A Framework for Analyzing Stochastic Optimization Algorithms Under Dependence

In this dissertation, a theoretical framework based on concentration inequalities for empirical processes is developed to better design iterative optimization algorithms and analyze their convergence properties in the presence of complex dependence between directions and step-sizes. Based on this framework, we proposed a stochastic away-step Frank-Wolfe algorithm and a stochastic pairwise-step Frank-Wolfe algorithm for solving strongly convex problems with polytope constraints and proved that both of those algorithms converge linearly to the optimal solution in expectation and almost surely. Numerical results showed that the proposed algorithms are faster and more stable than most of their competitors. This framework can be applied for designing and analyzing stochastic algorithms with adaptive step-sizes that are based on local curvature for self-concordant optimization problems. Notably, we proposed and analyzed a stochastic BFGS algorithm without line-search, and proved that it converges linearly globally and super-linearly locally using the framework mentioned above. This is the first work that analyzes a fully stochastic BFGS algorithm, which also avoids time consuming or even impossible line-search steps. A third class of problems that the empirical processes framework can be applied to is to study the optimization of compositions of stochastic functions. A multi-level Monte Carlo based unbiased gradient generation method is introduced into stochastic optimization algorithms for minimizing function compositions. Based on this, standard stochastic optimization algorithms can be applied to these problems directly

Research on lethal levels of buildings based on historical seismic data

Due to the influences of buildings, geographical and geomorphological environments, road conditions, etc., the probabilities and numbers of casualties in different areas after an earthquake are different. Accordingly, we propose the concept of the lethal level, which attains different grades representing the mortality rate of differing intensities. Different regions have unique lethal levels, and regional lethal levels are affected mainly by the proportion of each building type and the corresponding lethal level, as different types of buildings also have unique lethal levels. Based on data of 52 historical earthquake disasters, we constructed a lethal level calculation model and obtained the lethal level of each building type. The results reveal that the lethal level ranges of different building types are fixed and unequal; moreover, the ranges of different building types overlap each other. The lethal level range of adobe structures is 0.85–1, that of civil structures is 0.75–0.95, that of brick-wood structures is 0.6–0.9, that of brick-concrete structures is 0.33–0.6, that of wood structures is 0.2–0.35, and that of reinforced concrete structures is 0.1–0.25. Based on the lethal levels of these building types, the overall level of a region can be quantified and graded, and this classification does not depend on the geographical location or administrative boundaries. In pre-earthquake evaluation efforts, the lethal level of an area can be derived through field research. After an earthquake, the number of casualties can be quickly assessed based on the mortality rate corresponding to the intensity of the area. This approach can further provide scientific support for risk zoning and risk assessment research

Marginal models with random weighting method

When using marginal models to analyse longitudinal or clustered data, the estimation methods based on each marginal model are often readily available. However, combining them to obtain a more accurate or possibly optimal estimate under certain criterion, could be difficult. The main reason is that the objective functions based on marginal models may be not differentiable. Moreover, the estimating functions based on marginal models might have variances that are difficult to compute or approximate, preventing the direct use of the method of generalized estimating equations. To circumvent these difficulties, a random weighting method is proposed to use. A general theorem on validity of random weighting method is given and an example that bootstrap fails to consistently estimate the estimator's variance but random weighting does is provided. The main advantage of this approach is that it is computationally straightforward even when no particular structure of dependence among marginal models is available. The resulting estimator achieves certain optimality in terms of asymptotic variance. We illustrate the method with median regression, Mann-Whitney-Gehan's estimation, Buckely-James estimation and multivariate proportional hazards model as examples. Supportive empirical evidence is shown in the simulation studies. Application is illustrated with a well-known medical study

Research on the application of mobile phone location signal data in earthquake emergency work: A case study of Jiuzhaigou earthquake.

After an earthquake, the important task of emergency rescue work is to minimize casualties, but due to the suddenness of earthquake disasters, it is difficult to obtain enough disaster information immediately, especially personnel distribution and movement information. The traditional methods of obtaining disaster data are through reports from the disaster area or field investigations by the emergency rescue team; this work lags, and its efficiency is low. This paper analyzes the feasibility of using mobile phone location signal data in earthquake emergency rescue work in several respects, such as quantity, location, change rate, and epicentral distance. The results show that mobile phone location signal data can quickly obtain the situation of personnel distribution and quantity after an earthquake, and we find the change rate, distance, etc., can determine the approximate range of the earthquake impact field. Through the data distribution in different time periods, the movement of personnel after the earthquake can be obtained. Based on several situations, we can determine the basic situation of the disaster-stricken areas in times after the earthquake, especially the personnel relevant to the situation, and these data can provide a scientific basis for emergency rescue decision making