  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 $\langle A_t, \theta_\star\rangle^2$ where $\theta_\star \in \mathbb
R^d$ is an unknown parameter vector. We prove that the minimax cumulative
regret in this problem is $\smash{\tilde \Theta(d \sqrt{n})}$, which improves
on the best known bounds 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
are not sufficient for optimal regret.
