We present the validity of stochastic averaging principle for non-autonomous slow–fast differential equations (SDEs) whose fast motions admit random periodic solutions. Our investigation is motivated by some problems arising from multi-scale dynamical systems, where configurations are time dependent due to nonlinearity underlying vector fields and onset invariant sets. Averaging with respect un...

Considering the constrained stochastic optimization problem over a time-varying random network, where agents are to collectively minimize sum of objective functions subject common constraint set, we investigate asymptotic properties distributed algorithm based on dual averaging gradients. Different from most existing works algorithms that mainly focused their non-asymptotic properties, prove no...

The aim of the paper is to establish a convergence theorem for multi-dimensional stochastic approximation when the “innovations” satisfy some “light” averaging properties in the presence of a pathwise Lyapunov function. These averaging assumptions allow us to unify apparently remote frameworks where the innovations are simulated (possibly deterministic like in Quasi-Monte Carlo simulation) or e...

Bayesian predictions are stochastic just like predictions of any other inference scheme that generalize from a finite sample. While a simple variational argument shows that Bayes averaging is generalization optimal given that the prior matches the teacher parameter distribution the situation is less clear if the teacher distribution is unknown. I define a class of averaging procedures, the temp...

In this paper we compare ruin functions for two risk processes with respect to stochastic ordering, stop-loss ordering and ordering of adjustment coeecients. The risk processes are as follows: in the Markov-modulated environment and the associated averaged compound Poisson model. In the latter case the arrival rate is obtained by averaging over time the arrival rate in the Markov modulated mode...

A theory of iterated averaging is developed for a class of highly oscillatory ordinary differential equations (ODEs) with three well separated time scales. The solutions of these equations are assumed to be (almost) periodic in the fastest time scales. It is proved that the dynamics on the slowest time scale can be approximated by an effective ODE obtained by averaging out oscillations. In part...

We consider ”nonconventional” averaging setup in the form dX(t) dt = ǫB ` X(t), ξ(q1(t)), ξ(q2(t)), ..., ξ(ql(t)) ́ where ξ(t), t ≥ 0 is either a stochastic process or a dynamical system (i.e. then ξ(t) = F x) with sufficiently fast mixing while qj(t) = αjt, α1 < α2 < ... < αk and qj , j = k+1, ..., l grow faster than linearly. We show that the properly normalized error term in the ”nonconventio...

A stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general measure is considered. The averaging principle for the established. convergence rate compared with other results related topics.

Couplings of microscopic stochastic models to deterministic macroscopic ordinary and partial differential equations are commonplace in numerous applications such as catalysis, deposition processes, polymeric flows, biological networks and parametrizations of tropical and open ocean convection. In this paper we continue our study of the class of prototype hybrid systems presented in [8]. These m...

We present an algorithm for fast stochastic gradient descent that uses a nonlinear adaptive momentum scheme to optimize the late time convergence rate. The algorithm makes eeective use of curvature information, requires only O(n) storage and computation, and delivers convergence rates close to the theoretical optimum. We demonstrate the technique on linear and large nonlinear back-prop networks...