This paper resolves a longstanding open question pertaining to the design of near-optimal first-order algorithms for smooth and strongly-convex-strongly-concave minimax problems. Current state-of-the-art first-order algorithms find an approximate Nash equilibrium using or $\tild…
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48 results for “strongly concave function”
Two new algorithms solve nonconvex-strongly concave problems efficiently.
problem Solving nonconvex-strongly concave minimax problems.
method Proposed MINIMAX-TR and MINIMAX-TRACE algorithms.
result Find -second order stationary points within iterations.
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.
A new method solves a complex optimization problem efficiently.
problem Nonconvex-strongly-concave constrained minimax optimization.
method First-order augmented Lagrangian method with a first-order subproblem solver.
result Achieves improved operation complexity for finding solutions.
Paper improves risk bounds for nonconvex-strongly-concave minimax problems.
problem Achieving sharper risk bounds for nonconvex-strongly-concave minimax problems.
method Using uniform localized convergence to derive high probability generalization error bounds.
result Derives n times faster excess primal risk bounds for popular algorithms.
A generalized optimistic method for saddle point problems with improved complexity.
problem Solving convex-concave saddle point problems efficiently.
method Proposes a generalized optimistic method that includes the optimistic gradient method as a special case, handling constrained saddle point problems with composite objective functions and arbitrary norms.
result Best-known global iteration complexity bounds for first-, second-, and higher-order methods.
Paper analyzes complexity of solving nonconvex-strongly-concave problems.
problem Finding approximate stationary points of nonconvex-strongly-concave minimax problems.
method Introduces a generic acceleration scheme to solve crafted subproblems.
result Algorithm nearly matches lower complexity bounds in general setting.
Zigzag sampling algorithm efficiently samples from strongly log-concave distributions with low computational cost.
problem Sampling from strongly log-concave distributions efficiently and with low computational complexity.
method Zigzag sampling algorithm with warm start assumption, focusing on gradient evaluations.
result Achieves ε error in chi-square divergence with computational cost of O(κ²d^(1/2)(log(1/ε))^(3/2)) gradient evaluations.
New algorithms solve DR-submodular maximization with faster convergence.
problem Maximizing monotone DR-submodular functions under convex constraints.
method Introduced strongly DR-submodular functions and proposed SDRFW and PGA algorithms.
result SDRFW achieves optimal approximation ratio after fewer iterations.
SA algorithms control dynamic regret in non-stationary settings with strong convexity or exp-concavity.
problem Non-stationary Online Convex Optimization with dynamic regret control.
method Strongly Adaptive (SA) algorithms view dynamic regret as path variation of the comparator sequence.
result SA algorithms achieve and dynamic regret for strongly convex and exp-concave losses, respectively.
PURE-CD algorithm proves complexity bounds for convex-concave problems.
problem Solving convex-concave min-max problems with bilinear coupling.
method Primal-dual algorithm with random extrapolation and coordinate descent (PURE-CD).
result Complexity bounds match or improve existing results for dense and sparse problems.
Introduces CSLC models to bridge deep generative models and classical algorithms.
problem Mode collapse and memorization issues in deep generative models and restrictive assumptions in classical algorithms.
method Introduces conditionally strongly log-concave (CSLC) models, factorizing data distribution into strongly log-concave conditional distributions.
result Efficient parameter estimation and sampling algorithms with theoretical guarantees for non-log-concave data distributions.
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…
New algorithms solve complex minimax problems efficiently.
problem Nonconvex-strongly concave minimax problems in machine learning.
method Gradient norm regularized trust-region (GRTR) and Levenberg-Marquardt (LMNegCur) algorithms.
result Proved iteration complexities matching best known results.
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.
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.
Paper establishes lower bounds for finite-sum optimization problems using novel construction methods.
problem Lower complexity bounds for finite-sum optimization problems with various component functions.
method Developed novel approach to construct hard instances and analyzed PIFO algorithms.
result Established lower complexity bounds for convex-concave and nonconvex-strongly-concave objectives.
