The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.
problem Convergence of Lipschitz functions and sets defined by equations.
method Painlevé-Kuratowski convergence applied to Lipschitz functions and sets defined by equations.
result Generalizations and reverses of classical theorems on convergence of functions and sets.
Study shows convergence speed for Fekete points on specific sets.
problem Understanding convergence speed for Fekete points on certain sets.
method Demonstrates (Cα,Cα′)-regularity for uniformly polynomially cuspidal sets. result Established convergence speed for Fekete points on these sets.
Paper revisits set membership estimation for linear systems with relaxed disturbance bounds.
problem Set membership estimation for linear systems with disturbances bounded by convex sets.
method Adopted block-martingale small-ball condition and random perturbed control policies to establish convergence rates.
result Established convergence rates for disturbances bounded by general convex sets.
In this paper, we will show that Hausdorff convergence and varifold convergence coincide on the class of almost minimal sets.
We propose a distributed approach to train deep neural networks (DNNs), which has guaranteed convergence theoretically and great scalability empirically: close to 6 times faster on instance of ImageNet data set when run with 6 machines. The proposed scheme is close to optimally scalable in terms of number of machines, …
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…
We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where eve…
Policy gradient converges linearly with Hadamard parameterization in tabular settings.
problem Convergence of policy gradient methods under Hadamard parameterization.
method Studied convergence rate and established linear convergence after k0 iterations. result Algorithm converges linearly with rate $O( rac{1}{k})$ and faster locally after k0. Uniform counting formulas for orthogeodesics in Kleinian groups converge.
problem Counting orthogeodesics in Kleinian groups converging to a limit.
method Spectral gap of the limit manifold and geodesic flow mixing property.
result Asymptotically uniform counting formulas for orthogeodesics.
We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…
Efficient algorithm converges to Nash equilibrium in bilinear problems with bandit feedback.
problem Learning dynamics in bilinear saddle-point problems with bandit feedback.
method Uncoupled learning algorithm combining experimental design and FTRL with a tailored regularizer.
result Last-iterate convergence rate of ildeO(T−1/4) in high probability. We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.
Algorithm converges to Nash equilibria in competitive games.
problem Finding Nash equilibria in decentralized, competitive Markov games.
method Decentralized Optimistic Gradient Descent/Ascent with a critic.
result Converges to the set of Nash equilibria under self-play.
Study convergence and approximations of entropic regularized Wasserstein distances for Gaussian and RKHS measures.
problem Convergence and approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
method Analysis of convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
result Strictly weaker convergence in 2-Sinkhorn divergence for Gaussian measures compared to exact 2-Wasserstein distance.
Paper proposes a working set algorithm for non-convex sparse regression with provable convergence.
problem Estimating sparse linear models from high-dimensional data using non-convex regularizers.
method FireWorks algorithm based on non-convex reformulation and leveraging residual geometry.
result Convergence to a stationary point of the full problem with provable guarantees.
This work establishes uniform convergence of subdifferentials in stochastic optimization.
problem Understanding how empirical stationary points approximate population ones in nonsmooth, nonconvex stochastic optimization.
method Reduction principle for weakly convex stochastic objectives, focusing on subgradient convergence.
result Sharp uniform convergence rates for subdifferential mappings in stochastic convex-composite optimization.
Study shows thresholding scheme converges for mean curvature flow of convex sets.
problem Analyzing convergence of thresholding scheme for mean curvature flow.
method Time discretization using Merriman, Bence and Osher's scheme, focusing on two-phase mean convex settings.
result Time-integrated energy of approximation converges to limit's energy in minimizing movements interpretation.
MSGD outperforms SGD in overparametrized settings with faster convergence rates.
problem Optimization of non-convex functions with momentum.
method Momentum Stochastic Gradient Descent (MSGD) with rigorous analysis.
result MSGD converges exponentially faster than SGD in overparametrized settings.
This paper presents a convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We pr…
Study on Nesterov's method in stochastic settings, revealing divergence under certain conditions.
problem Understanding Nesterov's method in stochastic settings, especially finite-sum.
method Analysis of Nesterov's accelerated gradient method in stochastic and finite-sum settings.
result Nesterov's method may diverge in finite-sum settings without additional conditions.
Screening rules help identify active sets in optimization problems.
problem Identifying active sets in optimization problems.
method Screening rules based on subdifferential sets and optimality conditions.
result The number of iterations needed depends only on the convergence rate.
The paper examines convergence of distances in Lipschitz structures on manifolds.
problem Convergence of distances in Lipschitz vector fields and norms on manifolds.
method Analysis of convergence of distances associated to converging structures of Lipschitz vector fields and norms.
result Under mild controllability assumption, distances converge locally uniformly to the limit Carnot-Carathéodory distance.
Study continuity equation on Hopf and Inoue surfaces, proving estimates and convergence.
problem Analyzing the continuity equation on specific complex surfaces.
method Extended La Nave-Tian's continuity equation to Hermitian setting, proving estimates and Gromov-Hausdorff convergence.
result Proved a priori estimates for solutions on Hopf and Inoue surfaces, and convergence of Inoue surfaces to a circle.
Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to nois…
The paper reconstructs Lorentzian spacetimes from causal sets.
problem Reconstructing Lorentzian spacetimes from causal sets.
method Introduced a concept of isomorphy and three types of convergence.
result Established Gromov's reconstruction theorem in Lorentzian geometry.
