Near-optimal Per-Action Regret Bounds for Sleeping Bandits
This work addresses a theoretical gap in online learning for sleeping bandits, which is incremental as it improves upon existing bounds but does not introduce a new paradigm.
The paper tackles the problem of deriving near-optimal per-action regret bounds for sleeping bandits, where arms and losses are adversarially chosen, by directly minimizing per-action regret using generalized algorithms like EXP3, EXP3-IX, and FTRL with Tsallis entropy, achieving bounds of order O(√(TA ln K)) and O(√(T√(AK))) that close the gap from previous O(K√(TA ln K)) bounds.
We derive near-optimal per-action regret bounds for sleeping bandits, in which both the sets of available arms and their losses in every round are chosen by an adversary. In a setting with $K$ total arms and at most $A$ available arms in each round over $T$ rounds, the best known upper bound is $O(K\sqrt{TA\ln{K}})$, obtained indirectly via minimizing internal sleeping regrets. Compared to the minimax $Ω(\sqrt{TA})$ lower bound, this upper bound contains an extra multiplicative factor of $K\ln{K}$. We address this gap by directly minimizing the per-action regret using generalized versions of EXP3, EXP3-IX and FTRL with Tsallis entropy, thereby obtaining near-optimal bounds of order $O(\sqrt{TA\ln{K}})$ and $O(\sqrt{T\sqrt{AK}})$. We extend our results to the setting of bandits with advice from sleeping experts, generalizing EXP4 along the way. This leads to new proofs for a number of existing adaptive and tracking regret bounds for standard non-sleeping bandits. Extending our results to the bandit version of experts that report their confidences leads to new bounds for the confidence regret that depends primarily on the sum of experts' confidences. We prove a lower bound, showing that for any minimax optimal algorithms, there exists an action whose regret is sublinear in $T$ but linear in the number of its active rounds.