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We study the problem of online resource allocation, where customers arrive sequentially, and the seller must irrevocably allocate resources to each incoming customer while also facing a prespecified procurement cost function over the total allocation. The objective is to maximize the reward obtained from fulfilling the customers’ requests sans the cumulative procurement cost. We analyze the competitive ratio of a primal-dual algorithm in this setting and develop an optimization framework for designing a surrogate function for the procurement cost to be used by the algorithm to improve the competitive ratio of the primal-dual algorithm. We use the optimal surrogate function for polynomial procurement cost functions to improve on previous bounds. For general procurement cost functions, our design method uses quasiconvex optimization to find optimal design parameters. We then implement the design techniques and show the improved performance of the algorithm in numerical examples. Finally, we extend the analysis by devising a posted pricing mechanism in which the algorithm does not require the customers’ preferences to be revealed. Funding: M. Fazel’s work was supported in part by the National Science Foundation [Awards 2023166, 2007036, and 1740551]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2021.0012 . 

The growing interest in complex decision-making and language modeling problems highlights the importance of sample-efficient learning over very long horizons. This work takes a step in this direction by investigating contextual linear bandits where the current reward depends on at most s prior actions and contexts (not necessarily consecutive), up to a time horizon of h. In order to avoid polynomial dependence on h, we propose new algorithms that leverage sparsity to discover the dependence pattern and arm parameters jointly. We consider both the data-poor (T= h) regimes and derive respective regret upper bounds O(d square-root(sT) +min(q, T) and O( square-root(sdT) ), with sparsity s, feature dimension d, total time horizon T, and q that is adaptive to the reward dependence pattern. Complementing upper bounds, we also show that learning over a single trajectory brings inherent challenges: While the dependence pattern and arm parameters form a rank-1 matrix, circulant matrices are not isometric over rank-1 manifolds and sample complexity indeed benefits from the sparse reward dependence structure. Our results necessitate a new analysis to address long-range temporal dependencies across data and avoid polynomial dependence on the reward horizon h. Specifically, we utilize connections to the restricted isometry property of circulant matrices formed by dependent sub-Gaussian vectors and establish new guarantees that are also of independent interest.