Paper develops algorithms for solving non-convex non-concave problems with applications in GAN training.
problem Solving non-convex non-concave min-max saddle-point problems.
method Inexact proximal point method with strongly monotone mappings.
result First-order convergence to a nearly stationary solution of the original min-max problem.
Improved sampling guarantees for weakly log-concave distributions.
problem Sampling from distributions that are not strongly log-concave.
method Proximal sampler with convergence guarantees under weaker assumptions.
result New state-of-the-art sampling guarantees for various target distributions.
Decentralized method solves saddle point problems with theoretical guarantees.
problem Solving saddle point problems in a decentralized network.
method Proximal point method adapted for decentralized networks.
result Converges to approximate stationarity with rate of O(1/√T).
New algorithm solves complex non-convex problems efficiently.
problem Non-smooth non-convex problems with weakly convex and strongly concave components.
method Stochastic Moreau envelope approximate gradient method (SMAG).
result First single-loop algorithm with state-of-the-art convergence rate.
New method improves sampling for weakly log-concave posteriors.
problem Sampling from weakly log-concave posterior distributions.
method Stochastic Langevin Monte Carlo with over-damped diffusion.
result Simulation horizon is (dlog(n)2)(1+r)2 with Poisson subsampling. The paper extends risk measures to two-step approximations and studies log-concave distributions.
problem Extending classical risk measures to two-step approximations.
method Optimization problem for determining optimal regime thresholds and values for log-concave distributions.
result Conditions for the uniqueness of regime changing in log-concave distributions.
We establish C2,α estimates for PDE of the form convex + a sum of weakly concave functions of the Hessian, thus generalising a recent result of Collins which is in turn inspired by a theorem of Caffarelli and Yuan. Independently, we also prove an existence result for a certain generalised Monge-Ampère PDE.
Proves Calabi-Yau theorem for certain nonnegative curvature manifolds.
problem Proving a Calabi-Yau type theorem for specific manifolds.
method Existence result for bounded regions with weakly mean-concave boundary.
result Proves contractibility of certain manifolds with positive scalar curvature.
This paper optimizes Bayesian estimation for log-concave models using Langevin Monte-Carlo.
problem Optimizing Bayesian estimators for log-concave models with Langevin Monte-Carlo.
method Quantitative statistical bounds and numerical approximation of Gibbs measures.
result Established optimal numerical strategy and its cost for Bayesian posterior mean approximation.
We consider the problem of sampling from a strongly log-concave density in Rd, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin dif…
Paper provides exponential convergence guarantees for Iterative Markovian Fitting.
problem Addressing the Schrödinger Bridge problem in computational optimal transport and generative modeling.
method Develops non-asymptotic exponential convergence guarantees for Iterative Markovian Fitting.
result First non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions.
We study revenue optimization learning algorithms for repeated posted-price auctions where a seller interacts with a single strategic buyer that holds a fixed private valuation for a good and seeks to maximize his cumulative discounted surplus. For this setting, first, we propose a novel algorithm that never decreases …
New method samples from non-log-concave distributions with weak dissipativity.
problem Sampling from distributions that are not log-concave and weakly dissipative.
method Taming scheme tailored to growth and decay properties of the target distribution.
result Explicit non-asymptotic guarantees for KL, TV, and Wasserstein distances.
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. In this paper, we consider an online optimization process, where the objective functions are not convex (nor concave) but instead belong to a broad class of continuous submodular functions. We first propose a variant of the Frank-Wolfe algorithm that has access to the full gradient of the objective functions. We show t…
The study identifies criteria for 3-manifolds to be boundaries of exotic 4-manifolds.
problem Determining which 3-manifolds can be boundaries of exotic 4-manifolds.
method Provided criteria and examples of 3-manifolds that can be boundaries of 4-manifolds with infinitely many distinct smooth structures.
result Identified specific types of 3-manifolds (weakly fillable contact, non-vanishing Heegaard Floer invariant) that are boundaries of 4-manifolds with infinitely many distinct smooth structures.
