On Interpolating Experts and Multi-Armed Bandits
This work addresses a theoretical gap in online decision-making for researchers in machine learning and optimization, providing foundational insights into feedback structures.
The paper tackles the problem of interpolating between expert advice and multi-armed bandits by introducing a family of problems called m-MAB, where arms are grouped and pulling an arm reveals losses for its entire group. It proves tight minimax regret bounds of Θ(√(T∑ log(m_k+1))) for m-MAB and optimal PAC algorithm bounds of Θ((1/ε²)∑ log(m_k+1)) for its pure exploration version, extending results to bandit with graph feedback.
Learning with expert advice and multi-armed bandit are two classic online decision problems which differ on how the information is observed in each round of the game. We study a family of problems interpolating the two. For a vector $\mathbf{m}=(m_1,\dots,m_K)\in \mathbb{N}^K$, an instance of $\mathbf{m}$-MAB indicates that the arms are partitioned into $K$ groups and the $i$-th group contains $m_i$ arms. Once an arm is pulled, the losses of all arms in the same group are observed. We prove tight minimax regret bounds for $\mathbf{m}$-MAB and design an optimal PAC algorithm for its pure exploration version, $\mathbf{m}$-BAI, where the goal is to identify the arm with minimum loss with as few rounds as possible. We show that the minimax regret of $\mathbf{m}$-MAB is $Θ\left(\sqrt{T\sum_{k=1}^K\log (m_k+1)}\right)$ and the minimum number of pulls for an $(ε,0.05)$-PAC algorithm of $\mathbf{m}$-BAI is $Θ\left(\frac{1}{ε^2}\cdot \sum_{k=1}^K\log (m_k+1)\right)$. Both our upper bounds and lower bounds for $\mathbf{m}$-MAB can be extended to a more general setting, namely the bandit with graph feedback, in terms of the clique cover and related graph parameters. As consequences, we obtained tight minimax regret bounds for several families of feedback graphs.