  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
can be reduced to the regret bound of embedded adversarial learning algorithms.
Based on this framework, we obtain new results under various settings. 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. We then
develop another algorithm that works when no machine-learned predictions are
given and show the performances.
