arXiv research

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 …

The paper analyzes deep neural networks using control theory to set a time limit for their convergence.

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 …

We investigate finite-time decoupled convergence in nonlinear two-time-scale stochastic approximation.

First-order method solves stochastic bilevel optimization with linear constraints.

Study shows how a curve shortens to a half-circle under specific flow.

Paper analyzes finite-time convergence of double Q-learning.

Curve shortening flow converges to a point with entropy bound.

Theoretical justification for asymmetric actor-critic algorithms in reinforcement learning.

Paper analyzes convergence of dynamic policy gradient for MDPs, improving performance in finite-time problems.

Given any embedded Lagrangian on a four dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class which develops a finite time singularity under mean curvature flow. This contradicts a weaker version of the Thomas-Yau conjecture regarding long time existence and convergence of Lag…

We study finite-time collapsing limits of the continuity method. When the continuity method starting from a rational initial Kähler metric on a projective manifold encounters a finite-time volume collapsing, this projective manifold admits a Fano fibration over a lower dimensional base. In this case, we prove the conti…

This work analyzes $Q$-learning with adaptive stepsizes for finite-time convergence.

Q-learning with neural network function approximation (neural Q-learning for short) is among the most prevalent deep reinforcement learning algorithms. Despite its empirical success, the non-asymptotic convergence rate of neural Q-learning remains virtually unknown. In this paper, we present a finite-time analysis of a…

The Kähler-Ricci flow's singularities are analyzed with bounds and convergence results.

Improved TD learning with tail averaging and regularization achieves optimal convergence rates.

Cyclic coordinate descent identifies models in finite time and converges linearly.

The article explores Helfrich flow with spontaneous curvature, finding singularities and convergence behaviors.

We show that strictly convex surfaces expanding by the inverse Gauss curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.

New scalable MARL framework for dynamic networked systems.

Study on SA with heavy-tailed and LRD noise, establishing finite-time bounds.

Constructing eigenfunctions for finite-time singularities in Lagrangian mean curvature flow

We show some results for the $L^2$ curvature flow linked by the theme of addressing collapsing phenomena. First we show long time existence and convergence of the flow for $SO(3)$-invariant initial data on $S^3$, as well as a long time existence and convergence statement for three-manifolds with initial $L^2$ norm of c…

New analysis shows actor-critic method converges efficiently in practical settings.

This work analyzes actor-critic methods for faster convergence.

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametr…

We use the momentum construction of Calabi to study the conical Kähler-Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov-Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts the cone divisor to a single point …

Paper analyzes SVGD algorithm for non-asymptotic convergence.

Stochastic gradient descent (SGD) is almost ubiquitously used for training non-convex optimization tasks. Recently, a hypothesis proposed by Keskar et al. [2017] that large batch methods tend to converge to sharp minimizers has received increasing attention. We theoretically justify this hypothesis by providing new pro…

Simulated annealing is a popular method for approaching the solution of a global optimization problem. Existing results on its performance apply to discrete combinatorial optimization where the optimization variables can assume only a finite set of possible values. We introduce a new general formulation of simulated an…

Paper analyzes convergence of decentralized algorithms with noise and bias.

Paper shows no finite time singularities for smooth conformal heat flow of harmonic maps.

Sharp chord-arc estimates for curve shortening flow on spheres.

Study extends convergence results to noncompact Ricci flows.

Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC (SGHMC), and the stochastic gradient thermostat. While finite-time convergence properties of th…

We show that strictly convex surfaces contracting with normal velocity equal to |A|^2 shrink to a point in finite time. After appropriate rescaling, they converge to spheres. We indicate how we used a computer to find the main test function.

Inverse mean curvature flow converges to a disk in hyperbolic space.

We study the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time $T$ or as $t \to +\infty$.

We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. Firstly, we prove an analogue to Huisken's distance comparison principle for curve shortening flow for initial curves whose local total cu…

NSGLD improves SGLD for non-convex optimization problems.

Flow approach solves Toda system equations.

In this work, convergence of evolving Finslerian metrics first in a general flow next under Finslerian Ricci flow is studied. More intuitively it is proved that a family of Finslerian metrics $g(t)$ which are solutions to the Finslerian Ricci flow converge in $C^{\infty}$ to a smooth limit Finslerian metric as $t$ ap…

Several results on existence and convergence of the Yang-Mills flow in dimension four are given. We show that a singularity modeled on an instanton cannot form within finite time. Given low initial self-dual energy, we then study convergence of the flow at infinite time. If an Uhlenbeck limit is anti-self-dual and has …

We study the behaviour of the Kähler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable Kähler class, then the fibers collapse in finite time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base.

Classifies self-similar solutions for heat equations with positive speed.

New method improves generalization in deep learning models.

The paper studies how convex hypersurfaces evolve under curvature flows in space forms.

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a pinching condition in a hyperbolic space form to a round point in finite time.