  We investigate a nonstochastic bandit setting in which the loss of an action
is not immediately charged to the player, but rather spread over the subsequent
rounds in an adversarial way. The instantaneous loss observed by the player at
the end of each round is then a sum of many loss components of previously
played actions. This setting encompasses as a special case the easier task of
bandits with delayed feedback, a well-studied framework where the player
observes the delayed losses individually.
  Our first contribution is a general reduction transforming a standard bandit
algorithm into one that can operate in the harder setting: We bound the regret
of the transformed algorithm in terms of the stability and regret of the
original algorithm. Then, we show that the transformation of a suitably tuned
FTRL with Tsallis entropy has a regret of order $\sqrt{(d+1)KT}$, where $d$ is
the maximum delay, $K$ is the number of arms, and $T$ is the time horizon.
Finally, we show that our results cannot be improved in general by exhibiting a
matching (up to a log factor) lower bound on the regret of any algorithm
operating in this setting.
