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 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. New algorithms solve stochastic variational inequalities without bounded variance assumption.
problem Solving stochastic variational inequalities without bounded variance assumption.
method Developed algorithms for two classes of problems: monotone and structured nonmonotone VIs.
result Oracle complexity of O(ε^-4) for solving VIs with unbounded domains and possibly unbounded variance.
New algorithms reduce variance in solving complex mathematical problems.
problem Solving convex-concave saddle point problems, variational inequalities, and inclusions.
method Stochastic variance reduction for extragradient, forward-backward-forward, and forward-reflected-backward methods.
result All proposed methods converge with complexities matching or improving deterministic counterparts.
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…
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. Robust VB framework handles contamination using min-max median aggregation.
problem Handling contamination and outliers in datasets.
method Partition data into subsets, formulate robust optimization problem, use min-max median KL divergence.
result Min-max median formulation improves robustness and statistical rates.
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.
In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus g≥2. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a …
In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in mach…
Variational method for eigenvalues on manifolds.
problem Optimizing functionals involving eigenvalues of Riemannian manifolds.
method New Palais-Smale sequences and min-max methods for locally-Lipschitz functionals.
result Convergence of Palais-Smale sequences in Laplace and Steklov eigenvalues.
Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.
problem Existence of Yamabe metrics on conical 4-manifolds with singular points.
method Min-max scheme adapted to singular setting, leveraging recent positive mass theorems.
result Existence of Yamabe metrics on conical 4-manifolds with finitely-many singular points.
Study min-max theory for hypersurfaces with boundary constraints.
problem Finding minimal hypersurfaces with boundary constraints.
method Schoen-Simon-type regularity result for integral varifolds, proving existence of closed hypersurfaces with specific properties.
result Existence of a closed C1,1 hypersurface with codimension ≥7 singular set in the interior. 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.
Improved variational inequality algorithms using adaptive step sizes.
problem Solving monotone variational inequalities and convex-concave min-max problems efficiently.
method Adaptive step sizes that eliminate hyperparameters and global Lipschitz continuity requirements.
result Eliminated the need for the golden ratio in the algorithm and improved complexity bounds.
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.
The study bounds the topology of free boundary minimal surfaces in 3D manifolds.
problem Understanding the topology of free boundary minimal surfaces in compact 3D manifolds.
method Establishing general bounds on the topology via min-max methods and analyzing varifolds.
result The first Betti number is lower semicontinuous in the limit of min-max sequences.
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.
Constructs a unique surface in a ball with specific properties.
problem Creating a minimal surface with specific topological and geometric constraints.
method Variational methods, equivariant min-max theory, nontrivial sweepout.
result First genus one critical catenoid in a unit ball.
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.
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.
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.
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. New insights into stochastic methods for solving variational inequalities.
problem Understanding convergence behaviors of stochastic algorithms in variational inequalities.
method Re-casting SEG/SGDA as Markov Chains to analyze their probabilistic structures.
result The average iterate is asymptotically normal with a unique invariant distribution for various VIPs.
We study global variational properties of the space of solutions to −ε2Δu+W′(u)=0 on any closed Riemannian manifold M. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show t…
The paper characterizes gaps in minimal foliations on tori using energy criteria.
problem Characterizing gaps in minimal foliations on tori.
method Introduced an energy to study min-max theory and applied it to Almgren-Pitts min-max theory.
result For a generic metric, if a lamination contains a gap, there exists a non-area-minimizing minimal hypersurface inside the gap.
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.
Paper defines a new functional for spinors on Euclidean manifolds.
problem No specific problem stated; focuses on a new functional.
method Variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.
result Ricci flow is the gradient flow of the new functional.
New nonlocal minimal surfaces on manifolds, proving Yau's conjecture.
problem Proving Yau's conjecture for nonlocal minimal surfaces.
method Introducing nonlocal minimal surfaces and applying min-max variational methods.
result Construction of infinitely many nonlocal s-minimal surfaces on closed manifolds. New framework models algorithmic decisions affecting data distribution, enabling efficient learning.
problem Algorithmic decisions can alter data distribution, affecting model performance.
method Model performative effects as push-forward measures, estimating gradients under shift operators.
result Prove convexity of performative risk, allowing more accurate models to be harder to classify.
