Abstract: We develop two new stochastic Gauss-Newton algorithms for solving a class of stochastic non-convex compositional optimization problems frequently arising in practice. We consider both the expectation and finite-sum settings under standard assumptions. We use both standard stochastic and SARAH estimators for approximating function values and Jacobians. In the expectation case, we establish $O(\varepsilon^{-2})$ iteration complexity to achieve a stationary point in expectation and estimate the total number of stochastic oracle calls for both function values and its Jacobian, where $\varepsilon$ is a desired accuracy. In the finite sum case, we also estimate the same iteration complexity and the total oracle calls with high probability. To our best knowledge, this is the first time such global stochastic oracle complexity is established for stochastic Gauss-Newton methods. We illustrate our theoretical results via numerical examples on both synthetic and real datasets.