Related papers: Stabilization of stochastic approximation by step …

In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations, by means of tools…

Recently, many machine learning optimizers have been analysed considering them as the asymptotic limit of some differential equations when the step size goes to zero. In other words, the optimizers can be seen as a finite difference scheme…

Stochastic approximation is a framework unifying many random iterative algorithms occurring in a diverse range of applications. The stability of the process is often difficult to verify in practical applications and the process may even be…

We develop a practical approach to establish the stability, that is, the recurrence in a given set, of a large class of controlled Markov chains. These processes arise in various areas of applied science and encompass important numerical…

We study systems on time scales that are generalizations of classical differential or difference equations. In this paper we consider linear systems and their small nonlinear perturbations. In terms of time scales and of eigenvalues of…

The performance of standard stochastic approximation implementations can vary significantly based on the choice of the steplength sequence, and in general, little guidance is provided about good choices. Motivated by this gap, in the first…

This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use…

This paper presents new sufficient conditions for convergence and asymptotic or exponential stability of a stochastic discrete-time system, under which the constructed Lyapunov function always decreases in expectation along the system's…

We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much…

We consider the subgradient method with constant step size for minimizing locally Lipschitz semi-algebraic functions. In order to analyze the behavior of its iterates in the vicinity of a local minimum, we introduce a notion of discrete…

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of…

The problem of designing adaptive stepsize sequences for the gradient descent method applied to convex and locally smooth functions is studied. We take an adaptive control perspective and design update rules for the stepsize that make use…

We develop a method to prove almost global stability of stochastic differential equations in the sense that almost every initial point (with respect to the Lebesgue measure) is asymptotically attracted to the origin with unit probability.…

We present a theoretical analysis of stochastic optimization methods in terms of their sensitivity with respect to the step size. We identify a key quantity that, for each method, describes how the performance degrades as the step size…

For stochastic approximation algorithms with discontinuous dynamics, it is shown that under suitable distributional assumptions, the interpolated iterates track a Fillipov solution of the limiting differential inclusion. In addition, we…

We decide the stability and compute the Lyapunov exponent of continuous-time linear switching systems with a guaranteed dwell time. The main result asserts that the discretization method with step size~$h$ approximates the Lyapunov exponent…

In this paper, we study numerical approximations for stochastic differential equations (SDEs) that use adaptive step sizes. In particular, we consider a general setting where decisions to reduce step sizes are allowed to depend on the…

This paper analyzes a (1, $\lambda$)-Evolution Strategy, a randomized comparison-based adaptive search algorithm, optimizing a linear function with a linear constraint. The algorithm uses resampling to handle the constraint. Two cases are…

We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much…

Stiff and chaotic differential equations are challenging for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution…