Non-ergodic measures found in horocycle flow on Abelian differentials.
problem Finding non-ergodic measures in the horocycle flow on Abelian differentials.
method Analyzing weak convergence of ergodic measures to non-ergodic invariant measures.
result Existence of points with non-equidistributing horocycle flow orbits.
Paper improves Heavy-ball method convergence in convex settings.
problem Convergence analysis of Heavy-ball method in convex optimization.
method Improved convergence complexity results for Heavy-ball method with constant step size.
result First non-ergodic O(1/k) rate result for coercive objective functions.
Non-ergodic geodesic flow on Cantor tree surfaces found.
problem Determining when geodesic flow on Cantor tree surfaces is non-ergodic.
method Interpolating between two rates of convergence of cuff lengths to zero to prove non-ergodicity.
result Cantor tree surfaces with certain rates of cuff length convergence are non-parabolic.
Strong stability of ergodic iterations proven without ergodic driving sequence.
problem Ensuring strong stability of ergodic iterations under non-ergodic driving sequences.
method Revisiting processes driven by stationary ergodic sequences, proving strong stability under mild conditions on recursive maps.
result Strong stability of iterations proven without ergodic driving sequence.
This paper analyzes the convergence of dynamic HMC and NUTS methods.
problem Theoretical understanding of dynamic HMC and NUTS convergence.
method General class of MCMC algorithms, NUTS as a particular case, geometric ergodicity, irreducibility.
result NUTS is geometrically ergodic under certain conditions and ergodic without bounded stepsize.
Actor-critic converges globally in LQR with ergodic cost.
problem Theoretical understanding of actor-critic algorithm's global convergence.
method Nonasymptotic convergence analysis of actor-critic in linear quadratic regulator (LQR) setting.
result Actor-critic finds globally optimal policy and value function at a linear rate.
The paper studies harmonic map heat flow to flat tori, proving ergodic behavior and convergence to hyperbolic measure.
problem Analyzing the behavior of harmonic map heat flow to moduli space of flat tori.
method Investigates stability and ergodic behavior of harmonic map heat flow using hyperbolic structure and relative entropy.
result The flow converges weak--∗ to the normalized hyperbolic measure on the moduli space. We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…
This paper compares two NUTS variants and analyzes their convergence and mixing times.
problem Theoretical comparison and convergence guarantees of NUTS variants.
method Deriving necessary and sufficient conditions for geometric ergodicity, and analyzing mixing times.
result NUTS-mul and NUTS-BPS have nearly identical qualitative behavior but differ quantitatively in convergence rates.
Improves estimation of financial market models using limited data.
problem Limited data and computational constraints in estimating financial market models.
method Analyzed ergodic properties of moment functions and used Monte Carlo experiments.
result Understanding ergodic properties can improve estimation of financial market models.
New concentration inequality for U-statistics of Markov chains.
problem Proving a concentration inequality for U-statistics of order two in uniformly ergodic Markov chains.
method Inductive analysis using martingale techniques, uniform ergodicity, Nummelin splitting, and Bernstein's inequality.
result Recovery of convergence rate for U-statistics of independent random variables and canonical kernels, with improved results for dependent kernels.
Elliptical slice sampling converges geometrically, providing reliable sampling for Bayesian learning.
problem Sampling from posterior distributions in Bayesian learning.
method Elliptical slice sampling, geometric ergodicity.
result Elliptical slice sampling yields geometric convergence guarantees under weak regularity assumptions.
We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the foldin…
The paper establishes CLTs for Markov chains and improves sampling algorithms for heavy-tailed distributions.
problem Establishing central limit theorems for ergodic averages of Markov chains.
method Drift conditions to provide necessary and sufficient conditions for CLTs, including lower bounds on convergence rates.
result Sharp conditions and convergence rates for various MCMC algorithms on heavy-tailed targets.
We generalize stochastic subgradient descent methods to situations in which we do not receive independent samples from the distribution over which we optimize, but instead receive samples that are coupled over time. We show that as long as the source of randomness is suitably ergodic---it converges quickly enough to a …
Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or fat-shattering) dimension of F is a combinatorial quantity that measures the extent to which functions f in F can separate finite sets of points at a predefined resolution gamma > 0. We establish a connection between the g…
Convergence of Siegel-Veech constants for weakly convergent measures on translation surfaces.
problem Convergence of Siegel-Veech constants for weakly convergent measures on translation surfaces.
method Recurrence result related to Eskin-Masur techniques, measure equidistribution result.
result Convergence of sequences of Siegel-Veech constants associated to Teichmüller curves in genus two.
