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This article studies the dynamics of the mean-field approximation of continuous random networks. These networks are stochastic integrodifferential equations driven by Gaussian noise. The kernels in the integral operators are realizations of…

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…

Reinforcement learning in non-stationary environments is challenging due to abrupt and unpredictable changes in dynamics, often causing traditional algorithms to fail to converge. However, in many real-world cases, non-stationarity has some…

This article is devoted to the analysis of semilinear, parabolic, Stochastic Partial Differential Equations, with slow and fast time scales. Asymptotically, an averaging principle holds: the slow component converges to the solution of…

We develop a worst-case complexity theory for stochastically preconditioned stochastic gradient descent (SPSGD) and its accelerated variants under heavy-tailed noise, a setting that encompasses widely used adaptive methods such as Adam,…

In this work, we investigate the McKean-Vlasov stochastic partial differential equations driven by Poisson random measure. By adapting the variational framework, we prove the well-posedness and large deviation principle for a class of…

Stochastic Gradient Descent (SGD) is commonly modeled as a Langevin process, assuming that minibatch noise acts as Brownian motion. However, this approximation relies on a continuous-time limit and a sqrt(eta) noise scaling that does not…

We study the almost sure convergence of the Stochastic Approximation algorithm to the fixed point $x^\star$ of a nonlinear operator under a negative drift condition and a general noise sequence with finite $p$-th moment for some $p > 1$.…

We construct space-time stationary solutions of the 1D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poissonian point field in…

We consider the problem of approximation of the solution of the backward stochastic differential equation in the Markovian case. We suppose that the trend coefficient of the diffusion process depends on some unknown parameter and the…

We study the long-time behavior of fully discretized semilinear SPDEs with additive space-time white noise, which admit a unique invariant probability measure $\mu$. We show that the average of regular enough test functions with respect to…

Stochastic approximation (SA) with multiple coupled sequences has found broad applications in machine learning such as bilevel learning and reinforcement learning (RL). In this paper, we study the finite-time convergence of nonlinear SA…

In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially…

In this paper, we investigate an ill-posed Cauchy problem involving a stochastic parabolic equation. We first establish a Carleman estimate for this equation. Leveraging this estimate, we derive the conditional stability and convergence…

This paper studies the stability properties of stochastic differential equations subject to persistent noise (including the case of additive noise), which is noise that is present even at the equilibria of the underlying differential…

The well-established practice of time series analysis involves estimating deterministic, non-stationary trend and seasonality components followed by learning the residual stochastic, stationary components. Recently, it has been shown that…

This paper studies a risk minimization problem with decision dependent data distribution. The problem pertains to the performative prediction setting in which a trained model can affect the outcome estimated by the model. Such dependency…

Stochastic approximation is one of the effective approach to deal with the large-scale machine learning problems and the recent research has focused on reduction of variance, caused by the noisy approximations of the gradients. In this…

Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for…

Stochastic optimization is a vital field in the realm of mathematical optimization, finding applications in diverse areas ranging from operations research to machine learning. In this paper, we introduce a novel first-order optimization…