Proposes active sampling for improving fairness in machine learning.
problem Improving fairness in machine learning models, especially for disadvantaged groups.
method Simple active sampling and reweighting strategies for min-max fairness.
result Proves the rate of convergence to a min-max fair solution for convex problems.
A new method for fair PCA ensures balanced error across groups.
problem Balancing approximation error across different groups in multi-group data.
method Iterative method to compute fair principal components minimizing max group-wise reconstruction error.
result Preserves the containment property of standard PCA and reduces to standard PCA for single-group data.
This work optimizes model performance while ensuring fairness through AUC constraints.
problem Ensuring fairness in machine learning models, especially for protected populations.
method Formulates fairness-aware machine learning model training as AUC optimization subject to fairness constraints, solves using stochastic first-order methods.
result Demonstrates effectiveness of the approach on real-world data under different fairness metrics.
Study introduces statistical mechanics for min-max problems.
problem Understanding the properties of min-max problems in high dimensions.
method Statistical mechanical formalism for analyzing min-max problems.
result Derives the relationship between training data and generalization error.
Survey of advances in non-convex min-max optimization for applications.
problem Finding optimal solutions in non-convex, non-concave min-max problems.
method Selective review of theoretical and algorithmic advances.
result Exciting recent advances in solving non-convex min-max problems.
Develops fair classifiers robust to training distribution perturbations.
problem Ensuring fairness in classifiers robust to training data perturbations.
method Formulates a min-max objective function to minimize distributionally robust training loss while maintaining fairness for perturbed distributions. Uses an iterative online learning algorithm to find a fair and robust classifier.
result Our classifier maintains fairness and accuracy for a wide range of perturbations compared to state-of-the-art fair classifiers.
Machine learning algorithms have been increasingly deployed in critical automated decision-making systems that directly affect human lives. When these algorithms are only trained to minimize the training/test error, they could suffer from systematic discrimination against individuals based on their sensitive attributes…
New NMF method aims to improve fairness in machine learning.
problem Fairness and bias in machine learning algorithms.
method Modification of NMF objective function using min-max formulation, with two minimization methods.
result The method can sometimes improve fairness but may increase error for some individuals.
We propose a fair principal component analysis method that balances reconstruction error and subgroup fairness.
problem Fairness and robustness in principal component analysis for consequential domains.
method Distributionally robust optimization over the Stiefel manifold with a Riemannian subgradient descent.
result The proposed method achieves better performance on real-world datasets compared to state-of-the-art baselines.
Study shows strong min-max principle for phase transitions.
problem Understanding nodal sets near minimal hypersurfaces.
method Analogous to White's principle, applies to Allen-Cahn energy.
result Strong min-max principle for phase transitions.
Equity-Transformer solves NP-hard min-max routing problems efficiently.
problem Min-max routing problems with multiple agents and large-scale applications.
method Sequential planning approach with Transformer and equitable workload distribution inductive biases.
result Significant runtime and cost reductions in min-max mTSP and min-max mPDP tasks.
Upper bound for Morse index of min-max varifolds.
problem Bounding Morse index of varifolds.
method Proving upper bound for Morse index of min-max stationary integral varifolds.
result Upper bound for Morse index of min-max stationary integral varifolds.
Localized min-max method proves minimal hypersurface existence.
problem Existence of minimal hypersurfaces in complete manifolds.
method Localized min-max approach to prove existence.
result Existence of complete embedded minimal hypersurface with index at most one.
The paper solves min-max widths on a 3-sphere and strengthens multiplicity theorems.
problem Which min-max widths of the unit 3-sphere lie between 2π2 and 8π? method Homological min-max theory and stronger versions of multiplicity one theorems.
result Proves the 10th to 13th min-max widths of the unit 3-sphere lie between 2π2 and 8π. Adaptive momentum method solves non-convex min-max problems.
problem Non-convex min-max optimization problems in training generative adversarial networks.
method Proposes an adaptive momentum algorithm for non-convex min-max optimization.
result Establishes non-asymptotic convergence rates for the proposed algorithm.
Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.
problem Existence and finiteness of G-invariant minimal hypersurfaces. method Equivariant min-max theory, compactness theorem, bumpy metrics theorem.
result Generalization of Morse index estimates to equivariant setting.
New proof of Smale conjecture for RP^3 and lens spaces using min-max theory.
problem Proving the Smale conjecture for specific spaces.
method Minimal surfaces and min-max theory.
result New proof of Smale conjecture for RP3 and lens spaces. Recent applications that arise in machine learning have surged significant interest in solving min-max saddle point games. This problem has been extensively studied in the convex-concave regime for which a global equilibrium solution can be computed efficiently. In this paper, we study the problem in the non-convex reg…
Paper improves Morse index bound for hypersurfaces.
problem Improving Morse index bound for hypersurfaces.
method Construction of hierarchical deformations and restrictive min-max theory.
result Generalizes a result by X. Zhou for 3≤n+1≤7. New methods solve min-max problems on manifolds using Riemannian Hamiltonians.
problem Min-max optimization on Riemannian manifolds.
method Riemannian Hamiltonian methods (RHM) to minimize the Hamiltonian function.
result RHM leads to correct search directions and global optimality in min-max problems.
Bound on equivariant index for min-max surfaces.
problem Bounding the index of equivariant min-max surfaces.
method Equivariant min-max procedure with group action.
result Equivariant index bound by number of parameters.
We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each…
We reformulate LIPs as min-max problems for easier solution.
problem Recovering signals from few linear measurements.
method Proposed a min-max reformulation of LIPs.
result Saddle points characterize solutions to LIPs.
