Abstract: The performance of a bandit algorithm is usually measured by the cumulative rewards of the actions chosen by the algorithm. However, in many real-world applications, the rewards in each round should be good enough for reasons such as safety and fairness. In this paper, we investigate the contextual conservative interleaving bandit problem, which has a performance constraint that requires the chosen actions to be not much worse than given baseline actions in each round. This work is the first to simultaneously consider the following practical situations: (1) multiple actions are chosen in a round, (2) the feature vectors associated with given actions depend on the round, and (3) the performance constraints in each round that depend only on the actions chosen in that round. We propose a meta-algorithm, Greedy on Confidence Widths (GCW), that satisfies the performance constraints with high probability. GCW uses a standard bandit algorithm and achieves minimax optimal regret up to logarithmic factors if the algorithm used is also minimax optimal. We improve the existing analyses for the C${}^2$UCB algorithm and the Thompson sampling to combine with GCW. We show that these algorithms achieve near-optimal regret when the feasible sets of given actions are the bases of a matroid. Our numerical experiments on a real-world dataset demonstrate that GCW with the standard bandit algorithms efficiently improves performance while satisfying the performance constraints.