Nearly Minimax Optimal Submodular Maximization with Bandit Feedback

• URL: http://arxiv.org/abs/2310.18465v2
• Date: Thu, 12 Dec 2024 17:24:47 GMT
• Title: Nearly Minimax Optimal Submodular Maximization with Bandit Feedback
• Authors: Artin Tajdini, Lalit Jain, Kevin Jamieson,
• Abstract summary: We minimize the learner's regret with respect to an approximation of the maximum $f(S_*)$ with $|S_*| = k$. In this work, we establish the first minimax lower bound for this setting that scales like $tildeOmega(min_L le k(T2/3 + sqrtn choose k - LT)$. For a slightly restricted algorithm class, we prove a stronger regret lower bound of $tildeOmega(min_L
• Score: 12.28389976959093
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• Abstract: We consider maximizing an unknown monotonic, submodular set function $f: 2^{[n]} \rightarrow [0,1]$ with cardinality constraint under stochastic bandit feedback. At each time $t=1,\dots,T$ the learner chooses a set $S_t \subset [n]$ with $|S_t| \leq k$ and receives reward $f(S_t) + \eta_t$ where $\eta_t$ is mean-zero sub-Gaussian noise. The objective is to minimize the learner's regret with respect to an approximation of the maximum $f(S_*)$ with $|S_*| = k$, obtained through robust greedy maximization of $f$. To date, the best regret bound in the literature scales as $k n^{1/3} T^{2/3}$. And by trivially treating every set as a unique arm one deduces that $\sqrt{ {n \choose k} T }$ is also achievable using standard multi-armed bandit algorithms. In this work, we establish the first minimax lower bound for this setting that scales like $\tilde{\Omega}(\min_{L \le k}(L^{1/3}n^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. For a slightly restricted algorithm class, we prove a stronger regret lower bound of $\tilde{\Omega}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. Moreover, we propose an algorithm Sub-UCB that achieves regret $\tilde{\mathcal{O}}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$ capable of matching the lower bound on regret for the restricted class up to logarithmic factors.

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