Near-optimal Per-Action Regret Bounds for Sleeping Bandits
Quan Nguyen Nishant A. Mehta
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 $\Omega(\sqrt{TA})$ 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.