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• W4309758889 abstract "Abstract This paper studies early-stopped mirror descent applied to noisy sparse phase retrieval, which is the problem of recovering a $k$-sparse signal $textbf{x}^star in{mathbb{R}}^n$ 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 hypentropy 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 $|textbf{x}^star |_2/sqrt{k}$. 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." @default.
• W4309758889 created "2022-11-29" @default.
• W4309758889 creator A5059190563 @default.
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• W4309758889 date "2022-11-20" @default.
• W4309758889 modified "2023-10-07" @default.
• W4309758889 title "Nearly minimax-optimal rates for noisy sparse phase retrieval via early-stopped mirror descent" @default.
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• W4309758889 doi "https://doi.org/10.1093/imaiai/iaac024" @default.
• W4309758889 hasPublicationYear "2022" @default.
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