  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.
