  We consider a general online stochastic optimization problem with multiple
budget constraints over a horizon of finite time periods. In each time period,
a reward function and multiple cost functions are revealed, and the decision
maker needs to specify an action from a convex and compact action set to
collect the reward and consume the budget. Each cost function corresponds to
the consumption of one budget. In each period, the reward and cost functions
are drawn from an unknown distribution, which is non-stationary across time.
The objective of the decision maker is to maximize the cumulative reward
subject to the budget constraints. This formulation captures a wide range of
applications including online linear programming and network revenue
management, among others. In this paper, we consider two settings: (i) a
data-driven setting where the true distribution is unknown but a prior estimate
(possibly inaccurate) is available; (ii) an uninformative setting where the
true distribution is completely unknown. We propose a unified
Wasserstein-distance based measure to quantify the inaccuracy of the prior
estimate in setting (i) and the non-stationarity of the system in setting (ii).
We show that the proposed measure leads to a necessary and sufficient condition
for the attainability of a sublinear regret in both settings. For setting (i),
we propose a new algorithm, which takes a primal-dual perspective and
integrates the prior information of the underlying distributions into an online
gradient descent procedure in the dual space. The algorithm also naturally
extends to the uninformative setting (ii). Under both settings, we show the
corresponding algorithm achieves a regret of optimal order. In numerical
experiments, we demonstrate how the proposed algorithms can be naturally
integrated with the re-solving technique to further boost the empirical
performance.
