Log-concave coefficient sequences for two-bridge knots proved.
problem Proving log-concavity of Alexander polynomial coefficient sequences for alternating knots.
method Introducing a polynomial Δ ( t ) Δ(t) Δ ( t ) associated to Christoffel words and proving its log-concavity. result Strong Fox conjecture for two-bridge knots proved.
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.
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 ) O(1/T) O ( 1/ T ) for the duality gap of general SCSC min-max problems. Paper solves minimax optimization gap with near-optimal algorithms.
problem Designing efficient algorithms for smooth and strongly-convex-strongly-concave minimax problems.
method Accelerated proximal point method and accelerated solver for minimax proximal steps.
result First algorithm with gradient complexity matching the lower bound up to logarithmic factors.
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.
Algorithm tackles constrained reinforcement learning with concave-convex and knapsack constraints.
problem Constrained episodic reinforcement learning with concave rewards and convex constraints.
method Modular analysis with strong theoretical guarantees for concave-convex and knapsack settings.
result Significantly outperforms existing approaches in constrained episodic environments.
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 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.
Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamenta…
Policy evaluation is a crucial step in many reinforcement-learning procedures, which estimates a value function that predicts states' long-term value under a given policy. In this paper, we focus on policy evaluation with linear function approximation over a fixed dataset. We first transform the empirical policy evalua…
The study improves fundamental gap estimates for surfaces with non-constant positive curvature.
problem Estimating the fundamental gap for surfaces with non-constant positive curvature.
method Using a two-point maximum principle, the study establishes log-concavity and fundamental gap estimates.
result Corresponding log-concavity and fundamental gap estimates for surfaces with non-constant positive curvature are derived.
Algorithm samples from composite log-concave distributions using gradient evaluations and restricted Gaussian oracles.
problem Sampling from composite log-concave distributions with limited gradient evaluations.
method Proximal gradient algorithm with RGO for g g g and strong/strongly convex conditions for f f f . result Achieves ε ε ε error in total variation distance in O ~ ( κ d log 4 ( 1 / ε ) ) \widetilde{\mathcal O}(κ\sqrt d \log^4(1/ε)) O ( κ d log 4 ( 1/ ε )) iterations. New bounds for generative models under weaker assumptions.
problem Establishing convergence guarantees for generative models under weak assumptions.
method Non-asymptotic 2-Wasserstein distance bounds for probability flow ODEs under weak log-concavity and Lipschitz continuity.
result Concrete convergence rates for generative models, including non-log-concave distributions.
Recent successes of game-theoretic formulations in ML have caused a resurgence of research interest in differentiable games. Overwhelmingly, that research focuses on methods and upper bounds on their speed of convergence. In this work, we approach the question of fundamental iteration complexity by providing lower boun…
The goal of online prediction with expert advice is to find a decision strategy which will perform almost as well as the best expert in a given pool of experts, on any sequence of outcomes. This problem has been widely studied and O ( T ) O(\sqrt{T}) O ( T ) and O ( log T ) O(\log{T}) O ( log T ) regret bounds can be achieved for convex losses (\cite{zin…
We consider the convex-concave saddle point problem min x max y f ( x ) + y ⊤ A x − g ( y ) \min_{x}\max_{y} f(x)+y^\top A x-g(y) min x max y f ( x ) + y ⊤ A x − g ( y ) where f f f is smooth and convex and g g g is smooth and strongly convex. We prove that if the coupling matrix A A A has full column rank, the vanilla primal-dual gradient method can achieve linear convergence even if f f f is not stron…
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.
Enhances SGLD for log-concave posteriors with asynchronous computation.
problem Sampling log-concave posterior distributions efficiently.
method Integrates asynchronous computation into SGLD with delayed gradients.
result Convergence in measure is not significantly affected by delayed gradient information.
Improved log-concave sampling to O ( d 1 / 2 ) O(d^{1/2}) O ( d 1/2 ) with warm starts.
problem Sampling from strongly log-concave distributions efficiently.
method Warm starts and discretized underdamped Langevin diffusion.
result Achieved O ( d 1 / 2 ) O(d^{1/2}) O ( d 1/2 ) complexity for high-accuracy sampling. 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.
The paper studies how quickly samples from Langevin dynamics become independent.
problem Understanding the dependence between samples along Langevin dynamics and related algorithms.
method Measures dependence via Φ Φ Φ -mutual information and proves strong data processing inequalities. result The Φ Φ Φ -mutual information between samples decreases exponentially to zero. New method uses momentum to converge in DC optimization with small batches.
problem Lack of convergence properties for stochastic difference-of-convex optimization with small batch sizes.
method Introduces momentum to enable convergence under standard assumptions for any batch size.
result Proves convergence of the algorithm under smoothness and bounded variance assumptions.