Strongly log-concave (SLC) distributions are a rich class of discrete probability distributions over subsets of some ground set. They are strictly more general than strongly Rayleigh (SR) distributions such as the well-known determinantal point process. While SR distributions offer elegant models of diversity, they lac…
New algorithm AG-OG optimizes separable convex-concave problems efficiently.
problem Efficiently solving separable convex-concave minimax optimization problems.
method Leverages Nesterov acceleration and optimistic gradient on component and coupling parts of the problem.
result Achieves optimal convergence rate for various settings including bilinearly coupled problems.
Epoch gradient descent method (a.k.a. Epoch-GD) proposed by Hazan and Kale (2011) was deemed a breakthrough for stochastic strongly convex minimization, which achieves the optimal convergence rate of with iterative updates for the {\it objective gap}. However, its extension to solving stochastic min-max pr…
RSGDA improves convergence rates for nonconvex-strongly concave optimization.
problem Optimization of nonconvex-strongly concave problems.
method Randomized Stochastic Gradient Descent Ascent (RSGDA) with optimal loop sizes.
result First almost sure convergence rates for SGDA algorithms on nonconvex-strongly concave settings.
We consider the convex-concave saddle point problem where is smooth and convex and is smooth and strongly convex. We prove that if the coupling matrix has full column rank, the vanilla primal-dual gradient method can achieve linear convergence even if is not stron…
New algorithms solve nonconvex-concave minimax problems without parameter knowledge.
problem Solving nonconvex-concave minimax problems efficiently.
method Three completely parameter-free single-loop algorithms.
result Achieve optimal iteration complexity for nonconvex-concave minimax problems.
This paper tackles bandit optimization with a new pairwise comparison oracle for unknown strongly concave functions.
problem Maximizing an unknown strongly concave function over T periods with a biased pairwise comparison oracle.
method Introduced a discretization technique and local polynomial approximation to relate the problem to linear bandits. Developed a tournament successive elimination technique to localize the discretized cell and run LinUCB algorithm on cells.
result Established optimal regret bounds and improved state-of-the-art results in operations management problems.
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.
Drago: Primal-Dual Coupled Variance Reduction for Faster Distributionally Robust Optimizationstat.ML
Drago optimizes DRO problems with faster convergence.
problem Distributionally robust optimization with closed, convex uncertainty sets.
method Primal-dual coupled variance reduction algorithm with cyclic and randomized updates.
result Achieves state-of-the-art linear convergence rate on strongly convex-strongly concave problems.
A distributed optimization method solves saddle point problems with strong concavity and convexity.
problem Solving saddle point problems with distributed and heterogeneous data.
method GT-GDA, a distributed first-order method using gradient tracking and consensus over coupling matrices.
result GT-GDA converges linearly to the unique saddle point solution under specific conditions.
Paper improves algorithms for convex-concave minimax optimization problems.
problem Minimizing convex-concave functions with strong convexity and concavity properties.
method Proposes a new algorithm with improved gradient complexity.
result Improves gradient complexity upper bound for minimax optimization.
The paper analyzes generalization bounds for NC-SC/NC-C stochastic minimax optimization.
problem Generalization analysis of nonconvex-(strongly)-concave stochastic minimax optimization.
method Established algorithm-agnostic and algorithm-dependent generalization bounds via uniform convergence and stability arguments.
result Sample complexities and generalization bounds for NC-SC and NC-C settings.
Gibbs sampler contracts entropy under strong log-concavity, improving mixing time.
problem Improving the mixing time of Gibbs sampler under strong log-concavity.
method Analyzing Gibbs sampler contraction under strong log-concavity, providing sharp contraction rate.
result Gibbs sampler contracts entropy linearly with condition number and independent of dimension under strong log-concavity.
This research accelerates sampling methods using Nesterov's Acceleration.
problem Improving sampling efficiency in MCMC methods.
method Developed a Hessian-Free High-Resolution ODE reformulation of NAG-SC, injected noise, and discretized the diffusion process.
result Quantified acceleration beyond underdamped Langevin in distance for log-strongly-concave targets.
Improved sampling algorithm with state-of-the-art complexity bounds.
problem Efficient sampling from various probability distributions.
method Proximal sampler with inexact restricted Gaussian oracle.
result State-of-the-art complexity bounds in almost all settings.
