We consider the online setting, where the input arrives over time, and irrevocable decisions must be made without knowledge of the future. For all these problems, any online algorithm must incur a cost that is approximately log|I| times the optimal cost in the worst-case, where |I| is the length of theinput.

Compositional generalization is a basic and essential intellective capability of human beings, which allows us to recombine known parts readily.

Proposition 2. Let X be some set andC be any collection of subsets ofX.

We study the preference-based pure exploration problem for bandits with vector-valued rewards.