New algorithms improve convergence of minimax optimization.
problem Minimax optimization convergence issues in nonconvex problems.
method Established new convergence results for two single-loop algorithms.
result Improved convergence rates for minimax optimization.
This paper tackles robust control of noisy systems with uncertain distributions.
problem Optimal control of sampled-data stochastic systems with multiplicative noise and distributional ambiguity.
method Develops a convex relaxation to handle the ``concave-max'' geometry and derives a probabilistic performance guarantee.
result Derives an explicit, non-asymptotic bound on the duality gap and proves robust viability conditions.
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.
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 i l d e O ( T V T ∨ log T ) ilde O(\sqrt{TV_T} \vee \log T) i l d e O ( T V T ∨ log T ) and i l d e O ( d T V T ∨ d log T ) ilde O(\sqrt{dTV_T} \vee d\log T) i l d e O ( d T V T ∨ d log T ) dynamic regret for strongly convex and exp-concave losses, respectively. Near-logarithmic regret per switch achieved for mixable/exp-concave losses.
problem Online optimization of mixable loss functions with dynamic environments.
method Online mixture framework using static solvers and hyper-expert creations.
result Near-logarithmic regret per switch with sub-polynomial complexity.
This work proposes new methods for variational inference using gradient flows on Gaussian measures.
problem Developing algorithmic guarantees for variational inference.
method Proposes principled methods for variational inference using gradient flows on the Bures--Wasserstein space of Gaussian measures.
result Strong theoretical guarantees for log-concave posteriors.
We consider empirical risk minimization of linear predictors with convex loss functions. Such problems can be reformulated as convex-concave saddle point problems, and thus are well suitable for primal-dual first-order algorithms. However, primal-dual algorithms often require explicit strongly convex regularization in …
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.
Paper analyzes complexity of PSGLA for sampling log-concave distributions.
problem Sampling from log-concave distributions with composite potentials.
method Uses primal-dual interpretation and duality gap to analyze PSGLA complexity.
result Complexity of PSGLA is O ( 1 / ε 2 ) O(1/\varepsilon^2) O ( 1/ ε 2 ) for strongly convex potentials. 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.
pHMC converges on infinite-dimensional spaces with bounds.
problem Convergence of pHMC on Hilbert spaces.
method Coupling of two pHMC copies, adapted from arXiv:1805.00452.
result Proven convergence bounds in 1-Wasserstein distance.
New algorithm solves saddle point problems in Banach spaces.
problem Solving saddle point problems in real reflexive Banach spaces.
method Stochastic Bregman Primal-Dual Splitting Algorithm with relative smoothness and strong convexity assumptions.
result Almost sure convergence to saddle points under various conditions.
We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in the high tensor powers of the line bund…
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.
Study counterfactuals in combinatorial choice using a representative agent model.
problem Analyzing decision-making from aggregated binary polytope data.
method Nonparametric approach based on a representative agent model, solving polynomial and mixed-integer convex programs.
result Developed a method for counterfactual prediction that works even under model misspecification.
Proves log-concavity of cluster algebra coefficients for type A n A_n A n .
problem Log-concavity of cluster algebra coefficients.
method Introduced atomic theta basis and proved log-concavity for type A n A_n A n . result Proved log-concavity of coefficients for cluster algebra variables of type A n A_n A n . 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.
Previous studies on stochastic primal-dual algorithms for solving min-max problems with faster convergence heavily rely on the bilinear structure of the problem, which restricts their applicability to a narrowed range of problems. The main contribution of this paper is the design and analysis of new stochastic primal-d…
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 log u \log u log u . A hybrid impurity measure balances theoretical soundness and computational efficiency.
problem Developing a robust impurity measure for decision trees.
method Integrates Tsallis entropy with an exponential polarization component.
result Simple parametric measures outperform ITC, but ITC variants are competitive with strong theoretical guarantees.
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.
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.
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.
We define a class of L-convex-concave subsets of R P n \Bbb{R}P^n R P n , where L is a projective subspace of dimension l in R P n \Bbb{R}P^n R P n . 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-…
Geodesic concavity and hypersymplectic structures in G 2 G2 G 2 -structures space.
problem Analyzing the geodesic concavity and hypersymplectic structures in the space of closed G 2 G2 G 2 -structures. method Utilising the geodesic constructed in the previous article, we show geodesic concavity and decrease in length of G 2 G2 G 2 Laplacian flow. result Hitchin's volume functional is geodesically concave and the G 2 G2 G 2 Laplacian flow decreases the length.