The paper tackles robust statistical methods using Wasserstein DRO formulations.
problem Distributional uncertainty in learning from limited samples.
method Min-max distributionally robust optimization with Wasserstein DRO formulations.
result Error bounds free from the curse of dimensionality.
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.
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.
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.
Proves conjecture about sphere widths under rotational symmetry.
problem Width stability of rotationally symmetric metrics.
method Proof of conjecture and extensions to higher dimensions.
result Stability of min-max width under rotational symmetry.
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.
Wasserstein distributionally robust optimization estimators are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing the worst-case loss among all probability models within a certain distance (in a Wasserstein sense) from the underlying empirical measure. While motivated by…
Develops robust learning methods for datasets with sub-populations.
problem Robust performance and generalization to unseen testing populations in datasets with sub-populations.
method Min-max-regret (MMR) formulation for distribution-free robust hierarchical model.
result Empirical MMR enjoys regret guarantees on training and unseen testing populations.
We consider the problem of two-player zero-sum games. This problem is formulated as a min-max Markov game in the literature. The solution of this game, which is the min-max payoff, starting from a given state is called the min-max value of the state. In this work, we compute the solution of the two-player zero-sum game…
New saddle network architectures preserve convex-concave geometry in optimization problems.
problem Optimization models with convex x and concave y components.
method Structured separable decomposition and saddle network architectures.
result Proven one-dimensional approximation theorem and high accuracy on various test functions.
Framework for worst-case generation using Wasserstein space optimization.
problem Evaluating robustness and stress-testing systems under distribution shifts.
method Min-max optimization over continuous probability distributions in Wasserstein space.
result Global convergence guarantees for the proposed Gradient Descent Ascent scheme.
Adversarial training improves linear regression solutions, revealing sparsity and abrupt interpolation.
problem Adversarial attacks on linear regression models.
method Formulated as a convex problem, adversarial training is used to find robust solutions that are sparse and interpolate data.
result Adversarial training with small disturbances gives the solution with the minimum-norm that interpolates the training data, revealing abrupt transition into interpolation.
Reward suffices for convex MDPs, expanding RL to new problems.
problem Capturing goals as convex functions of stationary distribution.
method Reformulated as a min-max game using Fenchel duality.
result Convex MDPs require non-stationary reward functions.
In recent years, Generative Adversarial Networks (GANs) have drawn a lot of attentions for learning the underlying distribution of data in various applications. Despite their wide applicability, training GANs is notoriously difficult. This difficulty is due to the min-max nature of the resulting optimization problem an…
Model-free approach to hedge path-dependent options using min-max optimization.
problem Hedging path-dependent options with maturity T using a static portfolio of vanilla options.
method Model-free approach based on primal-dual Martingale Optimal Transport (MOT) problem, solving a min-max optimization problem.
result Provides theoretical bounds on hedging error at maturity T.
Deep-learning-based methods for different applications have been shown vulnerable to adversarial examples. These examples make deployment of such models in safety-critical tasks questionable. Use of deep neural networks as inverse problem solvers has generated much excitement for medical imaging including CT and MRI, b…
The paper is devoted to modeling optimal exercise strategies of the behavior of investors and issuers working with convertible bonds. This implies solution of the problems of stock price modeling, payoff computation and min-max optimization. Stock prices (underlying asset) were modeled under the assumption of the geome…
Bayesian optimization for function-valued responses, addressing worst case deviations.
problem Optimizing expensive functions with functional responses, focusing on worst case performance.
method Min-Max Functional Bayesian Optimization (MM-FBO) using Gaussian process surrogates and functional principal component analysis.
result MM-FBO consistently outperforms existing methods in synthetic and real-world applications.
Min-max formulations have attracted great attention in the ML community due to the rise of deep generative models and adversarial methods, while understanding the dynamics of gradient algorithms for solving such formulations has remained a grand challenge. As a first step, we restrict to bilinear zero-sum games and giv…
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.
New method improves PCA robustness using Wasserstein distances.
problem Uncertainty in probability distribution affects PCA robustness.
method Distributionally robust optimization with Wasserstein distances.
result Explicit reformulation leads to efficient smoothing algorithm.
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. A framework for federated adversarial learning with convergence analysis.
problem Unique vulnerabilities to adversarial attacks in federated learning.
method Formulates a general federated adversarial learning framework with inner and outer loops for client-side adversarial training and server-side model aggregation.
result The minimum loss under the proposed algorithm can converge to ε with chosen learning rate and communication rounds.
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.
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. 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. Distributionally robust optimization (DRO) problems are increasingly seen as a viable method to train machine learning models for improved model generalization. These min-max formulations, however, are more difficult to solve. We therefore provide a new stochastic gradient descent algorithm to efficiently solve this DR…
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…
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.
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.
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.
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.
The min-max problem, also known as the saddle point problem, is a class of optimization problems which minimizes and maximizes two subsets of variables simultaneously. This class of problems can be used to formulate a wide range of signal processing and communication (SPCOM) problems. Despite its popularity, most exist…
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 …
Study evaluates adversarial training for deep learning IDSs against various attacks.
problem Evasion attacks against deep learning-based IDSs.
method Investigated adversarial training using min-max approach on CNN and RNN.
result Adversarial training improves robustness against five attack methods.
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.
A dictionary is a database of standard vectors, so that other vectors / signals are expressed as linear combinations of dictionary vectors, and the task of learning a dictionary for a given data is to find a good dictionary so that the representation of data points has desirable features. Dictionary learning and the re…