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 calculating an approximation 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.

冲突函数的优化在决策中至关重要，而现实世界的应用程序经常涉及不确定或未知的数据，从而导致出现随机类型的多目标优化（MOO）问题。我们研究了随机多梯度（SMG）方法，该方法被视为对单目标优化的经典随机梯度方法的扩展。在SMG方法的每次迭代中，通过求解二次子问题来计算随机的多梯度方向，并且表明即使在所有单独的梯度估计量都没有偏向的情况下，该方向也是有偏向的。我们建立速率以计算Pareto前沿的点，其顺序与凸和强凸情况下的随机梯度相似。该分析处理了多梯度中的偏差以及极限帕累托点的未知先验权重。SMG方法被构架为Pareto-front类型算法，用于计算整个Pareto前沿的近似值。Pareto-front SMG算法能够针对许多综合测试问题来稳健地确定Pareto前沿。可以将其应用于由监督机器学习引起的任何随机MOO问题，并且我们报告逻辑二元分类的结果，其中多个目标对应于不同来源的数据组。