Policy gradient methods achieve linear convergence in simple MDPs.
problem Analyzing convergence rates of policy gradient methods in finite MDPs.
method Connections with policy iteration to show linear convergence with large step-sizes.
result Policy gradient methods succeed with large step-sizes and achieve linear rate of convergence.
We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss equation and some natural characterizations of the regular part of the Gaussian …
Study shows gap between uniform convergence and test error in random feature models.
problem Understanding the gap between uniform convergence and test error in random feature models.
method Analytical expressions for uniform convergence over norm balls, interpolators, and minimum norm interpolator risk derived and proved.
result Uniform convergence over interpolators still gives a non-trivial bound of test error even when classical uniform convergence is vacuous.
Paper studies stochastic optimization methods with momentum, proving convergence and avoiding traps.
problem Optimizing non-convex functions with momentum.
method Unified analysis of stochastic gradient descent variants, including S-NAG and Adam.
result Convergence to critical points and avoidance of undesired critical points like local maxima or saddle points.
Study continuity of limit sets in symmetric spaces.
problem Continuity of limit sets for geometrically finite subgroups in symmetric spaces.
method Extended geometrically finite representations theory.
result Limit sets vary continuously with respect to Hausdorff distance under strong convergence.
Recent years have witnessed exciting progress in the study of stochastic variance reduced gradient methods (e.g., SVRG, SAGA), their accelerated variants (e.g, Katyusha) and their extensions in many different settings (e.g., online, sparse, asynchronous, distributed). Among them, accelerated methods enjoy improved conv…
This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean…
Unified analysis of Federated Averaging and Nesterov FedAvg for linear speedup.
problem Understanding convergence of FL algorithms under non-i.i.d. data and partial participation.
method Systematic study of convergence guarantees for FedAvg and Nesterov FedAvg under different conditions.
result Unified analysis of linear speedup for FedAvg and Nesterov FedAvg in various settings.
New insights into continual learning for deep models, showing convergence issues but local linear solutions.
problem Challenges in continual learning for homogeneous deep models.
method Sequential projections onto task margin sets, leveraging nonconvex projection theory.
result Local linear convergence under certain conditions for homogeneous deep networks.
Paper proves EM algorithm convergence for mixtures of discrete and continuous parameters.
problem Nontrivial convergence analysis for EM algorithms with mixed-integer parameters.
method Introduces conditions for EM convergence in mixed-integer optimization.
result Proves convergence of EM-based sparse Bayesian learning algorithm.
The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.
problem Stability in compact finite dimensional Alexandrov spaces.
method Equivariant Gromov--Hausdorff convergence and almost commutative diagrams.
result Stability result in compact finite dimensional Alexandrov spaces.
In our previous article [Rad16], we investigated the asymptotic behaviour of orthogonal Bianchi class B perfect fluids close to the initial singularity and proved the Strong Cosmic Censorship conjecture in this setting. In several of the statements, the case of a stiff fluid had to be excluded. The present paper fills …
Rare Teichmüller disks converge to small limit sets.
problem Understanding limit sets of Teichmüller disks.
method Analyzing Thurston boundary of Teichmüller space.
result Teichmüller disks with smallest limit sets are exceptional.
Gradient descent achieves exact linear convergence rate for symmetric matrix completion.
problem Low-rank symmetric matrix completion using gradient descent.
method Local analysis of gradient descent for symmetric matrices without additional assumptions.
result Closed-form expression of exact linear convergence rate matches practice.
Paper analyzes TD(λ) convergence rates for arbitrary features.
problem Convergence rates for linear TD(λ) under arbitrary features. method Developed a novel stochastic approximation result for arbitrary features.
result Established L2 convergence rates for linear TD(λ) without linearly independent features assumption. Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we cons…
Improved convergence rates for saddle-point optimization algorithms.
problem Understanding last-iterate convergence rates for saddle-point optimization algorithms in constrained settings.
method Expanding the understanding of last-iterate convergence for Optimistic Gradient Descent Ascent (OGDA) and Optimistic Multiplicative Weights Update (OMWU) in the constrained setting.
result Linear last-iterate convergence achieved with a universal constant learning rate for OMWU in bilinear games over the simplex.
Study Gromov-Hausdorff convergence of metric pairs and tuples.
problem Understanding convergence in metric spaces.
method Prove equivalence of definitions, embedding, completeness, and compactness theorems.
result Relative version of Fukaya's theorem and finiteness theorem for stratified spaces.
In this note we give necessary and sufficient conditions for the validity of the local spectral convergence, in balls, on the RCD∗-setting.
In this paper we prove that for a given Kähler-Ricci flow with uniformly bounded Ricci curvatures in an arbitrary dimension, for every sequence of times ti converging to infinity, there exists a subsequence such that (M,g(ti+t))→(Y,gˉ(t)) and the convergence is smooth outside a singular set (which is a …
Variational inference methods for latent variable statistical models have gained popularity because they are relatively fast, can handle large data sets, and have deterministic convergence guarantees. However, in practice it is unclear whether the fixed point identified by the variational inference algorithm is a local…
New study shows faster convergence of SGD and Kaczmarz methods.
problem Improving convergence rates of iterative linear system solvers.
method Last-iterate convergence analysis of SGD with greedy step size over smooth quadratics.
result The t-th iterate attains an O(1/t3/4) convergence rate. Study proves existence and convergence of discrete-time Kyle models with multiple insiders.
problem Existence and convergence of discrete-time Kyle models with multiple informed traders.
method Proves existence and convergence of discrete-time Kyle models with multiple informed traders using mathematical proofs.
result Equilibrium exists and converges to continuous-time equilibrium as the number of trading times increases.