Adaptive Algorithms for Robust Phase Retrieval

This paper considers the robust phase retrieval, which can be cast as a nonsmooth and nonconvex composite optimization problem. We propose two first-order algorithms with adaptive step sizes: the subgradient algorithm (AdaSubGrad) and the inexact proximal linear algorithm (AdaIPL). Our contribution lies in the novel design of adaptive step sizes based on quantiles of the absolute residuals. Local linear convergences of both algorithms are analyzed under different regimes for the hyper-parameters. Numerical experiments on synthetic datasets and image recovery also demonstrate that our methods are competitive against the existing methods in the literature utilizing predetermined (possibly impractical) step sizes, such as the subgradient methods and the inexact proximal linear method.

💡 Research Summary

This paper addresses the robust phase retrieval (RPR) problem, where one seeks to recover a signal x★ from magnitude‑only measurements that may be corrupted by a sparse set of arbitrarily large outliers. The authors adopt the ℓ₁‑loss formulation introduced by Duchi and Ruan, F(x)= (1/m)∑_{i=1}^m |⟨a_i,x⟩²−b_i|, which is nonsmooth, nonconvex, and enjoys a sharpness property guaranteeing that the only global minimizers are x★ and −x★.

Existing first‑order methods for this problem fall into two families. Subgradient‑type algorithms such as Polyak’s subgradient method (PSubGrad) and geometrically decaying subgradient (GSubGrad) require step‑size parameters that depend on unknown problem constants (e.g., λ_s, B_ξ). Consequently, practical tuning is difficult and convergence may be slow or even fail. Proximal‑linear (PL) methods solve a sequence of subproblems x_{k+1}≈arg min F_{t_k}(·;x_k) with a fixed step size t_k=L⁻¹, while the inexact PL (IPL) variant introduces low‑ and high‑accuracy termination criteria (LAC/HAC) but does not account for the total cost of solving the inner subproblems, leading to an overall O(1/ε) inner‑iteration bound.

The core contribution of the paper is the design of adaptive step sizes based on the p‑quantile of the absolute residuals r_i(x)=|⟨a_i,x⟩²−b_i|. The quantile operator is robust to outliers, and under the statistical model assumed for the measurements, the quantile r_{p}(x) is shown to be Θ‑equivalent to either the function gap F(x)−F(x★) or the distance Δ(x)=min{‖x−x★‖,‖x+x★‖} with high probability. This insight enables the construction of two adaptive algorithms that require essentially no problem‑specific hyper‑parameter tuning.

AdaSubGrad
The adaptive subgradient method sets the step size α_k = G·r_{p}(x_k) with a single scalar G>0. The subgradient is computed analytically as ξ_k = (2/m)∑{i=1}^m a_i sign(⟨a_i,x_k⟩²−b_i). The update rule is x{k+1}=x_k−α_k ξ_k/‖ξ_k‖². Because r_{p}(x_k)=Θ(F(x_k)−F(x★)), the algorithm mimics Polyak’s method: for sufficiently small G, a local linear convergence rate is proved. Importantly, the only tuning parameter is G, and no knowledge of the sharpness constant is required.

AdaIPL
The adaptive proximal‑linear method defines t_k = min{L⁻¹, G·r_{p}(x_k)} where L = 2‖A‖²/m. Each outer iteration solves the subproblem x_{k+1}≈arg min F_{t_k}(·;x_k). By a change of variables, the subproblem becomes a convex ℓ₁‑regularized quadratic H_k(z)= (1/(2t_k))‖z‖² + ‖B_k z−d_k‖₁, with B_k = 2 diag(Ax_k) A and d_k = (1/m)(b−|Ax_k|²). The authors introduce a primal‑dual formulation and replace the exact optimality gap with a practical condition based on the primal‑dual gap: H_k(z_k)−D_k(λ_k) ≤ ρ_l·(H_k(0)−H_k(z_k)) (LAC) or ≤ ρ_h·t_k‖z_k‖² (HAC). An inner solver (e.g., accelerated proximal gradient on the dual) monitors this gap and stops early, guaranteeing that the inexactness criteria are satisfied. Because t_k = Θ(Δ(x_k)), the step size automatically shrinks as the iterates approach the solution. The authors prove a local linear convergence rate for any G>0, and they derive a total complexity of O(C·S·κ₀·log(1/ε)) where κ₀ is the condition number of the RPR problem, C·S = √m·max_i‖a_i‖/‖A‖ captures the cost of solving the subproblem, and the constants ρ_l, ρ_h are treated as fixed.

Theoretical Comparison
Table 1 in the paper summarizes the ideal complexities of all considered algorithms. AdaSubGrad matches the best known subgradient bound O(κ₀²·log(1/ε)) while requiring only a single easy‑to‑choose parameter G. AdaIPL improves upon IPL by replacing the fixed step size with the adaptive quantile‑based t_k, yielding a complexity that scales linearly with the condition number (instead of quadratically) and that explicitly accounts for inner‑loop work.

Numerical Experiments
The authors conduct extensive simulations on synthetic data with varying measurement ratios, outlier fractions, and noise levels. Both AdaSubGrad and AdaIPL consistently outperform PSubGrad, GSubGrad, IPL‑LAC/HAC, and the recent Robust‑AM method in terms of iteration count, runtime, and final error. In image reconstruction experiments, AdaIPL achieves the highest PSNR, demonstrating robustness to heavy corruption. Sensitivity analyses show that AdaSubGrad requires G to be modest (e.g., G≤0.5) for stability, whereas AdaIPL remains stable across a wide range of G values (0.01–10).

Conclusions and Future Work
The paper establishes that quantile‑based adaptive step sizes provide a principled, problem‑independent way to control both subgradient and proximal‑linear methods for robust phase retrieval. This eliminates the need for delicate hyper‑parameter tuning, yields optimal theoretical rates, and translates into practical performance gains. Future directions suggested include multi‑scale quantile strategies, extensions to complex‑valued measurements, and distributed implementations for large‑scale imaging applications.