Variational Auto-Encoders enforce their learned intermediate latent-space data distribution to be a simple distribution, such as an isotropic Gaussian. However, this causes the posterior collapse problem and loses manifold structure which can be important for datasets such as facial images. A GAN can transform a simple…
Study finds existence of Q-curvature metrics on even-dimensional manifolds with conical singularities.
problem Existence of Q-curvature metrics on manifolds with conical singularities. method Blow-up analysis of a 2mth-order PDE and variational min-max argument. result First existence result for supercritical conic manifolds (except spheres).
New study shows min-max algorithms can converge to non-stationary points.
problem Challenges in min-max optimization due to periodic cycles and spurious attractors.
method Analyzed state-of-the-art algorithms and heuristics in non-convex/non-concave problems.
result Spurious attractors can prevent min-max algorithms from reaching true optima.
The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the min-max minimal hypersurface. We advance the theory further and prove the first general Morse index bounds for minimal hypersurface…
This paper shows how to learn variational inequalities fast with strong monotonicity.
problem Learning variational inequalities efficiently.
method Extending convex optimization techniques to variational inequalities with strong monotonicity.
result Fast generalization rates of Θ(1/ε) for learning variational inequalities. Paper tackles multi-block min-max optimization with applications in deep AUC maximization.
problem Multi-block min-max bilevel optimization with non-convex strongly-concave upper level and strongly convex lower level.
method Single-loop randomized stochastic algorithm for constant number of blocks per iteration.
result Sample complexity of O(1/ε^4) for finding ε-stationary point, matching optimal complexity.
New minimal hypersphere found in 4-sphere solving Bernstein problem.
problem Spherical Bernstein problem in S4 method Equivariant min-max theory for G-invariant minimal hypersurfaces result Construction of embedded non-equatorial minimal hypersphere
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.
Expands learning paradigm to stochastic orders using Choquet-Toland distance and Variational Dominance Criterion.
problem Learning high-dimensional distributions with stochastic orders.
method Introduces Choquet-Toland distance and Variational Dominance Criterion, uses input convex maxout networks (ICMNs).
result Proposes surrogates for Choquet-Toland distance and Variational Dominance Criterion with parametric rates.
A novel feature selection method for SVM improves model accuracy and interpretability.
problem Feature selection in nonlinear SVM classification problems.
method Embedded min-max optimization problem, leveraging duality theory.
result Improves model accuracy and interpretability on benchmark data sets.
General fuzzy min-max (GFMM) neural network is a generalization of fuzzy neural networks formed by hyperbox fuzzy sets for classification and clustering problems. Two principle algorithms are deployed to train this type of neural network, i.e., incremental learning and agglomerative learning. This paper presents a comp…
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.
New methods reduce communication in distributed training for variational inequalities.
problem Reducing communication in distributed training for high-dimensional models.
method Distributed methods with compressed communication for solving variational inequalities.
result Theoretical guarantees and practical algorithms for compressed communication.
Paper tackles fast convergence for non-convex strongly-concave min-max problems.
problem Non-convex strongly-concave min-max problems in deep learning.
method Proximal stage-based method with PL condition for faster convergence.
result Established fast convergence in primal objective gap and duality gap.
In this paper, we study the problem of constrained robust (min-max) optimization ina black-box setting, where the desired optimizer cannot access the gradients of the objective function but may query its values. We present a principled optimization framework, integrating a zeroth-order (ZO) gradient estimator with an a…
We obtain in arbitrary codimension a removability result on the order of singularity of Willmore surfaces realising the width of Willmore min-max problems on spheres. As a consequence, out of the twelve families of non-planar minimal surfaces in R3 of total curvature greater than −12π, only three of them …
Riemannian algorithms converge at Euclidean rates for geodesically convex-concave problems.
problem Min-max optimization on Riemannian manifolds.
method RCEG method and RGDA for geodesically strongly-convex-concave problems.
result RCEG achieves linear convergence rate in geodesically strongly-convex-concave cases.
Lower bounds found for nonconvex-strongly-concave min-max optimization problems.
problem Finding stationary points in nonconvex-strongly-concave min-max optimization.
method Provided lower bounds for first-order oracle complexity.
result Lower bounds of Ω(√κε⁻²) for deterministic oracles and Ω(√κε⁻² + κ¹/₃ε⁻⁴) for stochastic oracles.