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Properties of Multivariate q-Gaussian Distribution and its application to Smoothed Functional Algorithms for Stochastic Optimization

The importance of the q-Gaussian distribution is due to their power-law nature, and close association with the popular Gaussian distribution. This distribution arises from nonextensive information theory, and it has an interesting property that uncorrelated q-Gaussian random variates show a special kind of inter-dependence. In this work, we study some key properties related to higher order moments and comoments of the multivariate q-Gaussian distribution. We use the important features of this distribution to improve upon the smoothing properties of a Gaussian kernel. Based on this, we propose a Smoothed Functional scheme for gradient and Hessian estimation using q-Gaussian distribution. Using the above estimation technique, we propose four two-timescale algorithms for optimization of a stochastic objective function using gradient descent and Newton based search methods. We prove that the proposed algorithms converge to a local optimum. Performance of the algorithms is shown by simulation results on a queuing model. The results show that the q-Gaussian based schemes provide a better tuning of the algorithms, which improve their performance compared to their Gaussian counterparts.