  We consider an online two-stage stochastic optimization with long-term
constraints over a finite horizon of $T$ periods. At each period, we take the
first-stage action, observe a model parameter realization and then take the
second-stage action from a feasible set that depends both on the first-stage
decision and the model parameter. We aim to minimize the cumulative objective
value while guaranteeing that the long-term average second-stage decision
belongs to a set. We develop online algorithms for the online two-stage problem
from adversarial learning algorithms. Also, the regret bound of our algorithm
cam be reduced to the regret bound of embedded adversarial learning algorithms.
Based on our framework, we obtain new results under various settings. When the
model parameter at each period is drawn from identical distributions, we derive
\textit{state-of-art} $O(\sqrt{T})$ regret that improves previous bounds under
special cases. Our algorithm is also robust to adversarial corruptions of model
parameter realizations. When the model parameters are drawn from unknown
non-stationary distributions and we are given machine-learned predictions of
the distributions, we develop a new algorithm from our framework with a regret
$O(W_T+\sqrt{T})$, where $W_T$ measures the total inaccuracy of the
machine-learned predictions.
