Active Classification with Comparison Queries

Abstract

We study an extension of active learning in which the learning algorithm may ask the annotator to compare the distances of two examples from the boundary of their label-class. For example, in a recommendation system application (say for restaurants), the annotator may be asked whether she liked or disliked a specific restaurant (a label query); or which one of two restaurants did she like more (a comparison query).We focus on the class of half spaces, and show that under natural assumptions, such as large margin or bounded bit-description of the input examples, it is possible to reveal all the labels of a sample of size n using approximately O(log n) queries. This implies an exponential improvement over classical active learning, where only label queries are allowed. We complement these results by showing that if any of these assumptions is removed then, in the worst case, Ω(n) queries are required.Our results follow from a new general framework of active learning with additional queries. We identify a combinatorial dimension, called the inference dimension, that captures the query complexity when each additional query is determined by O(1) examples (such as comparison queries, each of which is determined by the two compared examples). Our results for half spaces follow by bounding the inference dimension in the cases discussed above.

Near-optimal Linear Decision Trees for k-SUM and Related Problems

“…Our results are based on the notion of "inference dimension," which was recently introduced by the authors of [16] in the context of active learning.…”

“…The first one is the inference dimension of H = R × {1} ⊂ R 2 , and the second one is of H = R 2 × {1} ⊂ R 3 . Both examples are from [16], where they are described using machinelearning terminology: The first one corresponds to (affine) thresholds in R, and the second one to (affine) thresholds in R 2 . We note that in the first example, only label queries are used, and in the second example, both label and comparison queries are used.…”

“…However, the following theorem rules it out. Theorem 1.9 ( [16]). The inference dimension of H = R 3 is ∞.…”

“…It is natural to ask whether the techniques used in this article, and in particular the inference-dimension, can be used to improve the state-of-the-art upper bounds for general point location problems. Unfortunately, unless the set of hyperplanes H has some combinatorial structure, its inference dimension may be unbounded: In [16] we construct examples of H ⊂ R 3 whose inference dimension is unbounded. Nevertheless, we conjecture that by generalizing comparison queries (which are ±1 linear combinations of two elements in H ) to arbitrary linear combinations of two elements from H might solve the problem.…”

Interactive Linear Regression with Pairwise Comparisons

“…On the other hand, although we have seen great progress on using single or multiple oracles with the same form of interaction [9,16], classification using both comparison and labeling queries remains an interesting open problem. Independently of our work, Kane et al [23] concurrently analyzed a similar setting of learning to classify using both label and comparison queries. However, their algorithms work only in the noise-free setting.…”