121 research outputs found

### 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

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