Beyond Disagreement-based Agnostic Active Learning

We study agnostic active learning, where the goal is to learn a classifier in a pre-specified hypothesis class interactively with as few label queries as possible, while making no assumptions on the true function generating the labels. The main algorithms for this problem are {\em{disagreement-based active learning}}, which has a high label requirement, and {\em{margin-based active learning}}, which only applies to fairly restricted settings. A major challenge is to find an algorithm which achieves better label complexity, is consistent in an agnostic setting, and applies to general classification problems. In this paper, we provide such an algorithm. Our solution is based on two novel contributions -- a reduction from consistent active learning to confidence-rated prediction with guaranteed error, and a novel confidence-rated predictor

A directed isoperimetric inequality with application to Bregman near neighbor lower bounds

Bregman divergences $D_\phi$ are a class of divergences parametrized by a convex function $\phi$ and include well known distance functions like $\ell_2^2$ and the Kullback-Leibler divergence. There has been extensive research on algorithms for problems like clustering and near neighbor search with respect to Bregman divergences, in all cases, the algorithms depend not just on the data size $n$ and dimensionality $d$, but also on a structure constant $\mu \ge 1$ that depends solely on $\phi$ and can grow without bound independently. In this paper, we provide the first evidence that this dependence on $\mu$ might be intrinsic. We focus on the problem of approximate near neighbor search for Bregman divergences. We show that under the cell probe model, any non-adaptive data structure (like locality-sensitive hashing) for $c$-approximate near-neighbor search that admits $r$ probes must use space $\Omega(n^{1 + \frac{\mu}{c r}})$. In contrast, for LSH under $\ell_1$ the best bound is $\Omega(n^{1+\frac{1}{cr}})$. Our new tool is a directed variant of the standard boolean noise operator. We show that a generalization of the Bonami-Beckner hypercontractivity inequality exists "in expectation" or upon restriction to certain subsets of the Hamming cube, and that this is sufficient to prove the desired isoperimetric inequality that we use in our data structure lower bound. We also present a structural result reducing the Hamming cube to a Bregman cube. This structure allows us to obtain lower bounds for problems under Bregman divergences from their $\ell_1$ analog. In particular, we get a (weaker) lower bound for approximate near neighbor search of the form $\Omega(n^{1 + \frac{1}{cr}})$ for an $r$-query non-adaptive data structure, and new cell probe lower bounds for a number of other near neighbor questions in Bregman space.Comment: 27 page