New stability bounds for Sinkhorn's algorithm in entropic optimal transport.
problem Stability and convergence of Sinkhorn's algorithm for entropic optimal transport.
method Semiconcavity approach to analyze stability and convergence.
result Exponential convergence of Sinkhorn's algorithm under semiconcavity conditions.
PAPAL algorithm finds mixed Nash equilibria in continuous games.
problem Finding mixed Nash equilibria in non-convex, non-concave games.
method Particle-based Primal-Dual Algorithm (PAPAL) for weakly entropy-regularized min-max optimization.
result PAPAL offers non-asymptotic convergence guarantees for ε-mixed Nash equilibrium. Improved bounds for MALA in non-convex sampling problems.
problem Sampling from non-convex distributions in high dimensions.
method Metropolis-adjusted Langevin algorithm (MALA) with improved bounds.
result MALA is faster than competitors in many challenging scenarios.
Unified analysis of KL divergence using shifted composition for sampling.
problem Sampling from target distributions with KL divergence guarantees.
method Shifted composition rule applied to KL divergence, combining local error analysis and Girsanov's theorem.
result Unified KL guarantees for strongly log-concave, weakly log-concave, and log-Sobolev distributions.
CD methods tackle nonconvex optimization with three terms, achieving critical points.
problem Minimizing nonconvex functions with specific structure.
method Developed randomized CD, randomly permuted CD, and accelerated CD methods.
result CD methods converge to critical points with sublinear complexity.
Concave elliptic operators yield concave functions on cohomology.
problem Understanding concave functions on cohomology.
method General construction of concave elliptic operators.
result Generalized Khovanskii-Teissier inequalities.
The paper establishes conditions for strict power concavity in convolutions.
problem Conditions for strict power concavity in convolutions.
method Analyzes sufficient conditions for strict parabolic power concavity of convolutions.
result Establishes sufficient conditions for strict power concavity of convolutions.
Minimal graph level sets are concave if boundary is concave.
problem Understanding curvature of minimal graph level sets.
method Proved an inequality and showed geometric properties.
result Level sets of minimal graphs are concave if boundary is concave.
Simple connection between Harnack inequalities and concavity of arrival time functions.
problem Proving differential Harnack inequalities for various flows.
method Directly proving concavity properties of time-of-arrival functions for a class of flows using a concavity maximum principle.
result Short proof of Hamilton's and Andrews' differential Harnack inequalities.
Proves log-concavity of cluster algebra coefficients for type An.
problem Log-concavity of cluster algebra coefficients.
method Introduced atomic theta basis and proved log-concavity for type An. result Proved log-concavity of coefficients for cluster algebra variables of type An. Study improves sampling from non-log-concave distributions using Fisher information.
problem Sampling from non-log-concave distributions with high Fisher information guarantees.
method Proximal sampler with RGO implementation, leveraging log-concave sampling results.
result Improved complexity guarantee in relative Fisher information for non-log-concave sampling.
Weakly Einstein Kähler surfaces are characterized and classified.
problem Characterizing and classifying weakly Einstein Kähler surfaces.
method Several conditions and constructions to characterize and classify weakly Einstein Kähler surfaces.
result Classification of weakly Einstein Kähler surfaces with specific properties and construction of new examples.
Established concavity principle for curved spaces.
problem Solving equations on curved spaces with nonnegative curvature.
method Applied concavity principle to elliptic and parabolic equations on locally symmetric spaces with nonnegative curvature.
result First general concavity principle on spaces with non-constant sectional curvature.
Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.
problem Quantifying the Hessian of log-concave eigenfunctions on convex domains.
method Analyzes log-concavity properties of the first Dirichlet eigenfunction on convex domains.
result Obtains quantitative estimates for the Hessian of logu. The study examines weakly Einstein Lie groups and proves non-existence for certain types.
problem Characterizing and proving the non-existence of weakly Einstein Lie groups.
method Analyzing left-invariant metrics on Lie groups and using algebraic properties.
result No weakly Einstein non-abelian 2-step nilpotent Lie groups exist.
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.
Investigates concavity of spacetimes, showing conditions for local concavity.
problem Understanding the concavity of spacetimes in Finsler geometry.
method Analyzes flag curvature and future capsules to characterize concavity.
result Berwald spacetimes are locally concave if and only if their flag curvature is nonnegative in timelike directions.
Heat flow fails to preserve concavity in curved spaces.
problem Non-preservation of concavity properties in curved spaces.
method Analysis of Dirichlet heat flow on Riemannian manifolds.
result No concavity properties are preserved unless curvature is zero.
Classifies weakly Einstein submanifolds in space forms satisfying specific equalities.
problem Characterizing submanifolds in space forms with certain geometric properties.
method Classification based on Chen's equality and semisymmetric conditions.
result Classification of weakly Einstein submanifolds in space forms.
The paper constructs random concave functions on the unit simplex.
problem Understanding probability measures on spaces of concave functions.
method Constructing random concave functions via a scaled minimum of random hyperplanes.
result There is a transition from deterministic to non-trivial limiting distributions as the number of hyperplanes increases.
The study explores weakly p-Kähler hyperbolic manifolds.
problem Generalization and application of weakly p-Kähler hyperbolic manifolds. method Investigation of generalizations and applications.
result Exploration of weakly p-Kähler hyperbolic manifolds. We define a class of L-convex-concave subsets of RPn, where L is a projective subspace of dimension l in RPn. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on L'. We introduce an L-duality for these sets, and prove that the L-dual to an L-…
Improved sampling guarantees for underdamped Langevin Monte Carlo without restrictive assumptions.
problem Sampling from unnormalized densities with improved guarantees and acceleration.
method Novel analysis relaxing assumptions on log-Sobolev inequality and Hessian smoothness, using Rényi discretization bounds.
result First KL divergence guarantees for ULMC without Hessian smoothness under strong log-concavity.
Geodesic concavity and hypersymplectic structures in G2-structures space.
problem Analyzing the geodesic concavity and hypersymplectic structures in the space of closed G2-structures. method Utilising the geodesic constructed in the previous article, we show geodesic concavity and decrease in length of G2 Laplacian flow. result Hitchin's volume functional is geodesically concave and the G2 Laplacian flow decreases the length. Gradient methods converge exponentially in concave network games.
problem Finding Nash equilibria in concave network zero-sum games.
method Gradient Ascent and Optimistic Gradient Ascent analyses.
result Exponential convergence rates in various game settings.
This study examines how earnings announcements affect option volatility and pricing.
problem The impact of earnings announcements on option volatility and pricing.
method Analysis of extremely short-term options data to study bimodality and concavity in IV curves.
result Investors pay a premium to hedge against extreme volatility during earnings announcements in the presence of concave IV smiles.
Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.
problem Proving log-concavity of eigenfunctions on curved surfaces.
method Analyzing the Laplacian eigenfunctions on positively curved surfaces.
result Strong log-concavity of the first eigenfunction on positively curved surfaces.
Establishes a concavity property for positive Hessian quotient operators.
problem Analyzing positive Hessian quotient operators on Riemannian manifolds.
method Proves a special concavity property and a Jacobi inequality.
result Proves a Jacobi inequality for symmetric tensors.
Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.
problem Characterizing weakly weighted Einstein-Finsler metrics.
method Showed isotropic S-curvature under certain conditions. Characterized via navigation expressions and α and β. result Weakly weighted Einstein-Kropina metrics have isotropic S-curvature and can be completely characterized.
Unified routing and arbitrage with concave continuation.
problem Combining routing and arbitrage in financial markets.
method Extending AMM trade functions to negative inputs via concave continuation.
result Unified approach unifies routing and arbitrage.
The study proves non-existence of concave functions on specific metric spaces.
problem Proving the non-existence of concave functions on certain metric spaces.
method Analogue theorems for Alexandrov spaces and Cα-Hölder Riemannian manifolds. result Proves non-existence of concave functions on complete manifolds with finite volume and specific metric spaces.
Log-concavity proven for multinomial likelihoods under specific constraints.
problem Log-concavity of multinomial likelihoods under interval censoring constraints.
method Proved log-concavity by showing M-convex subsets of the discrete simplex.
result Likelihood function is completely log-concave.