This paper considers stochastic linear bandits with general nonlinear constraints. The objective is to maximize the expected cumulative reward over horizon $T$ subject to a set of constraints in each round $\tau\leq T$. We propose a pessimistic-optimistic algorithm for this problem, which is efficient in two aspects. First, the algorithm yields $\tilde{\cal O}\left(\left(\frac{K^{0.75}}{\delta}+d\right)\sqrt{\tau}\right)$ (pseudo) regret in round $\tau\leq T,$ where $K$ is the number of constraints, $d$ is the dimension of the reward feature space, and $\delta$ is a Slater's constant; and zero constraint violation in any round $\tau>\tau',$ where $\tau'$ is independent of horizon $T.$ Second, the algorithm is computationally efficient. Our algorithm is based on the primal-dual approach in optimization and includes two components. The primal component is similar to unconstrained stochastic linear bandits (our algorithm uses the linear upper confidence bound algorithm (LinUCB)). The computational complexity of the dual component depends on the number of constraints, but is independent of the sizes of the contextual space, the action space, and the feature space. Thus, the overall computational complexity of our algorithm is similar to that of the linear UCB for unconstrained stochastic linear bandits.

翻译：本文用一般的非线性限制来考虑随机线性线性土匪。 目的是在每回合$\tau\\\leqT$中, 在一系列限制的情况下, 最大限度地增加预期在地平线$T$的累积奖励。 我们建议对此问题采用悲观- 乐观的算法, 它在两个方面是有效的。 首先, 算法产生 $tdelde_ acal O ⁇ left( left)( left (\frac{K> 0. 0. 75\ delta\\ d\\right)\ sqrt\ täright) $( 假) 。 我们的算法基于在优化中的原始- dial- dal leq T $ 回合中采用的直线性方法, $dd$ 是奖励地平线性地平线性地平线性地平面空间平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面, 。