Survey of methods for solving smooth stochastic variational inequalities.
problem Solving smooth (strongly) monotone stochastic variational inequalities.
method Deterministic foundation, general stochastic formulation, finite sum setup, recent advances.
result Review of various methods for solving smooth stochastic variational inequalities.
Neural networks solve variational inequalities for optimal stopping problems.
problem Solving variational inequalities for optimal stopping problems in finance.
method Proposed neural network approach using loss functions directly incorporating variational inequality on whole domain.
result Existence and convergence of neural networks whose losses converge to zero.
A variational inequality for pricing the perpetual American option and the corresponding difference equation are considered. First, the maximum principle and uniqueness of the solution to variational inequality for pricing the perpetual American option are proved. Then the maximum principle, the existence and uniquenes…
This paper combines three techniques to reduce communications in distributed variational inequalities.
problem Efficiently communicating solutions in large-scale distributed variational inequalities.
method Combining similarity, compression, and local steps to reduce communication rounds and cost.
result Best theoretical guarantees of communication complexity and superior performance in adversarial learning experiments.
The extragradient method fails for hypomonotone variational inequalities.
problem The convergence of the extragradient method for hypomonotone variational inequalities.
method Application of the extragradient method to hypomonotone linear operators.
result The extragradient method diverges for hypomonotone variational inequalities.
Extends tracking guarantees for time-varying variational inequalities.
problem Tracking solutions of time-varying variational inequalities.
method Extends existing results to sublinear solution paths and periodic problems.
result Discrete dynamical systems of periodic time-varying VI can exhibit chaotic behavior or converge to the solution.
Study finds loops with specific curvature exist using Hardy's inequality.
problem Existence of closed planar loops with prescribed curvature.
method Variational approach, Hardy's inequality and associated functional space.
result Existence of loops with specific curvature proven.
New proof of Gaffney's inequality for differential forms on manifolds with boundary.
problem Proving Gaffney's inequality for differential forms on manifolds with boundary.
method Variational approach combined with Bochner's technique.
result New proof of Gaffney's inequality for differential forms.
A new method solves variational inequality problems with multiple constraints without needing optimal Lagrange multipliers.
problem Solving variational inequality problems with multiple functional constraints efficiently.
method Constrained Gradient Method (CGM) for Minty variational inequality problems.
result The Constrained Gradient Method achieves complexity similar to projection-based methods but with cheaper oracles.
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 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.
Unified analysis of efficient local training methods for distributed variational inequalities.
problem Efficient distributed/federated learning for variational inequality problems.
method Unified convergence analysis of communication-efficient local training methods.
result First local gradient descent-accent algorithms with improved communication complexity.
Unified approach for first-order methods with Markovian noise in stochastic optimization and variational inequalities.
problem Stochastic optimization problems with Markovian noise.
method Unified theoretical analysis of first-order gradient methods using randomized batching and multilevel Monte Carlo.
result Optimal (linear) dependence on the mixing time of the noise sequence, eliminating previous limiting assumptions.
Simplified proof for Frank and Lieb's inequality on Heisenberg group.
problem Proving the sharp Frank-Lieb inequality on the Heisenberg group.
method Simpler proof based on 2nd variation of subcritical functionals.
result A simpler proof of the inequality without the need for minimizer existence.
Through the direct study of the analysis estimator we derive oracle inequalities with fast and slow rates by adapting the arguments involving projections by Dalalyan, Hebiri and Lederer (2017). We then extend the theory to the square root analysis estimator. Finally, we focus on (square root) total variation regularize…
Market makers optimize trading with a new implicit scheme for complex inequalities.
problem Optimizing trading in a limit order book with stochastic and impulse control.
method Implicit numerical scheme coupled with policy iteration algorithm.
result Convergence to the unique viscosity solution of the HJBQVI.
This paper examines challenges and solutions for solving variational inequalities.
problem Stability issues in solving variational inequalities, especially in multi-objective scenarios.
method Continuous-time analysis to understand and improve stability of algorithms.
result Understanding continuous-time dynamics can help in designing more stable algorithms for variational inequalities.
