This paper discusses a stochastic equilibrium problem for which the function is in the form of the expectation of nonmonotone bifunctions and the constraint set is closed and convex. This problem includes various applications such as stochastic variational inequalities, stochastic Nash equilibrium problems, and nonconvex stochastic optimization problems. For solving this stochastic equilibrium problem, we propose an inexact stochastic subgradient projection method. The proposed method sets a random realization of the bifunction and then updates its approximation by using both its stochastic subgradient and the projection onto the constraint set. The main contribution of this paper is to present a convergence analysis showing that, under certain assumptions, any accumulation point of the sequence generated by the proposed method using a constant step size almost surely belongs to the solution set of the stochastic equilibrium problem. A convergence rate analysis of the method is also provided to illustrate the method’s efficiency. Another contribution of this paper is to show that a machine learning algorithm based on the proposed method achieves the expected risk minimization for a class of least absolute selection and shrinkage operator (lasso) problems in statistical learning with sparsity. Numerical comparisons of the proposed machine learning algorithm with existing machine learning algorithms for the expected risk minimization using LIBSVM datasets demonstrate the effectiveness and superior classification accuracy of the proposed algorithm.

本文讨论了一个随机均衡问题，该函数的形式为非单调双函数的期望形式，并且约束集是封闭的和凸的。此问题包括各种应用，例如随机变分不等式，随机Nash平衡问题和非凸随机优化问题。为了解决这一随机均衡问题，我们提出了一种不精确的随机次梯度投影方法。所提出的方法设置双功能的随机实​​现，然后通过使用其随机次梯度和对约束集的投影来更新其逼近。本文的主要贡献是提出了收敛分析，表明在某些假设下，由所提出的方法使用恒定步长生成的序列的任何累加点几乎肯定属于随机均衡问题的解集。还提供了该方法的收敛速度分析，以说明该方法的效率。本文的另一项贡献是表明，基于该方法的机器学习算法对于具有稀疏性的统计学习中的一类最小绝对选择和收缩算子（套索）问题实现了预期的风险最小化。使用LIBSVM数据集将拟议的机器学习算法与现有的机器学习算法进行数值比较，以期将期望的风险最小化，从而证明了该算法的有效性和优越的分类精度。