  Stochastic optimization of continuous objectives is at the heart of modern
machine learning. However, many important problems are of discrete nature and
often involve submodular objectives. We seek to unleash the power of stochastic
continuous optimization, namely stochastic gradient descent and its variants,
to such discrete problems. We first introduce the problem of stochastic
submodular optimization, where one needs to optimize a submodular objective
which is given as an expectation. Our model captures situations where the
discrete objective arises as an empirical risk (e.g., in the case of
exemplar-based clustering), or is given as an explicit stochastic model (e.g.,
in the case of influence maximization in social networks). By exploiting that
common extensions act linearly on the class of submodular functions, we employ
projected stochastic gradient ascent and its variants in the continuous domain,
and perform rounding to obtain discrete solutions. We focus on the rich and
widely used family of weighted coverage functions. We show that our approach
yields solutions that are guaranteed to match the optimal approximation
guarantees, while reducing the computational cost by several orders of
magnitude, as we demonstrate empirically.

Machine Learning;Discrete Mathematics;Machine Learning