Sharp inequalities and eigenvalue problems on Finsler manifolds with nonnegative Ricci curvature.
problem Establishing sharp Morrey-Sobolev inequalities and eigenvalue problems on Finsler manifolds.
method Combining sharp isoperimetric inequality and anisotropic symmetrization argument.
result Existence and multiplicity of solutions for eigenvalue problems and elliptic PDEs.
In this paper we first introduce two new financial products: stock loan and capped stock loan. Then we develop a pure variational inequality method to establish explicitly the values of these stock loans. Finally, we work out ranges of fair values of parameters associated with the loans.
Study on new Monge-Ampère functionals and their variational problems.
problem Existence and uniqueness of solutions for nonlinear eigenvalue problems.
method Introduction of a family of real Monge-Ampère functionals and proving Sobolev type inequalities.
result Existence of solutions for a nonlinear eigenvalue problem.
The paper proves inequalities for submanifolds in Riemannian manifolds.
problem Proving geometric inequalities for submanifolds in Riemannian manifolds.
method Using Rauch's comparison theorem and first variation formula.
result General Li-Yau inequality applicable in bounded sectional curvature manifolds.
New mass inequalities and proofs for causal variational principles.
problem Proving new mass inequalities for causal variational principles.
method Proved a new inequality for minimizers of causal variational principles and applied it to prove the positive mass theorem.
result Introduced a positive quasilocal mass and proved new mass inequalities.
Sharp inequality between TV and Hellinger distances for Gaussian mixtures.
problem Understanding the relationship between total variation and Hellinger distances for Gaussian mixtures.
method Established a general upper bound on Hellinger distance in terms of TV distance raised to a power, demonstrating sharpness with specific examples.
result The Hellinger distance between two Gaussian mixtures is bounded by the TV distance raised to a power 1 − o ( 1 ) 1-o(1) 1 − o ( 1 ) , where o ( 1 ) o(1) o ( 1 ) is of order 1 / log log ( 1 / T V ) 1/\log\log(1/\mathrm{TV}) 1/ log log ( 1/ TV ) . Study on surfaces in product space with curvature inequality.
problem Characterizing surfaces in S n i m e s R \mathbb{S}^n imes \mathbb{R} S n im es R with total mean curvature. method Defined differential operators and proved integral inequalities.
result Integral inequality for closed stationary H \mathcal{H} H -surfaces in S n i m e s R \mathbb{S}^n imes \mathbb{R} S n im es R . 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 ) Θ(1/ε) Θ ( 1/ ε ) for learning variational inequalities. Paper presents a method to solve variational inequalities with general constraints without requiring analytic solutions.
problem Solving variational inequalities with general constraints.
method A primal-dual approach using approximate subproblem solutions and warm-starting.
result The method converges with a rate of O ( 1 K ) O(\frac{1}{\sqrt{K}}) O ( K 1 ) for L L L -Lipschitz and monotone operators. 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.
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.
Establishes a Li-Yau type inequality for curves in any codimension.
problem Finding a lower bound for the normalized bending energy of curves in Euclidean space of any codimension.
method Variational approach, Langer-Singer's classification of elasticae, André's algebraic-independence theorem.
result Optimal inequality for any codimension except for planar closed curves with odd multiplicity.
Sharp Sobolev inequality on circle proven with Lorentz invariance.
problem Proving a sharp Sobolev inequality on the circle.
method Rearrangement inequality, variational method, Lorentz transformation.
result Sharp inequality and global minimizer on circle.
Generative adversarial networks (GANs) form a generative modeling approach known for producing appealing samples, but they are notably difficult to train. One common way to tackle this issue has been to propose new formulations of the GAN objective. Yet, surprisingly few studies have looked at optimization methods desi…
Algorithm improves variational inference in Wasserstein distance.
problem Improving variational inference methods for complex models.
method Wasserstein contraction analysis of coordinate ascent.
result General and sharp convergence guarantees for various models.
