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# Online Learning with Vector Costs and Bandits with Knapsacks

*Proceedings of Thirty Third Conference on Learning Theory*, PMLR 125:2286-2305, 2020.

#### Abstract

We introduce online learning with vector costs ($OLVC_p$) where in each time step $t \in \{1,\ldots, T\}$, we need to play an action $i \in \{1,\ldots,n\}$ that incurs an unknown vector cost in $[0,1]^d$. The goal of the online algorithm is to minimize the $\ell_p$ norm of the sum of its cost vectors. This captures the classical online learning setting for $d=1$, and is interesting for general $d$ because of applications like online scheduling where we want to balance the load between different machines (dimensions). We study $OLVC_p$ in both stochastic and adversarial arrival settings, and give a general procedure to reduce the problem from $d$ dimensions to a single dimension. This allows us to use classical online learning algorithms in both full and bandit feedback models to obtain (near) optimal results. In particular, we obtain a single algorithm (up to the choice of learning rate) that gives sublinear regret for stochastic arrivals and a tight $O(\min\{p, \log d\})$ competitive ratio for adversarial arrivals. The $OLVC_p$ problem also occurs as a natural subproblem when trying to solve the popular Bandits with Knapsacks (BWK) problem. This connection allows us to use our $OLVC_p$ techniques to obtain (near) optimal results for BWK in both stochastic and adversarial settings. In particular, we obtain a tight $O(\log d \cdot \log T)$ competitive ratio algorithm for adversarial BWK, which improves over the $O(d \cdot \log T)$ competitive ratio algorithm of Immorlica et al. (2019).