  Interesting theoretical associations have been established by recent papers
between the fields of active learning and stochastic convex optimization due to
the common role of feedback in sequential querying mechanisms. In this paper,
we continue this thread in two parts by exploiting these relations for the
first time to yield novel algorithms in both fields, further motivating the
study of their intersection. First, inspired by a recent optimization algorithm
that was adaptive to unknown uniform convexity parameters, we present a new
active learning algorithm for one-dimensional thresholds that can yield minimax
rates by adapting to unknown noise parameters. Next, we show that one can
perform $d$-dimensional stochastic minimization of smooth uniformly convex
functions when only granted oracle access to noisy gradient signs along any
coordinate instead of real-valued gradients, by using a simple randomized
coordinate descent procedure where each line search can be solved by
$1$-dimensional active learning, provably achieving the same error convergence
rate as having the entire real-valued gradient. Combining these two parts
yields an algorithm that solves stochastic convex optimization of uniformly
convex and smooth functions using only noisy gradient signs by repeatedly
performing active learning, achieves optimal rates and is adaptive to all
unknown convexity and smoothness parameters.
