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 apparentl…