  In this paper, we study stochastic optimization of two-level composition of
functions without Lipschitz continuous gradient. The smoothness property is
generalized by the notion of relative smoothness which provokes the Bregman
gradient method. We propose three Stochastic Composition Bregman Gradient
algorithms for the three possible relatively smooth compositional scenarios and
provide their sample complexities to achieve an $\epsilon$-approximate
stationary point. For the smooth of relatively smooth composition, the first
algorithm requires $O(\epsilon^{-2})$ calls to the stochastic oracles of the
inner function value and gradient as well as the outer function gradient. When
both functions are relatively smooth, the second algorithm requires
$O(\epsilon^{-3})$ calls to the inner function value stochastic oracle and
$O(\epsilon^{-2})$ calls to the inner and outer functions gradients stochastic
oracles. We further improve the second algorithm by variance reduction for the
setting where just the inner function is smooth. The resulting algorithm
requires $O(\epsilon^{-5/2})$ calls to the inner function value stochastic
oracle, $O(\epsilon^{-3/2})$ calls to the inner function gradient and
$O(\epsilon^{-2})$ calls to the outer function gradient stochastic oracles.
Finally, we numerically evaluate the performance of these three algorithms over
two different examples.
