Nonstochastic Bandits with Composite Anonymous Feedback

Nicolò Cesa-Bianchi*, Tommaso Cesari, Roberto Colomboni, Claudio Gentile, Yishay Mansour

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 (formula presented) , 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.

Original languageEnglish
Article number277
JournalJournal of Machine Learning Research
StatePublished - 1 Aug 2022


FundersFunder number
EU Horizon 2020 ICT-48 research and innovation action951847, ANR-19-P3IA-0004
Yandex Initiative for Machine Learning
International Business Machines Corporation
Horizon 2020 Framework Programme882396
European Research Council
Agence Nationale de la Recherche
Ministero dell’Istruzione, dell’Università e della Ricerca
Israel Science Foundation993/17
Tel Aviv University


    • Multi-armed bandits
    • composite losses
    • delayed feedback
    • non-stochastic losses
    • online learning


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