Solving the Offline and Online Min-Max Problem of Non-smooth Submodular-Concave Functions: A Zeroth-Order Approach

We consider max-min and min-max problems with objective functions that are possibly non-smooth, submodular with respect to the minimiser and concave with respect to the maximiser. We investigate the performance of a zeroth-order method applied to this problem. The method is based on the subgradient of the Lovász extension of the objective function with respect to the minimiser and based on Gaussian smoothing to estimate the smoothed function gradient with respect to the maximiser. In expectation sense, we prove the convergence of the algorithm to an $ε$-saddle point in the offline case. Moreover, we show that, in the expectation sense, in the online setting, the algorithm achieves $O(\sqrt{N\bar{P}_N})$ online duality gap, where $N$ is the number of iterations and $\bar{P}_N$ is the path length of the sequence of optimal decisions. The complexity analysis and hyperparameter selection are presented for all the cases. The theoretical results are illustrated via numerical examples.

Min-Max Optimisation for Nonconvex-Nonconcave Functions Using a Random Zeroth-Order Extragradient Algorithm

This study explores the performance of the random Gaussian smoothing Zeroth-Order ExtraGradient (ZO-EG) scheme considering min-max optimisation problems with possibly NonConvex-NonConcave (NC-NC) objective functions. We consider both unconstrained and constrained, differentiable and non-differentiable settings. We discuss the min-max problem from the point of view of variational inequalities. For the unconstrained problem, we establish the convergence of the ZO-EG algorithm to the neighbourhood of an $\epsilon$-stationary point of the NC-NC objective function, whose radius can be controlled under a variance reduction scheme, along with its complexity. For the constrained problem, we introduce the new notion of proximal variational inequalities and give examples of functions satisfying this property. Moreover, we prove analogous results to the unconstrained case for the constrained problem. For the non-differentiable case, we prove the convergence of the ZO-EG algorithm to a neighbourhood of an $\epsilon$-stationary point of the smoothed version of the objective function, where the radius of the neighbourhood can be controlled, which can be related to the ($\delta,\epsilon$)-Goldstein stationary point of the original objective function.

Properties of Fixed Points of Generalised Extra Gradient Methods Applied to Min-Max Problems

This paper studies properties of fixed points of generalised Extra-gradient (GEG) algorithms applied to min-max problems. We discuss connections between saddle points of the objective function of the min-max problem and GEG fixed points. We show that, under appropriate step-size selections, the set of saddle points (Nash equilibria) is a subset of stable fixed points of GEG. Convergence properties of the GEG algorithm are obtained through a stability analysis of a discrete-time dynamical system. The results and benefits when compared to existing methods are illustrated through numerical examples.

MAGO-SP: Detection and Correction of Water-Fat Swaps in Magnitude-Only VIBE MRI

Volume Interpolated Breath-Hold Examination (VIBE) MRI generates images suitable for water and fat signal composition estimation. While the two-point VIBE provides water-fat-separated images, the six-point VIBE allows estimation of the effective transversal relaxation rate R2* and the proton density fat fraction (PDFF), which are imaging markers for health and disease. Ambiguity during signal reconstruction can lead to water-fat swaps. This shortcoming challenges the application of VIBE-MRI for automated PDFF analyses of large-scale clinical data and of population studies. This study develops an automated pipeline to detect and correct water-fat swaps in non-contrast-enhanced VIBE images. Our three-step pipeline begins with training a segmentation network to classify volumes as "fat-like" or "water-like," using synthetic water-fat swaps generated by merging fat and water volumes with Perlin noise. Next, a denoising diffusion image-to-image network predicts water volumes as signal priors for correction. Finally, we integrate this prior into a physics-constrained model to recover accurate water and fat signals. Our approach achieves a < 1% error rate in water-fat swap detection for a 6-point VIBE. Notably, swaps disproportionately affect individuals in the Underweight and Class 3 Obesity BMI categories. Our correction algorithm ensures accurate solution selection in chemical phase MRIs, enabling reliable PDFF estimation. This forms a solid technical foundation for automated large-scale population imaging analysis.