  We introduce the Stochastic Monotone Aggregated Root-Finding (SMART)
algorithm, a new randomized operator-splitting scheme for finding roots of
finite sums of operators. These algorithms are similar to the growing class of
incremental aggregated gradient algorithms, which minimize finite sums of
functions; the difference is that we replace gradients of functions with
black-boxes called operators, which represent subproblems to be solved during
the algorithm. By replacing gradients with operators, we increase our modeling
power, and we simplify the application and analysis of the resulting
algorithms. The operator point of view also makes it easy to extend our
algorithms to allow arbitrary sampling and updating of blocks of coordinates
throughout the algorithm. Implementing and running an algorithm like this on a
computing cluster can be slow if we force all computing nodes to be synched up
at all times. To take better advantage of parallelism, we allow computing nodes
to delay updates and break synchronization.
  This paper has several technical and practical contributions. We prove the
weak, almost sure convergence of a new class of randomized operator-splitting
schemes in separable Hilbert spaces; we prove that this class of algorithms
convergences linearly in expectation when a weak regularity property holds; we
highlight connections to other algorithms; and we introduce a few new
algorithms for large-scale optimization.
