  We consider learning problems over training sets in which both, the number of
training examples and the dimension of the feature vectors, are large. To solve
these problems we propose the random parallel stochastic algorithm (RAPSA). We
call the algorithm random parallel because it utilizes multiple parallel
processors to operate on a randomly chosen subset of blocks of the feature
vector. We call the algorithm stochastic because processors choose training
subsets uniformly at random. Algorithms that are parallel in either of these
dimensions exist, but RAPSA is the first attempt at a methodology that is
parallel in both the selection of blocks and the selection of elements of the
training set. In RAPSA, processors utilize the randomly chosen functions to
compute the stochastic gradient component associated with a randomly chosen
block. The technical contribution of this paper is to show that this minimally
coordinated algorithm converges to the optimal classifier when the training
objective is convex. Moreover, we present an accelerated version of RAPSA
(ARAPSA) that incorporates the objective function curvature information by
premultiplying the descent direction by a Hessian approximation matrix. We
further extend the results for asynchronous settings and show that if the
processors perform their updates without any coordination the algorithms are
still convergent to the optimal argument. RAPSA and its extensions are then
numerically evaluated on a linear estimation problem and a binary image
classification task using the MNIST handwritten digit dataset.
