Distributed dynamic stochastic approximation algorithm over time-varying networks

In this paper, a distributed stochastic approximation algorithm is proposed to track the dynamic root of a sum of time-varying regression functions over a network. Each agent updates its estimate by using the local observation, the dynamic information of the global root, and information received from its neighbors. Compared with similar works in optimization area, we allow the observation to be noise-corrupted, and the noise condition is much weaker. Furthermore, instead of the upper bound of the estimate error, we present the asymptotic convergence result of the algorithm. The consensus and convergence of the estimates are established. Finally, the algorithm is applied to a distributed target tracking problem and the numerical example is presented to demonstrate the performance of the algorithm.

An Introduction to Development of Centralized and Distributed Stochastic Approximation Algorithm with Expanding Truncations

The stochastic approximation algorithm (SAA), starting from the pioneer work by Robbins and Monro in 1950s, has been successfully applied in systems and control, statistics, machine learning, and so forth. In this paper, we will review the development of SAA in China, to be specific, the stochastic approximation algorithm with expanding truncations (SAAWET) developed by Han-Fu Chen and his colleagues during the past 35 years. We first review the historical development for the centralized algorithm including the probabilistic method (PM) and the ordinary differential equation (ODE) method for SAA and the trajectory-subsequence method for SAAWET. Then, we will give an application example of SAAWET to the recursive principal component analysis. We will also introduce the recent progress on SAAWET in a networked and distributed setting, named the distributed SAAWET (DSAAWET).

Quasi-maximum likelihood estimation of the Component-GARCH model using the stochastic approximation algorithm with application to the S&P 500

The component GARCH (CGARCH) is suitable to better capture the short and long term of the volatility dynamic. Nevertheless, the parameter space constituted by the constraints of the non-negativity of the conditional variance, stationary and existence of moments, is only ex-post defined via the GARCH representation of the CGARCH. This is due to the lack of a general method to determine a priori the relaxed constraints of non-negativity of the CGARCH($N$) conditional variance for any $N\geq 1$. In this paper, a CGARCH parameter space constructed from the GARCH(1,1) component parameter spaces is provided a priori to identifying its GARCH form. Such a space fulfils the relaxed constraints of the CGARCH conditional variance non-negativity to be pre-estimated ensuring the existence of a QML estimation in the sense of the stochastic approximation algorithm. Simulation experiment as well as empirical application to the S\&P500 index are presented and both show the performance of the proposed method.