarize the talk I gave at the property testing workshop at ITCS in January 2010or more accurately, the talk I would have given, had I not been sick with the flu. In my absence, Rocco Servedio actually gave the talk. Here I will attempt to summarize what I would have said, or what Rocco might have said, or some combination of the two. The purpose of the talk was to present the results from two papers, [13] and [12], regarding the testability of halfspaces and certain subclasses of halfspaces. A halfspace is a function of the form f(x) = sgn(w1x1+ · · · + wnxn − θ) where w1, ..., wn, θ ∈ R. The wi’s are called “weights,” and θ is called the “threshold.” The sgn function is 1 on arguments ≥ 0, and −1 otherwise. The inputs to f can be either Boolean or real. Here we will mainly be concerned with functions over the Boolean cube, i.e. functions of the form f : {−1, 1}n → {−1, 1}. Halfspaces are also known as threshold functions or linear threshold functions; for brevity we shall refer to them here as LTFs. LTFs are a simple yet powerful class of functions, which for decades have played an important role complexity theory, optimization, and perhaps especially machine learning (see e.g. [9, 18, 2, 15, 14, 17]). A lot of attention has been paid to the problem of learning LTFsthat is, given examples labeled according to an unknown LTF (either random examples or queries to the function), find an LTF that it is -close to. However, the question we want to address is that of testing LTFs. That is, given query access to a function, we would like to distinguish whether it is an LTF or whether it is -far from any LTF. Though any proper learning algorithm can be used as a testing algorithm (see, e.g., the observations of [8]), testing potentially requires fewer queries. Indeed, in situations where query access is available, a query-efficient testing algorithm can be used to check whether a function is close to an an LTF, before bothering to run a more intensive algorithm to learn which LTF it is close to.