Online Stochastic Optimization with Wasserstein Based Non-stationarity
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We consider a general online stochastic optimization problem with multiple budget constraints over a horizon of finite time periods. At each time period, a reward function and multiple cost functions, where each cost function is involved in the consumption of one corresponding budget, are drawn from an unknown distribution, which is assumed to be non-stationary across time. Then, a decision maker needs to specify an action from a convex and compact action set to collect the reward, and the consumption each budget is determined jointly by the cost functions and the taken action. The objective of the decision maker is to maximize the cumulative reward subject to the budget constraints. Our model captures a wide range of applications including online linear programming and network revenue management, among others. In this paper, we design near-optimal policies for the decision maker under the following two specific settings: a data-driven setting where the decision maker is given prior estimates of the distributions beforehand and a no information setting where the distributions are completely unknown to the decision maker. Under each setting, we propose a new Wasserstein-distance based measure to measure the non-stationarity of the distributions at different time periods and show that this measure leads to a necessary and sufficient condition for the attainability of a sublinear regret. For the first setting, we propose a new algorithm which blends gradient descent steps with the prior estimates. We then adapt our algorithm for the second setting and propose another gradient descent based algorithm. We show that under both settings, our polices achieve a regret upper bound of optimal order. Moreover, our policies could be naturally incorporated with a re-solving procedure which further boosts the empirical performance in numerical experiments.
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