We consider nonconvex-concave minimax optimization problems of the form , where is strongly-concave in but possibly nonconvex in and is a convex and compact set. We focus on the stochastic setting, where we can only access an…
Random extrapolation speeds up coordinate descent for sparse and dense data.
problem Efficiently solving primal-dual coordinate descent for sparse and dense data.
method Adapts to sparsity and uses large step sizes for dense data, proving linear convergence under metric subregularity.
result Linear convergence under metric subregularity and optimal sublinear convergence rates in general convex-concave problems.
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.
Develops shuffling gradient-based methods for nonconvex-concave minimax optimization.
problem Nonconvex-concave minimax optimization problems.
method Two shuffling gradient-based algorithms for nonconvex-linear and nonconvex-strongly concave settings.
result Achieves state-of-the-art oracle complexity in nonconvex optimization and best-known complexity bounds for nonconvex-strongly concave setting.
New algorithm solves minimax games with linear constraints.
problem Nonconvex minimax games with coupled linear constraints.
method Primal-dual alternating proximal gradient (PDAPG) algorithm.
result Achieves ε-stationary solution within O(ε^(-2)) iterations for strongly concave settings.
Two algorithms solve nonconvex minimax problems with linear constraints, achieving complexity guarantees.
problem Nonconvex minimax problems with coupled linear constraints.
method Zeroth-order primal-dual alternating projected gradient (ZO-PDAPG) and zeroth-order regularized momentum primal-dual projected gradient (ZO-RMPDPG) algorithms.
result Iteration complexity guarantees for solving nonconvex-(strongly) concave minimax problems with coupled linear constraints.
This paper studies GAIL's global convergence for general MDP and nonlinear rewards.
problem Understanding when GAIL algorithms achieve global convergence for general MDP and nonlinear rewards.
method Characterization of global convergence for various policy gradient algorithms applied to GAIL.
result First systematic theoretical study of GAIL for global convergence.
We consider the classical problem of sequential resource allocation where a decision maker must repeatedly divide a budget between several resources, each with diminishing returns. This can be recast as a specific stochastic optimization problem where the objective is to maximize the cumulative reward, or equivalently …
We consider convex-concave saddle point problems with a separable structure and non-strongly convex functions. We propose an efficient stochastic block coordinate descent method using adaptive primal-dual updates, which enables flexible parallel optimization for large-scale problems. Our method shares the efficiency an…
Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Casemath.PR
New algorithms sample from log concave distributions without gradient Lipschitz continuity.
problem Sampling from log concave distributions without gradient Lipschitz continuity.
method Two algorithms based on monotone polygonal (tamed) Euler schemes.
result Non-asymptotic 2-Wasserstein distance bounds between the process and target measure.
Study on Wasserstein distance for numerical approximations of stochastic differential equations.
problem Estimating the Wasserstein distance between stochastic differential equation distributions and their numerical approximations.
method Unified framework for analyzing different integrators and a novel splitting method for underdamped Langevin dynamics.
result A novel splitting method for underdamped Langevin dynamics with optimal complexity.
New Langevin method achieves third order convergence for strongly log-concave distributions.
problem Sampling from complex distributions efficiently.
method Underdamped Langevin diffusion with third order convergence.
result Achieves 2-Wasserstein error of ε in O(√d/ε^1/3) steps under additional Lipschitz condition.
We consider the problem of sampling from a strongly log-concave density in , and prove an information theoretic lower bound on the number of stochastic gradient queries of the log density needed. Several popular sampling algorithms (including many Markov chain Monte Carlo methods) operate by using stochas…
The study provides guarantees for diffusion-based models under log-concave data, offering best-known convergence rates.
problem Theoretical guarantees for convergence of diffusion-based generative models under log-concave data distributions.
method Assumption of strongly log-concave data distributions, Lipschitz continuous functions for score estimation, and novel auxiliary process.
result Best known upper bounds for Wasserstein-2 distance between Gaussian distribution and sampling algorithm.
The paper studies stability of mean-field variational inference for log-concave distributions.
problem Stability of mean-field variational inference for log-concave distributions.
method Novel approach via linearized optimal transport, lifting non-convex problem to convex optimization over transport maps.
result Dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in 2-Wasserstein distance.
While classic work in convex-concave min-max optimization relies on average-iterate convergence results, the emergence of nonconvex applications such as training Generative Adversarial Networks has led to renewed interest in last-iterate convergence guarantees. Proving last-iterate convergence is challenging because ma…