Bandit Phase Retrieval

21 May 2021, 20:51 (edited 28 Jan 2022)NeurIPS 2021 PosterReaders: Everyone
• Keywords: Bandit phase retrieval, minimax regret
• TL;DR: We study a bandit version of phase retrieval and prove the minimax cumulative regret as well as simple regret.
• Abstract: We study a bandit version of phase retrieval where the learner chooses actions $(A_t)_{t=1}^n$ in the $d$-dimensional unit ball and the expected reward is $\langle A_t, \theta_\star \rangle^2$ with $\theta_\star \in \mathbb R^d$ an unknown parameter vector. We prove an upper bound on the minimax cumulative regret in this problem of $\smash{\tilde \Theta(d \sqrt{n})}$, which matches known lower bounds up to logarithmic factors and improves on the best known upper bound by a factor of $\smash{\sqrt{d}}$. We also show that the minimax simple regret is $\smash{\tilde \Theta(d / \sqrt{n})}$ and that this is only achievable by an adaptive algorithm. Our analysis shows that an apparently convincing heuristic for guessing lower bounds can be misleading and that uniform bounds on the information ratio for information-directed sampling (Russo and Van Roy, 2014) are not sufficient for optimal regret.
• Supplementary Material: pdf
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