q-Gaussian based Smoothed Functional Algorithm for Stochastic Optimization

The q-Gaussian distribution results from maximizing certain generalizations of Shannon entropy under some constraints. The importance of q-Gaussian distributions stems from the fact that they exhibit power-law behavior, and also generalize Gaussian distributions. In this paper, we propose a Smoothed Functional (SF) scheme for gradient estimation using q-Gaussian distribution, and also propose an algorithm for optimization based on the above scheme. Convergence results of the algorithm are presented. Performance of the proposed algorithm is shown by simulation results on a queuing model.Comment: 5 pages, 1 figur

Newton based Stochastic Optimization using q-Gaussian Smoothed Functional Algorithms

We present the first q-Gaussian smoothed functional (SF) estimator of the Hessian and the first Newton-based stochastic optimization algorithm that estimates both the Hessian and the gradient of the objective function using q-Gaussian perturbations. Our algorithm requires only two system simulations (regardless of the parameter dimension) and estimates both the gradient and the Hessian at each update epoch using these. We also present a proof of convergence of the proposed algorithm. In a related recent work (Ghoshdastidar et al., 2013), we presented gradient SF algorithms based on the q-Gaussian perturbations. Our work extends prior work on smoothed functional algorithms by generalizing the class of perturbation distributions as most distributions reported in the literature for which SF algorithms are known to work and turn out to be special cases of the q-Gaussian distribution. Besides studying the convergence properties of our algorithm analytically, we also show the results of several numerical simulations on a model of a queuing network, that illustrate the significance of the proposed method. In particular, we observe that our algorithm performs better in most cases, over a wide range of q-values, in comparison to Newton SF algorithms with the Gaussian (Bhatnagar, 2007) and Cauchy perturbations, as well as the gradient q-Gaussian SF algorithms (Ghoshdastidar et al., 2013).Comment: This is a longer of version of the paper with the same title accepted in Automatic

Moment-Based Variational Inference for Markov Jump Processes

We propose moment-based variational inference as a flexible framework for approximate smoothing of latent Markov jump processes. The main ingredient of our approach is to partition the set of all transitions of the latent process into classes. This allows to express the Kullback-Leibler divergence between the approximate and the exact posterior process in terms of a set of moment functions that arise naturally from the chosen partition. To illustrate possible choices of the partition, we consider special classes of jump processes that frequently occur in applications. We then extend the results to parameter inference and demonstrate the method on several examples.Comment: Accepted by the 36th International Conference on Machine Learning (ICML 2019

A sequential sampling strategy for extreme event statistics in nonlinear dynamical systems

We develop a method for the evaluation of extreme event statistics associated with nonlinear dynamical systems, using a small number of samples. From an initial dataset of design points, we formulate a sequential strategy that provides the 'next-best' data point (set of parameters) that when evaluated results in improved estimates of the probability density function (pdf) for a scalar quantity of interest. The approach utilizes Gaussian process regression to perform Bayesian inference on the parameter-to-observation map describing the quantity of interest. We then approximate the desired pdf along with uncertainty bounds utilizing the posterior distribution of the inferred map. The 'next-best' design point is sequentially determined through an optimization procedure that selects the point in parameter space that maximally reduces uncertainty between the estimated bounds of the pdf prediction. Since the optimization process utilizes only information from the inferred map it has minimal computational cost. Moreover, the special form of the metric emphasizes the tails of the pdf. The method is practical for systems where the dimensionality of the parameter space is of moderate size, i.e. order O(10). We apply the method to estimate the extreme event statistics for a very high-dimensional system with millions of degrees of freedom: an offshore platform subjected to three-dimensional irregular waves. It is demonstrated that the developed approach can accurately determine the extreme event statistics using limited number of samples