Epoch-GDA achieves optimal convergence rate for SCSC min-max problems.
problem Solving stochastic min-max problems with strong convexity and strong concavity.
method Epoch-wise stochastic gradient descent ascent method (Epoch-GDA) without additional assumptions.
result Achieves the optimal rate of O(1/T) for the duality gap of general SCSC min-max problems. The paper bounds the min-max width of embedded circles on spheres and manifolds.
problem Bounding the min-max width of embedded circles on spheres and manifolds.
method Inducing a sweepout by pairs of points in embedded circles from a given sweepout of the sphere by closed curves.
result Lower bounds for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles.
Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
Bayesian optimization methods improved for min max optimization problems.
problem Min-max optimization for unknown functions.
method Extended Bayesian optimization to min-max problems with new acquisition functions.
result Improved acquisition functions lead to better solutions.
In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold (Mn+1,g) of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension 7. We characterize the …
The study proves a generic multiplicity one theorem for G-invariant minimal hypersurfaces.
problem Proving a generic multiplicity one theorem for G-invariant minimal hypersurfaces. method Equivariant min-max theory and analysis of G-homology classes. result Shows a generic multiplicity one theorem for G-invariant minimal hypersurfaces. Constructs cmc doublings of minimal surfaces via min-max theory.
problem Construct cmc doublings of minimal surfaces.
method Uses min-max theory and catenoid estimate.
result Constructs ε-cmc doublings of Σ for small ε > 0.
This research proves that two min-max theories for hypersurfaces are equivalent.
problem Comparing two min-max theories for hypersurfaces.
method Developed and proved the equivalence of Almgren-Pitts and Allen-Cahn min-max theories.
result The Almgren-Pitts widths and Allen-Cahn widths are equivalent.
This study reveals a Min-Max property in LeNet's convolutional layers, enhancing adversarial robustness.
problem Uncertainty in the connection weights of convolutional layers in neural networks.
method Demonstrates the Min-Max property through back propagation-based training and a simplified convolution formulation.
result The Min-Max property improves adversarial robustness, indicating a stronger uncertainty in the model parameters.
New Gaussian min-max theorem extends classical results to non-i.i.d. Gaussian matrices.
problem Extending classical Gaussian min-max theorems to non-i.i.d. Gaussian matrices.
method Identifying a new pair of Gaussian processes that satisfy comparison inequalities.
result New Gaussian min-max and convex Gaussian min-max theorems with applications in multi-source Gaussian regression and binary classification.
New algorithm solves non-convex, non-differentiable min-max games.
problem Limited theoretical understanding of non-smooth min-max games.
method Proximal gradient descent-ascent algorithm for convex-strongly convex games.
result Algorithm converges to ε-Nash equilibrium with polynomial gradient evaluations.
Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
problem Finding generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
method Weyl asymptotic law for G-equivariant volume spectrum, generic density result. result Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
We characterize the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coin…
New geometric invariant from min-max width of spheres on Riemannian 2-spheres.
problem Understanding the min-max width of spheres associated to distance functions.
method Application of min-max methods to pairs of points on Riemannian 2-spheres.
result The min-max width does not always equal half the length of a simple closed geodesic.
Given a Riemannian manifold and a closed submanifold, we find a geodesic segment with free boundary on the given submanifold. This is a corollary of the min-max theory which we develop in this article for the free boundary variational problem. In particular, we develop a modified Birkhoff curve shortening process to ac…
Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.
problem Constructing stable anisotropic minimal surfaces in 3-manifolds.
method Anisotropic min-max theory, removable singularity theorems.
result Constructs stable anisotropic minimal surfaces in 3-manifolds without singularities.
In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature c. Moreover…
The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.
problem Finding smooth anisotropic minimal surfaces in closed 3-manifolds.
method Min-max construction with elliptic integrands, uniform upper bound for density ratios.
result Obtains a smooth anisotropic minimal surface in a closed 3-manifold.
In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in \cite{A2}\cite{P} corresponding to the fundamental class of a Riemannian manifold (Mn+1,g) of positive Ricci curvature with 2≤n≤6. We characterize the Morse index, area and multiplicity of this min-max hyp…
We show that the sum of the Morse indices of the Willmore spheres realising the width of Willmore type sweep-outs is bounded by the number of the parameters of the min-max. As an application, we deduce that among the true Willmore spheres realising the min-max sphere eversion, at most one of them one has index 1, while…
Improved algorithms for convex-concave min-max optimization and monotone variational inequalities.
problem Efficiently solving constrained convex-concave min-max problems and monotone variational inequalities.
method Higher-order methods achieving iteration complexities of O(1/T^{ rac{p+1}{2}}) for p-th order derivatives.
result Achieved improved convergence rates for min-max and monotone variational inequalities.
New algorithm solves min-max optimization problems in a decentralized manner.
problem Solving min-max saddle point games in a decentralized and adaptive manner.
method Developed a decentralized adaptive momentum (DADAM3) algorithm for min-max optimization. result DADAM3 achieves non-asymptotic rates of convergence for finding Nash equilibrium points. Optimizes solving complex min-max problems with stochastic and nonconvex elements.
problem Min-max problems with stochastic and nonconvex elements.
method Combines conic nonexpansiveness, refined inexact Halpern iteration, and multilevel Monte Carlo estimator.
result Optimal or best-known complexity guarantees for $ρ< rac{1}{L}$, improving previous results.
New algorithm solves structured nonconvex-nonconcave min-max problems.
problem Min-max optimization challenges in deep learning.
method Generalized extragradient algorithm for structured nonconvex-nonconcave problems.
result Algorithm converges to stationary points in Euclidean and ℓp spaces. We prove optimal genus bounds for minimal surfaces arising from the min-max construction of Simon-Smith. This confirms a conjecture made by Pitts-Rubinstein in 1986.