Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.
problem Distance between Lorentzian manifolds.
method Intrinsic timed Hausdorff convergence.
result Intrinsic timed Hausdorff convergence implies Gromov-Hausdorff and big bang convergence.
We investigate finite-time decoupled convergence in nonlinear two-time-scale stochastic approximation.
problem Achieving decoupled convergence in nonlinear two-time-scale stochastic approximation.
method Nested local linearity assumption, suitable step size selection, convergence analysis of matrix cross term, fourth-order moment convergence rates.
result Finite-time decoupled convergence rates can be achieved in nonlinear two-time-scale stochastic approximation with proper step size selection.
In this paper, we propose two discontinuous dynamical systems in continuous time with guaranteed prescribed finite-time local convergence to strict local minima of a given cost function. Our approach consists of exploiting a Lyapunov-based differential inequality for differential inclusions, which leads to finite-time …
New method accelerates optimization in fixed time, improving convergence rates.
problem Optimization in large-scale data-driven problems.
method Gradient-based optimization framework with fixed-time stable dynamical systems.
result Achieves convergence to the optimizer in a fixed number of iterations, independent of initialization.
The paper analyzes convergence of Langevin dynamics with time-dependent metrics.
problem Analyzing convergence of Langevin dynamics with time-dependent metrics.
method Formulated a modified gradient flow of the Kullback-Leibler divergence, selected a time-dependent relative Fisher information functional, and developed a time-dependent Hessian matrix condition.
result Proved convergence conditions for various Langevin dynamics.
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.
Proves compactness for timed-metric spaces using new distance and maps.
problem Weak convergence of space-times using timed-Hausdorff distance.
method Uses Gromov's original compactness theorem and introduces addresses.
result Establishes compactness theorem for intrinsic timed-Hausdorff convergence.
Continuous-time SGD converges under certain conditions, useful for deep learning.
problem Minimizing population expected loss in learning problems.
method Continuous-time approximation of stochastic gradient descent.
result Establishes sufficient conditions for convergence, applicable to overparametrized neural networks.
In this paper, we consider multi-agent learning via online gradient descent in a class of games called λ-cocoercive games, a fairly broad class of games that admits many Nash equilibria and that properly includes unconstrained strongly monotone games. We characterize the finite-time last-iterate convergence rate for …
Continuous-time distributed mirror descent with integral feedback converges to global optimum.
problem Distributed optimization of a global strongly convex function with local convex components.
method Continuous-time distributed mirror descent with integral feedback.
result Asymptotic convergence to global optimum with constant step-size.
The paper analyzes deep neural networks using control theory to set a time limit for their convergence.
problem Understanding the finite-time convergence of deep neural networks.
method Lyapunov based analysis of the loss function, control theory framework, finite-time control of non-linear systems.
result A priori guarantees of finite-time convergence for deep neural networks are provided.
Study shows Ricci flow's convergence and harmonic map heat flow's long-time existence.
problem Analyzing convergence of Ricci flow and harmonic map heat flow.
method Established long-time existence of harmonic map heat flow between Ricci flow and shrinker.
result Ricci flow converges exponentially to compact integrable shrinkers and at singularities modelled on the shrinker.
Study approximates financial market with discrete-time models.
problem Approximating continuous-time financial market models with discrete-time.
method Constructs discrete-time market models with Markov switching and proves convergence.
result Discrete-time models converge to continuous-time Black-Scholes model with Markov switching.
Method calculates Parisian stopping times and option prices using Markov chains.
problem Computing distribution and pricing of Parisian stopping times under Markov processes.
method Continuous-time Markov chain approximation to solve for distribution and convergence analysis.
result Sharp convergence rate and efficient method for diffusion and jump models.
Develops a new parabolic equation for surfaces, proving long-time existence and convergence.
problem Extending elliptic equations to parabolic settings for surfaces.
method Introduces a parabolic analogue of the elliptic split-type Monge-Ampère equation.
result Proves long-time existence and convergence conditions for the new equation.
Temporal difference learning explained through gradient splitting, improving convergence times.
problem Learning value functions in Markov Decision Processes with linear approximations.
method Interpreting TD learning as gradient splitting and applying convergence proofs from gradient descent.
result Improved convergence times for TD learning, especially with a minor variation.
