The Computational Complexity of Estimating Convergence Time

An important problem in the implementation of Markov Chain Monte Carlo algorithms is to determine the convergence time, or the number of iterations before the chain is close to stationarity. For many Markov chains used in practice this time is not known. Even in cases where the convergence time is known to be polynomial, the theoretical bounds are often too crude to be practical. Thus, practitioners like to carry out some form of statistical analysis in order to assess convergence. This has led to the development of a number of methods known as convergence diagnostics which attempt to diagnose whether the Markov chain is far from stationarity. We study the problem of testing convergence in the following settings and prove that the problem is hard in a computational sense: Given a Markov chain that mixes rapidly, it is hard for Statistical Zero Knowledge (SZK-hard) to distinguish whether starting from a given state, the chain is close to stationarity by time t or far from stationarity at time ct for a constant c. We show the problem is in AM intersect coAM. Second, given a Markov chain that mixes rapidly it is coNP-hard to distinguish whether it is close to stationarity by time t or far from stationarity at time ct for a constant c. The problem is in coAM. Finally, it is PSPACE-complete to distinguish whether the Markov chain is close to stationarity by time t or far from being mixed at time ct for c at least 1

A new approach to estimating the expected first hitting time of evolutionary algorithms

AbstractEvolutionary algorithms (EA) have been shown to be very effective in solving practical problems, yet many important theoretical issues of them are not clear. The expected first hitting time is one of the most important theoretical issues of evolutionary algorithms, since it implies the average computational time complexity. In this paper, we establish a bridge between the expected first hitting time and another important theoretical issue, i.e., convergence rate. Through this bridge, we propose a new general approach to estimating the expected first hitting time. Using this approach, we analyze EAs with different configurations, including three mutation operators, with/without population, a recombination operator and a time variant mutation operator, on a hard problem. The results show that the proposed approach is helpful for analyzing a broad range of evolutionary algorithms. Moreover, we give an explanation of what makes a problem hard to EAs, and based on the recognition, we prove the hardness of a general problem

Estimating expected first passage times using multilevel Monte Carlo algorithm

In this paper we devise a method of numerically estimating the expected first passage times of stochastic processes. We use Monte Carlo path simulations with Milstein discretisation scheme to approximate the solutions of scalar stochastic differential equations. To further reduce the variance of the estimated expected stopping time and improve computational efficiency, we use the multi-level Monte Carlo algorithm, recently developed by Giles (2008a), and other variance-reduction techniques. Our numerical results show significant improvements over conventional Monte Carlo techniques

Mutual Dependence: A Novel Method for Computing Dependencies Between Random Variables

In data science, it is often required to estimate dependencies between different data sources. These dependencies are typically calculated using Pearson's correlation, distance correlation, and/or mutual information. However, none of these measures satisfy all the Granger's axioms for an "ideal measure". One such ideal measure, proposed by Granger himself, calculates the Bhattacharyya distance between the joint probability density function (pdf) and the product of marginal pdfs. We call this measure the mutual dependence. However, to date this measure has not been directly computable from data. In this paper, we use our recently introduced maximum likelihood non-parametric estimator for band-limited pdfs, to compute the mutual dependence directly from the data. We construct the estimator of mutual dependence and compare its performance to standard measures (Pearson's and distance correlation) for different known pdfs by computing convergence rates, computational complexity, and the ability to capture nonlinear dependencies. Our mutual dependence estimator requires fewer samples to converge to theoretical values, is faster to compute, and captures more complex dependencies than standard measures

Statistical and Computational Tradeoff in Genetic Algorithm-Based Estimation

When a Genetic Algorithm (GA), or a stochastic algorithm in general, is employed in a statistical problem, the obtained result is affected by both variability due to sampling, that refers to the fact that only a sample is observed, and variability due to the stochastic elements of the algorithm. This topic can be easily set in a framework of statistical and computational tradeoff question, crucial in recent problems, for which statisticians must carefully set statistical and computational part of the analysis, taking account of some resource or time constraints. In the present work we analyze estimation problems tackled by GAs, for which variability of estimates can be decomposed in the two sources of variability, considering some constraints in the form of cost functions, related to both data acquisition and runtime of the algorithm. Simulation studies will be presented to discuss the statistical and computational tradeoff question.Comment: 17 pages, 5 figure

On-line Non-stationary Inventory Control using Champion Competition

The commonly adopted assumption of stationary demands cannot actually reflect fluctuating demands and will weaken solution effectiveness in real practice. We consider an On-line Non-stationary Inventory Control Problem (ONICP), in which no specific assumption is imposed on demands and their probability distributions are allowed to vary over periods and correlate with each other. The nature of non-stationary demands disables the optimality of static (s,S) policies and the applicability of its corresponding algorithms. The ONICP becomes computationally intractable by using general Simulation-based Optimization (SO) methods, especially under an on-line decision-making environment with no luxury of time and computing resources to afford the huge computational burden. We develop a new SO method, termed "Champion Competition" (CC), which provides a different framework and bypasses the time-consuming sample average routine adopted in general SO methods. An alternate type of optimal solution, termed "Champion Solution", is pursued in the CC framework, which coincides the traditional optimality sense under certain conditions and serves as a near-optimal solution for general cases. The CC can reduce the complexity of general SO methods by orders of magnitude in solving a class of SO problems, including the ONICP. A polynomial algorithm, termed "Renewal Cycle Algorithm" (RCA), is further developed to fulfill an important procedure of the CC framework in solving this ONICP. Numerical examples are included to demonstrate the performance of the CC framework with the RCA embedded.Comment: I just identified a flaw in the paper. It may take me some time to fix it. I would like to withdraw the article and update it once I finished. Thank you for your kind suppor