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Stochastic alternating algorithms for bi-objective optimization are considered when optimizing two conflicting functions for which optimization steps have to be applied separately for each function. Such algorithms consist of applying a certain number of steps of gradient or subgradient descent on each single objective at each iteration. In this paper, we show that stochastic alternating algorithms achieve a sublinear convergence rate of $\mathcal{O}(1/T)$, under strong convexity, for the determination of a minimizer of a weighted-sum of the two functions, parameterized by the number of steps applied on each of them. An extension to the convex case is presented for which the rate weakens to $\mathcal{O}(1/\sqrt{T})$. These rates are valid also in the non-smooth case. Importantly, by varying the proportion of steps applied to each function, one can determine an approximation to the Pareto front.

In machine learning (ML) applications, unfair predictions may discriminate against a minority group. Most existing approaches for fair machine learning (FML) treat fairness as a constraint or a penalization term in the optimization of a ML model, which does not lead to the discovery of the complete landscape of the trade-offs among learning accuracy and fairness metrics, and does not integrate fairness in a meaningful way. Recently, we have introduced a new paradigm for FML based on Stochastic Multi-Objective Optimization (SMOO), where accuracy and fairness metrics stand as conflicting objectives to be optimized simultaneously. The entire trade-offs range is defined as the Pareto front of the SMOO problem, which can then be efficiently computed using stochastic-gradient type algorithms. SMOO also allows defining and computing new meaningful predictors for FML, a novel one being the Sharpe predictor that we introduce and explore in this paper, and which gives the highest ratio of accuracy-to-unfairness. Inspired from SMOO in finance, the Sharpe predictor for FML provides the highest prediction return (accuracy) per unit of prediction risk (unfairness).

In the application of data clustering to human-centric decision-making systems, such as loan applications and advertisement recommendations, the clustering outcome might discriminate against people across different demographic groups, leading to unfairness. A natural conflict occurs between the cost of clustering (in terms of distance to cluster centers) and the balance representation of all demographic groups across the clusters, leading to a bi-objective optimization problem that is nonconvex and nonsmooth. To determine the complete trade-off between these two competing goals, we design a novel stochastic alternating balance fair $k$-means (SAfairKM) algorithm, which consists of alternating classical mini-batch $k$-means updates and group swap updates. The number of $k$-means updates and the number of swap updates essentially parameterize the weight put on optimizing each objective function. Our numerical experiments show that the proposed SAfairKM algorithm is robust and computationally efficient in constructing well-spread and high-quality Pareto fronts both on synthetic and real datasets.

In the case of coordinated combat, Multi-Missiles Cooperative Targets Assignment(MMCTA)of anti-ship missiles attack aircraft carrier fleet, comprehensively consider various factors affecting the operational effectiveness of anti-ship missile, and construct a missile-target distribution optimization model and Combat effectiveness evaluation index calculation model. In addition, the offline-MMCTA can’t cope with the dynamic change of the operational environment. Through the instantaneous data interaction capability between missiles, the online-MMCTA is realized. To meet real-time requirements of online distribution, an improved hybrid genetic simulated annealing algorithm(GASA) is proposed. GASA by designing coding methods avoids the problem of multi-missile ballistic crossing and low search efficiency. The simulation results demonstrate that the constructed model and the proposed algorithm can effectively solve the online distribution problem of the multi-anti-ship missile cooperative attack aircraft carrier fleet target.

Optimization of conflicting functions is of paramount importance in decision making, and real world applications frequently involve data that is uncertain or unknown, resulting in multi-objective optimization (MOO) problems of stochastic type. We study the stochastic multi-gradient (SMG) method, seen as an extension of the classical stochastic gradient method for single-objective optimization. At each iteration of the SMG method, a stochastic multi-gradient direction is calculated by solving a quadratic subproblem, and it is shown that this direction is biased even when all individual gradient estimators are unbiased. We establish rates to compute a point in the Pareto front, of order similar to what is known for stochastic gradient in both convex and strongly convex cases. The analysis handles the bias in the multi-gradient and the unknown a priori weights of the limiting Pareto point. The SMG method is framed into a Pareto-front type algorithm for the computation of the entire Pareto front. The Pareto-front SMG algorithm is capable of robustly determining Pareto fronts for a number of synthetic test problems. One can apply it to any stochastic MOO problem arising from supervised machine learning, and we report results for logistic binary classification where multiple objectives correspond to distinct-sources data groups., Comment: 31 pages, 14 figures

早大学位記番号:新8049

Based on the survey and research of awareness, preferences and purchasing behavior of residents about Korean food and agricultural products in Beijing, Shanghai, Shenyang, Qingdao and Chongqing in 2014, this paper makes a statistical analysis of the influence of consumers' personal characteristics, awareness, buying habits and income on Chinese consumers' identification and purchase of Korean food and agricultural products, and performs an empirical analysis on the purchasing behavior using Tobit model. The results show that consumers' personal characteristics have no significant influence on the purchase of Korean food; consumers' income levels have a significant positive effect on the purchasing amount of Korean food; consumers' location has a significant effect on the purchasing amount; consumers' recognition has a positive effect on consumers' purchasing behavior, but it is not significant.