We employ both non-stationarity measures to derive upper andlowerbounds fortheproblem.

In this paper, we study the problem of bandits with knapsacks (BwK) in a non-stationary environment. The BwK problem generalizes the multi-arm bandit (MAB) problem to model the resource consumption associated with playing each arm. At each time, the decision maker/player chooses to play an arm, and s/he will receive a reward and consume certain amount of resource from each of the multiple resource types. The objective is to maximize the cumulative reward over a finite horizon subject to some knapsack constraints on the resources. Existing works study the BwK problem under either a stochastic or adversarial environment.

Neural Information Processing Systems http://nips.cc/

In this paper, we study the problem of bandits with knapsacks (BwK) in a non-stationary environment. The BwK problem generalizes the multi-arm bandit (MAB) problem to model the resource consumption associated with playing each arm. At each time, the decision maker/player chooses to play an arm, and s/he will receive a reward and consume certain amount of resource from each of the multiple resource types. The objective is to maximize the cumulative reward over a finite horizon subject to some knapsack constraints on the resources. Existing works study the BwK problem under either a stochastic or adversarial environment.

Neural Information Processing Systems http://nips.cc/

We study the contextual bandits with knapsack (CBwK) problem under the high-dimensional setting where the dimension of the feature is large. The reward of pulling each arm equals the multiplication of a sparse high-dimensional weight vector and the feature of the current arrival, with additional random noise. In this paper, we investigate how to exploit this sparsity structure to achieve improved regret for the CBwK problem. To this end, we first develop an online variant of the hard thresholding algorithm that performs the sparse estimation in an online manner. We further combine our online estimator with a primal-dual framework, where we assign a dual variable to each knapsack constraint and utilize an online learning algorithm to update the dual variable, thereby controlling the consumption of the knapsack capacity. We show that this integrated approach allows us to achieve a sublinear regret that depends logarithmically on the feature dimension, thus improving the polynomial dependency established in the previous literature. We also apply our framework to the high-dimension contextual bandit problem without the knapsack constraint and achieve optimal regret in both the data-poor regime and the data-rich regime. We finally conduct numerical experiments to show the efficient empirical performance of our algorithms under the high dimensional setting.

In this paper, we study the problem of bandits with knapsacks (BwK) in a non-stationary environment. The BwK problem generalizes the multi-arm bandit (MAB) problem to model the resource consumption associated with playing each arm. At each time, the decision maker/player chooses to play an arm, and s/he will receive a reward and consume certain amount of resource from each of the multiple resource types. The objective is to maximize the cumulative reward over a finite horizon subject to some knapsack constraints on the resources. Existing works study the BwK problem under either a stochastic or adversarial environment. Our paper considers a non-stationary environment which continuously interpolates between these two extremes. We first show that the traditional notion of variation budget is insufficient to characterize the non-stationarity of the BwK problem for a sublinear regret due to the presence of the constraints, and then we propose a new notion of global non-stationarity measure. We employ both non-stationarity measures to derive upper and lower bounds for the problem. Our results are based on a primal-dual analysis of the underlying linear programs and highlight the interplay between the constraints and the non-stationarity. Finally, we also extend the non-stationarity measure to the problem of online convex optimization with constraints and obtain new regret bounds accordingly.