  We study the multi-armed bandit (MAB) problem with composite and anonymous
feedback. In this model, the reward of pulling an arm spreads over a period of
time (we call this period as reward interval) and the player receives partial
rewards of the action, convoluted with rewards from pulling other arms,
successively. Existing results on this model require prior knowledge about the
reward interval size as an input to their algorithms. In this paper, we propose
adaptive algorithms for both the stochastic and the adversarial cases, without
requiring any prior information about the reward interval. For the stochastic
case, we prove that our algorithm guarantees a regret that matches the lower
bounds (in order). For the adversarial case, we propose the first algorithm to
jointly handle non-oblivious adversary and unknown reward interval size. We
also conduct simulations based on real-world dataset. The results show that our
algorithms outperform existing benchmarks.