New streaming methods improve convergence rates for optimization problems.
problem Optimizing large-scale, sequential data problems.
method Time-varying mini-batches and Polyak-Ruppert averaging for gradient-based algorithms.
result Time-varying mini-batches and averaging achieve optimal convergence and variance reduction.
Gradient-based temporal difference (GTD) algorithms are widely used in off-policy learning scenarios. Among them, the two time-scale TD with gradient correction (TDC) algorithm has been shown to have superior performance. In contrast to previous studies that characterized the non-asymptotic convergence rate of TDC only…
We analyze the convergence rate of various momentum-based optimization algorithms from a dynamical systems point of view. Our analysis exploits fundamental topological properties, such as the continuous dependence of iterates on their initial conditions, to provide a simple characterization of convergence rates. In man…
GenFlow optimizes faster, avoiding saddle points in fixed time.
problem Designing efficient optimization algorithms for convex and non-convex functions.
method Introduces GenFlow and momentum variants with fixed-time convergence guarantees.
result GenFlow and momentum variants converge to optimal solutions in fixed time for PL functions and evade saddle points uniformly.
Improved convergence rates for Stein Variational Gradient Descent in finite-particle settings.
problem Improving convergence rates for Stein Variational Gradient Descent in finite-particle settings.
method Analyzing the time derivative of relative entropy and splitting it into dominant and smaller parts.
result Finite-particle convergence rates of order 1/\sqrt{N} for Kernelized Stein Discrepancy and Wasserstein-2 metrics.
Study shows financial value of weak information converges in discrete vs continuous markets.
problem Analyzing financial value of weak information in discrete vs continuous markets.
method Defined minimal probability measure and financial value of weak information, then showed convergence.
result Financial value of weak information converges in discrete vs continuous markets.
This paper analyzes discrete diffusion models, deriving convergence bounds for their generated samples.
problem Theoretical guarantees for discrete-state diffusion models remain under-explored.
method Continuous Time Markov Chain (CTMC) framework and discrete-time sampling algorithm.
result Convergence bounds for KL divergence and TV distance are derived, showing linear dependence on dimension.
Compactness theorem for timed-metric spaces established.
problem Compactness of timed-metric spaces and causality.
method Timed-Gromov--Hausdorff distance and intrinsic timed-Hausdorff distance.
result Induces same notion of convergence as intrinsic timed-Hausdorff distance.
Paper analyzes convergence rates of two time-scale AC and NAC algorithms.
problem Finite-sample convergence rate analysis of two time-scale AC and NAC algorithms.
method Developed novel techniques for bias error and convergence rate analysis.
result Established non-asymptotic convergence rates for two time-scale AC and NAC.
Study shows long-term solutions for complex equations on curved spaces.
problem Long-term behavior of solutions to fully non-linear parabolic equations on Hermitian manifolds.
method Used general assumptions and derived a Harnack inequality for the linearized equation.
result Proved the long-time existence and convergence of solutions.
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
MixML unifies analysis of weakly consistent parallel learning.
problem Lack of insight into how communication structure affects convergence in parallel learning.
method Proposes MixML framework for analyzing convergence of weakly consistent parallel machine learning.
result Shows dependency of convergence on mixing time tmix.
Introduces generalized Yamabe flows with long-time existence and convergence results.
problem Yamabe flow and its limitations.
method Introduces a family of conformal flows generalizing the classical Yamabe flow and proves long-time existence and convergence.
result Long-time existence and convergence for a large class of generalized Yamabe flows.
New analysis shows GMD can converge linearly under PL-like conditions.
problem Establishing linear convergence for generalized mirror descent.
method PL-based analysis for time-dependent mirrors, Taylor-series approach for stochastic GMD.
result Linear convergence of stochastic GMD under PL-like conditions.
Model financial default cascades on sparse graphs via hitting times.
problem Capturing systemic risk in large, sparsely-connected financial networks.
method Dynamic particle systems with hitting times and convergence theory.
result Characterization of default time distribution in tree-like networks.
Paper proposes a mean-field gradient descent for zero-sum games, proving convergence to Nash equilibrium.
problem Finding mixed Nash equilibria in zero-sum games with multiple players.
method Mean-field gradient descent dynamics with time-averaging, incorporating exponentially discounted gradients.
result Exponential convergence rate to mixed Nash equilibrium with respect to total variation metric.
Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
problem Accelerating convex optimization
method Hamiltonian dynamics
result Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
Study shows how a curve shortens to a half-circle under specific flow.
problem Stability of a semi-circle under curve shortening flow.
method Sharp rate of convergence for a free-boundary curve shortening flow in a convex domain.
result Established a sharp rate of convergence to a round half-point.
This work analyzes Q-learning with adaptive stepsizes for finite-time convergence.
problem Finite-time convergence analysis for average-reward Q-learning with adaptive stepsizes. method Adaptive stepsizes as local clocks, time-inhomogeneous Markovian reformulation, almost-sure time-varying bounds, conditioning arguments, and Markov chain concentration inequalities.
result Convergence rates of ildeO(1/k) for mean-square and pointwise mean-square convergence. 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.
Geometric tempering improves sampling from distributions, with exponential convergence rates.
problem Sampling from probability distributions using gradient flow dynamics.
method Geometric tempering of the target distribution in Wasserstein and Fisher-Rao gradient flows.
result Exponential convergence in continuous and discrete time for geometric tempering.
Mutation improves FTRL convergence in zero-sum games.
problem Lack of last-iterate convergence in FTRL variants.
method Introduced mutation to perturb action probabilities in FTRL.
result M-FTRL converges to Nash equilibria under full-information feedback.
Paper analyzes convergence of two time-scale stochastic approximation using martingale approach.
problem Analyzing convergence of two time-scale stochastic approximation algorithms.
method Uses martingale approach to establish convergence conditions and rates.
result Establishes different rates of convergence for fast and slow subsystems.
HF-opt uses Hamiltonian dynamics to optimize functions, achieving accelerated rates with randomized integration time.
problem Optimizing functions efficiently and accelerating convergence rates.
method Randomized Hamiltonian flow (RHF) with accelerated convergence rates.
result RHGD achieves accelerated convergence rates similar to Nesterov's AGD.
This work accelerates gradient descent with anytime convergence guarantees.
problem Improving the convergence rate of gradient descent methods.
method Proposes a stepsize schedule for gradient descent that achieves anytime convergence rates.
result Gradient descent can achieve convergence rates of O(T−1.119) for any stopping time T. The paper improves theoretical bounds on deep neural networks' convergence.
problem Understanding convergence of over-parameterized deep neural networks.
method Surrogate network construction with fixed activation patterns.
result Convergence to a global minimum guaranteed for networks with quadratic width and linear depth.
Recent theoretical results establish that time-consistent valuations (i.e. pricing operators) can be created by backward iteration of one-period valuations. In this paper we investigate the continuous-time limits of well-known actuarial premium principles when such backward iteration procedures are applied. We show tha…
New examples of mean curvature flow converge to minimal surfaces with multiplicity 2.
problem Constructing mean curvature flow examples in closed manifolds.
method Constructing new examples of mean curvature flow with convergence to minimal surfaces with multiplicity 2.
result Mean curvature flow examples converge to minimal surfaces with multiplicity 2.
This study analyzes AdaGrad's stability and convergence in non-convex optimization.
problem Lack of theoretical analysis for AdaGrad in non-convex optimization.
method Novel stopping time-based techniques from probability theory.
result Established stability and derived convergence rates for AdaGrad.
Paper shows no finite time singularities for smooth conformal heat flow of harmonic maps.
problem Smoothness of conformal heat flow of harmonic maps.
method Combines harmonic map flow with metric evolution in conformal direction.
result No finite time singularity occurs for the flow, and under certain conditions, maps converge to a point.
Paper analyzes convergence of dynamic policy gradient for MDPs, improving performance in finite-time problems.
problem Optimal policies in finite-time MDPs are not stationary and require epoch-specific training.
method Introduces dynamic policy gradient combining dynamic programming and policy gradient, analyzes convergence for softmax parametrisation.
result Dynamic policy gradient training exploits finite-time structure, leading to better convergence bounds.
New example of surface flow converging to a plane with multiplicity 2.
problem Constructing mean curvature flows with specific convergence properties.
method Constructing a new example of a mean curvature flow in R3. result The flow converges to a plane with multiplicity 2 as time approaches infinity.