Bandit Phase Retrieval
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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 ⟨ A_t, θ_⋆⟩^2 where θ_⋆∈ℝ^d is an unknown parameter vector. We prove that the minimax cumulative regret in this problem is Θ̃(d √(n)), which improves on the best known bounds by a factor of √(d). We also show that the minimax simple regret is Θ̃(d / √(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 are not sufficient for optimal regret.
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