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تسجيل مستخدم جديدNearly Instance Optimal Sample Complexity Bounds for Top-k Arm Selection
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In the Best-$k$-Arm problem, we are given $n$ stochastic bandit arms, each associated with an unknown reward distribution. We are required to identify the $k$ arms with the largest means by taking as few samples as possible. In this paper, we make progress towards a complete characterization of the instance-wise sample complexity bounds for the Best-$k$-Arm problem. On the lower bound side, we obtain a novel complexity term to measure the sample complexity that every Best-$k$-Arm instance requires. This is derived by an interesting and nontrivial reduction from the Best-$1$-Arm problem. We also provide an elimination-based algorithm that matches the instance-wise lower bound within doubly-logarithmic factors. The sample complexity of our algorithm strictly dominates the state-of-the-art for Best-$k$-Arm (module constant factors).
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In the classical best arm identification (Best-$1$-Arm) problem, we are given $n$ stochastic bandit arms, each associated with a reward distribution with an unknown mean. We would like to identify the arm with the largest mean with probability at lea
st $1-delta$, using as few samples as possible. Understanding the sample complexity of Best-$1$-Arm has attracted significant attention since the last decade. However, the exact sample complexity of the problem is still unknown. Recently, Chen and Li made the gap-entropy conjecture concerning the instance sample complexity of Best-$1$-Arm. Given an instance $I$, let $mu_{[i]}$ be the $i$th largest mean and $Delta_{[i]}=mu_{[1]}-mu_{[i]}$ be the corresponding gap. $H(I)=sum_{i=2}^nDelta_{[i]}^{-2}$ is the complexity of the instance. The gap-entropy conjecture states that $Omegaleft(H(I)cdotleft(lndelta^{-1}+mathsf{Ent}(I)right)right)$ is an instance lower bound, where $mathsf{Ent}(I)$ is an entropy-like term determined by the gaps, and there is a $delta$-correct algorithm for Best-$1$-Arm with sample complexity $Oleft(H(I)cdotleft(lndelta^{-1}+mathsf{Ent}(I)right)+Delta_{[2]}^{-2}lnlnDelta_{[2]}^{-1}right)$. If the conjecture is true, we would have a complete understanding of the instance-wise sample complexity of Best-$1$-Arm. We make significant progress towards the resolution of the gap-entropy conjecture. For the upper bound, we provide a highly nontrivial algorithm which requires [Oleft(H(I)cdotleft(lndelta^{-1} +mathsf{Ent}(I)right)+Delta_{[2]}^{-2}lnlnDelta_{[2]}^{-1}mathrm{polylog}(n,delta^{-1})right)] samples in expectation. For the lower bound, we show that for any Gaussian Best-$1$-Arm instance with gaps of the form $2^{-k}$, any $delta$-correct monotone algorithm requires $Omegaleft(H(I)cdotleft(lndelta^{-1} + mathsf{Ent}(I)right)right)$ samples in expectation.
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