Related papers: An ODE Method to Prove the Geometric Convergence o…
For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives.…
We prove novel convergence results for a stochastic proximal gradient algorithm suitable for solving a large class of convex optimization problems, where a convex objective function is given by the sum of a smooth and a possibly non-smooth…
In this paper we address the convergence of stochastic approximation when the functions to be minimized are not convex and nonsmooth. We show that the "mean-limit" approach to the convergence which leads, for smooth problems, to the ODE…
Recent years have seen a growing interest in understanding acceleration methods through the lens of ordinary differential equations (ODEs). Despite the theoretical advancements, translating the rapid convergence observed in continuous-time…
Many machine learning and optimization algorithms are built upon the framework of stochastic approximation (SA), for which the selection of step-size (or learning rate) $\{\alpha_n\}$ is crucial for success. An essential condition for…
Theory and application of stochastic approximation (SA) have become increasingly relevant due in part to applications in optimization and reinforcement learning. This paper takes a new look at SA with constant step-size $\alpha>0$, defined…
We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show…
We study stochastic approximation algorithms with Markovian noise and constant step-size $\alpha$. We develop a method based on infinitesimal generator comparisons to study the bias of the algorithm, which is the expected difference between…
Adam is a popular variant of stochastic gradient descent for finding a local minimizer of a function. In the constant stepsize regime, assuming that the objective function is differentiable and non-convex, we establish the convergence in…
Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with simulation of dynamical systems. Their convergence rates have been established by a growing body of…
For minimizing a strongly convex objective function subject to linear inequality constraints, we consider a penalty approach that allows one to utilize stochastic methods for problems with a large number of constraints and/or objective…
We give a development of the ODE method for the analysis of recursive algorithms described by a stochastic recursion. With variability modelled via an underlying Markov process, and under general assumptions, the following results are…
In this paper, we consider multi-stage stochastic optimization problems with convex objectives and conic constraints at each stage. We present a new stochastic first-order method, namely the dynamic stochastic approximation (DSA) algorithm,…
Pseudospectral approximation provides a means to approximate the dynamics of delay differential equations (DDE) by ordinary differential equations (ODE). This article develops a computer-aided algorithm to determine the distance between the…
In this paper, we study ordinary differential equations (ODE) coupled with solutions of a stochastic nonsmooth convex optimization problem (SNCOP). We use the regularization approach, the sample average approximation and the time-stepping…
In this paper we consider stochastic composite convex optimization problems with the objective function satisfying a stochastic bounded gradient condition, with or without a quadratic functional growth property. These models include the…
We study the problem of estimating the coefficients in linear ordinary differential equations (ODE's) with a diverging number of variables when the solutions are observed with noise. The solution trajectories are first smoothed with local…
This paper considers a stochastic approximation algorithm, with decreasing step size and martingale difference noise. Under very mild assumptions, we prove the non convergence of this process toward a certain class of repulsive sets for the…
Stochastic optimization methods have been hugely successful in making large-scale optimization problems feasible when computing the full gradient is computationally prohibitive. Using the theory of modified equations for numerical…
We develop a new continuous-time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an…