arXiv research

Study on convergence of SDEs using entropy methods.

Paper proves convergence of SA algorithm via martingale and converse Lyapunov methods.

We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in SL(2, C), as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family.…

Unified framework for finite-sample RL algorithms using Lyapunov theory.

The paper refines optimization algorithms using Lyapunov functions and differential equations.

Operator calculus for population-based optimization provides a unified framework for analyzing convergence of various methods.

New methods accelerate gradient descent for convex and strongly convex functions.

Paper proves convergence of Gini index to equilibrium in Wasserstein distance.

Unified analysis of stochastic iterative algorithms using Lyapunov functions.

New analysis improves SGD for robust and quantile regression with sub-quadratic convergence.

In this paper, we propose a dynamical systems perspective of the Expectation-Maximization (EM) algorithm. More precisely, we can analyze the EM algorithm as a nonlinear state-space dynamical system. The EM algorithm is widely adopted for data clustering and density estimation in statistics, control systems, and machine…

Lyapunov analysis improves RNN performance prediction.

This paper analyzes and improves monotonic accelerated algorithms like M-NAG and M-FISTA.

The paper analyzes convergence of Langevin dynamics with time-dependent metrics.

Paper analyzes SA for fixed-point equations with noise, establishing convergence rates.

New algorithm reduces communication time in federated learning.

This paper introduces a novel framework to construct the region of attraction (ROA) of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics. Most existing works on estimating ROA rely on analytical Lyapunov functions, which are subject to two limitations: the analyti…

New continuous-time optimization algorithms converge in finite time to local minima.

New bounds show SGD can match deterministic gradient descent's convergence rate.

We study the dynamical behaviors of degenerate stochastic differential equations (SDEs). We select an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conduct the Lyapunov exponential convergence analysis of degenerate SDEs. We derive the convergence rate cond…

Momentum speeds up evolutionary processes in machine learning.

Neural network outperforms traditional methods in chaotic dynamics classification.

The paper analyzes how disturbances affect the convergence of algorithms in complex systems.

Gradient flossing stabilizes RNN training by controlling Lyapunov exponents.

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

This paper uses dynamical systems to analyze and ensure convergence of the Bayesian EM algorithm.

Paper achieves $ε^{-2}$ sample complexity for actor-critic methods with minimal assumptions.

The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.

The paper analyzes convergence rates for stochastic approximation and reinforcement learning.

Lyapunov 1-forms on orbifolds help understand flows on compact spaces.

Overview of non-stochastic-gradient SA algorithms in signal processing and ML.

In this paper, following J. Franks' work on Lyapunov graphs of nonsingular Smale flows on $S^3$, we study Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$. More precisely, we determine necessary and sufficient conditions on an abstract Lyapunov graph to be associated with a nonsingular Smale flow on $S^1 …

The paper accelerates ISTA and FISTA algorithms for composite optimization problems.

Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is…

Paper proves PI consensus algorithm converges exponentially under restricted secant inequality.

New factorial power constants improve optimization convergence rates.

AROS uses Lyapunov-stabilized embeddings to improve out-of-distribution detection robustness against adversarial attacks.

New method stabilizes deep neural networks by setting Lyapunov exponent to zero.

Study approximates top Lyapunov exponents for surface mapping classes.

We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov exponent on the unstable bundle then the manifold has constant negative curvature. We a…

Policy gradient algorithm with variable learning rates achieves near-optimal performance in multi-arm bandit problems.

In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in genus three with any zero Lyapunov exponents. Furthermore, if a Teichmüller curv…

Optimal algorithms for Riemannian optimization with reduced complexity.

Double Q-learning has the same mean-squared error as Q-learning under certain conditions.

New neural methods for stable control with provable guarantees.

Combines Lyapunov functions with controller synthesis for safe control policies.

The paper analyzes convergence of Riemannian SA schemes for stochastic optimization.

We determine the Lyapunov spectrum of ball quotients arising from cyclic coverings. The computations are performed by rewriting the sum of Lyapunov exponents as ratios of intersection numbers and by the analysis of the period map near boundary divisors. As a corollary, we complete the classification of commensurability…