Develops Patterson-Sullivan theory for coarse cocycles.
problem None explicitly stated in the abstract.
method Theory of Patterson--Sullivan measures for coarse cocycles of convergence groups.
result Existence, uniqueness, and ergodicity results for Patterson-Sullivan measures under geometric assumptions.
New rates for GLD and SGLD in infinite-dimensional spaces without dimensionality issues.
problem Gradient Langevin dynamics and SGLD convergence rates in high-dimensional spaces.
method Analysis of GLD and SGLD in infinite-dimensional Hilbert spaces, using stochastic differential equations and Markov chains.
result Derivation of dimension-free convergence rates for GLD and SGLD.
Deep neural networks improve online learning by ensuring convergence to best strategies.
problem Challenges in online learning due to dependencies between observations.
method Lipschitz regularized deep neural networks for online learning.
result Guaranteed convergence to the best prediction strategy.
The paper tackles learning to control systems with unknown parameters using Brownian noise.
problem Learning to control systems with unknown parameters.
method Proposes algorithms based on moving empirical averages and integrates statistical methods with stochastic control theory.
result Achieves a logarithmic expected regret rate.
New MCMC methods map high-dimensional problems to spheres for better mixing.
problem Mixing issues in high-dimensional distributions, especially heavy-tailed ones.
method Stereographic Markov Chain Monte Carlo (MCMC) methods that map high-dimensional problems to spheres.
result Uniformly ergodic samplers for various distributions, including heavy-tailed ones, with faster convergence in higher dimensions.
Last iterate of Extragradient algorithm converges slower than averaged iterates in saddle point problems.
problem Smooth convex-concave saddle point problems
method Analysis of Extragradient (EG) algorithm convergence rates
result The last iterate of EG converges at a rate of O(1/√T), compared to O(1/T) for averaged iterates
Using elements from the theory of ergodic backward stochastic differential equations (BSDE), we study the behavior of forward entropic risk measures. We provide their general representation results (via both BSDE and convex duality) and examine their behavior for risk positions of long maturities. We show that forward …
New method estimates convergence bounds for nonlinear Markov chains.
problem Difficulty in describing properties of nonlinear Markov chains.
method Coupling Markov chains to reconstitute distribution relationships and estimate convergence bounds.
result Estimation of convergence bounds is more precise than existing results.
Estimates mixing coefficients of geometrically ergodic Markov processes from a single sample path.
problem Estimating mixing coefficients of geometrically ergodic Markov processes.
method Proposes methods to estimate β-mixing coefficients from a single sample path under standard smoothness conditions. result Obtains a rate of convergence of order \(\mathcal{O}(\log(n) n^{-[s]/(2[s]+2)})\) for the expected error of the estimator.
Neural networks trained with actor-critic algorithms converge to ODEs under weak convergence analysis.
problem Challenges in convergence analysis due to changing data distributions in online learning.
method Geometric ergodicity of data samples, Poisson equation, weak convergence techniques.
result Actor and critic networks converge to solutions of ODEs with random initial conditions.
Paper analyzes complexity of proximal inertial gradient descent.
problem Computational complexity of proximal inertial gradient descent.
method Analyzed convergence rates and proved various rates under different conditions.
result Proved non-ergodic O(1/k) rate for coercive objective functions.
The paper studies convergence of kernel autocovariance operators for stationary processes.
problem Estimating autocovariance operators of stationary processes on Polish spaces.
method Investigates convergence of empirical estimates of autocovariance operators under various conditions.
result Provides consistency results for kernel PCA and spectral analysis methods.
The paper develops a model for sovereign debt dynamics with explicit maturity structure.
problem Analyzing the sustainability and risk of long-term sovereign debt issuance.
method Discrete-time model with explicit maturity structure, deterministic and stochastic extensions.
result The model identifies conditions for ergodic convergence and derives analytical formulas for key metrics.
Neural networks' weights don't converge to stationary points but training loss stabilizes.
problem The disconnect between theoretical analyses and neural network training practice.
method An invariant measure perspective inspired by ergodic theory of dynamical systems.
result The distribution of weights converges to an approximate invariant measure, explaining loss stabilization.