We propose a family of variational approximations to Bayesian posterior distributions, called α α α -VB, with provable statistical guarantees. The standard variational approximation is a special case of α α α -VB with α = 1 α=1 α = 1 . When α ∈ ( 0 , 1 ] α\in(0,1] α ∈ ( 0 , 1 ] , a novel class of variational inequalities are developed for linking the Bayes risk …
This paper explores geometric and analytic aspects of Lojasiewicz inequalities on vector bundles.
problem Understanding growth and stability conditions for real-analytic functions over vector bundles.
method Outline theory of functionals and variational problems over vector bundles, explore applications to real-analytic functionals.
result Describes the energy functional on $S^{n-1$ as a functional over a vector bundle.
Study measures inequality in social-economic systems using Fokker-Planck equations and Lotka-Volterra dynamics.
problem Measuring inequality in oscillatory social-economic systems described by Fokker-Planck equations and Lotka-Volterra dynamics.
method Used Fokker-Planck equations and Lotka-Volterra dynamics to model inequality, focusing on coefficient of variation as a measure.
result Inequality initially tends to decrease in oscillatory systems, contrary to steady-state models.
Nonnegative sectional curvature linked to matrix displacement convexity.
problem Nonnegative sectional curvature in Riemannian manifolds.
method Matrix displacement convexity as a criterion for nonnegative sectional curvature.
result Entropy functional matrix displacement convexity implies nonnegative sectional curvature.
Improved Bayesian inference via variational approximations of generalized rho-posteriors.
problem Robust Bayesian inference under model misspecification and data contamination.
method Introducing a modified ρ ρ ρ -posterior and using PAC-Bayesian analysis with variational approximations. result Theoretical guarantees for tractable inference with competitive robustness and computational efficiency.
Study optimal consumption with drawdown limits over a fixed time frame.
problem Maximizing utility with consumption limits during a fixed period.
method Extended utility maximization problem with drawdown constraint, using PDE arguments and dual transform.
result Existence and uniqueness of classical solution to HJB variational inequality, with explicit free boundaries.
Alternative neural network training using monotone variational inequality.
problem Training neural networks efficiently and with guarantees.
method Using monotone variational inequality to solve non-convex problems efficiently.
result Our approach leads to fast convergence and competitive performance compared to traditional methods.
We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish the existence of an optimal domain. In doing so, this proof bypasses the main d…
We introduce several novel change of measure inequalities for two families of divergences: f f f -divergences and α α α -divergences. We show how the variational representation for f f f -divergences leads to novel change of measure inequalities. We also present a multiplicative change of measure inequality for α α α -divergences …
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.
New adaptive methods for constrained convex optimization and variational inequalities.
problem Optimization of constrained convex problems and variational inequalities.
method AdaACSA and AdaAGD+ are accelerated methods that achieve nearly-optimal convergence rates for smooth and non-smooth functions.
result Achieve nearly-optimal convergence rates for both smooth and non-smooth functions, even with stochastic gradients.
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.
In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study ( κ , N ) (κ,N) ( κ , N ) -convex functions on metric spaces where κ κ κ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that …
The paper analyzes variational autoencoders for state space models with risk bounds.
problem Analyzing the risk associated with variational autoencoders for state space models.
method Backward factorization of variational distributions to analyze excess risk, providing oracle inequalities and upper bounds.
result Explicit upper bounds on variational estimation error for state space models under strong mixing assumptions.
MER algorithm speeds up VI solving with Markovian data.
problem Solving stochastic variational inequalities with Markovian data.
method MER algorithm using multi-scale sampling from a Markovian buffer.
result Achieves faster convergence without knowing Markov chain mixing time.
The paper studies constant mean curvature hypersurfaces in Finsler manifolds.
problem Understanding geometric properties of hypersurfaces in Finsler manifolds.
method Using volume preserving variation and homothetic navigation.
result Deduced a Heintze-Karcher type inequality and proved an Alexandrov type theorem.