  This paper studies early-stopped mirror descent applied to noisy sparse phase
retrieval, which is the problem of recovering a $k$-sparse signal
$\mathbf{x}^\star\in\mathbb{R}^n$ from a set of quadratic Gaussian measurements
corrupted by sub-exponential noise. We consider the (non-convex) unregularized
empirical risk minimization problem and show that early-stopped mirror descent,
when equipped with the hyperbolic entropy mirror map and proper initialization,
achieves a nearly minimax-optimal rate of convergence, provided the sample size
is at least of order $k^2$ (modulo logarithmic term) and the minimum (in
modulus) non-zero entry of the signal is on the order of
$\|\mathbf{x}^\star\|_2/\sqrt{k}$. Our theory leads to a simple algorithm that
does not rely on explicit regularization or thresholding steps to promote
sparsity. More generally, our results establish a connection between mirror
descent and sparsity in the non-convex problem of noisy sparse phase retrieval,
adding to the literature on early stopping that has mostly focused on
non-sparse, Euclidean, and convex settings via gradient descent. Our proof
combines a potential-based analysis of mirror descent with a quantitative
control on a variational coherence property that we establish along the path of
mirror descent, up to a prescribed stopping time.
