  We consider the task of learning a parametric Continuous Time Markov Chain
(CTMC) sequence model without examples of sequences, where the training data
consists entirely of aggregate steady-state statistics. Making the problem
harder, we assume that the states we wish to predict are unobserved in the
training data. Specifically, given a parametric model over the transition rates
of a CTMC and some known transition rates, we wish to extrapolate its steady
state distribution to states that are unobserved. A technical roadblock to
learn a CTMC from its steady state has been that the chain rule to compute
gradients will not work over the arbitrarily long sequences necessary to reach
steady state ---from where the aggregate statistics are sampled. To overcome
this optimization challenge, we propose $\infty$-SGD, a principled stochastic
gradient descent method that uses randomly-stopped estimators to avoid infinite
sums required by the steady state computation, while learning even when only a
subset of the CTMC states can be observed. We apply $\infty$-SGD to a
real-world testbed and synthetic experiments showcasing its accuracy, ability
to extrapolate the steady state distribution to unobserved states under
unobserved conditions (heavy loads, when training under light loads), and
succeeding in difficult scenarios where even a tailor-made extension of
existing methods fails.