We analyze the generalization and robustness of the batched weighted average algorithm for V-geometrically ergodic Markov data. This algorithm is a good alternative to the empirical risk minimization algorithm when the latter suffers from overfitting or when optimizing the empirical risk is hard. For the generalization…
New algorithm optimizes nonlinear SDEs online with convergence guarantees.
problem Optimizing nonlinear stochastic differential equations (SDEs) is computationally challenging.
method Forward propagation algorithm that solves an SDE derived using forward differentiation.
result Convergence theorem for nonlinear dissipative SDEs with bounds on stochastic fluctuations.
New method improves convergence of gradient descent for non-convex, non-reversible Markov chains.
problem Improving convergence of gradient descent for non-convex, non-reversible Markov chains.
method Introducing a new technique that varies the mixing levels of the Markov chains to establish non-ergodic convergence under wider step sizes.
result Established non-ergodic convergence for non-convex problems and non-reversible finite-state Markov chains.
Optimizes MCMC chains with neural control variates.
problem Reducing variance in Markov Chain Monte Carlo (MCMC) simulations.
method Uses neural networks as control variates to minimize asymptotic variance.
result Derives optimal convergence rate under various ergodicity assumptions.
Continuity of earthquake flow map transfers Teichmüller dynamics results.
problem Transfer results from Teichmüller dynamics to earthquake flow.
method Analyze continuity of earthquake flow map and its inverse.
result Transfer results from Teichmüller dynamics to earthquake flow.
Unified analysis of Langevin dynamics for nonconvex optimization with improved convergence rates.
problem Global convergence of Langevin dynamics based algorithms for nonconvex optimization.
method Unified framework analyzing numerical approximations to Langevin dynamics.
result Improved convergence rates for gradient Langevin dynamics and stochastic gradient Langevin dynamics.
New algorithm minimizes sum of three functions with linear operator.
problem Minimizing the sum of three convex functions with a linear operator.
method Proposes a new primal-dual algorithm for the problem.
result Proves convergence and provides convergence rates.
A learning algorithm achieves logarithmic regret in a market making model.
problem Learning the price sensitivity parameter in a market making model.
method Maximum-likelihood estimator with regularization, based on HJB equation.
result Regret upper bound of order ln^2 T in expectation.
Study on Langevin dynamics convergence rates and their application to GAN training.
problem Understanding the long-term behavior of Langevin dynamics equations.
method Analytical and numerical methods to study convergence rates of underdamped mean-field Langevin dynamics.
result Exponential convergence rate results for the Langevin dynamics under various conditions.
AdaBoost is one of the most popular ML algorithms. It is simple to implement and often found very effective by practitioners, while still being mathematically elegant and theoretically sound. AdaBoost's interesting behavior in practice still puzzles the ML community. We address the algorithm's stability and establish m…
Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of wh…
Study uniform learnability of binary classification networks with communication.
problem Learning a network with communication between vertices from uniform ergodic Random Graph Process.
method Introduced structural Rademacher complexity and used martingale method and Marton's coupling.
result Uniform learnability as worst-case theoretical limits for binary classification problems.
Unfolding paths in Outer space accumulate on a simplex, not converge.
problem Understanding accumulation points in Outer space.
method Constructing an unfolding path in Outer space.
result Unfolding paths accumulate on a 1-simplex, not converge.
This paper tackles online estimation of diffusion process parameters.
problem Estimating parameters of partially observed diffusion processes online.
method Stochastic gradient ascent on incomplete-data log-likelihood.
result Convergence of the algorithm proved under ergodicity conditions.
The paper introduces reservoir computing models for complex systems.
problem Modeling complex engineering systems using nonlinear autoregression.
method Introduces reservoir computing with output feedback as stationary and ergodic infinite-order nonlinear autoregressive models.
result Demonstrates versatility of classical and quantum reservoir computers in modeling synthetic and real data.
Riemann moduli spaces are quantum ergodic for certain dimensions.
problem Quantum ergodicity of Riemann moduli spaces.
method Analysis of Weil--Petersson metric and geodesic flow.
result Riemann moduli spaces Mg,n are quantum ergodic for 3g+n≥4. We introduce a deformation of Riemann surfaces and we are interested in the convergence of this deformation to a point of the Gardiner-masur boundary of Teichmueller space. This deformation, which we call the horocyclic deformation, is directed by a projective measured foliation and belongs to a certain horocycle in a …