The stochastic multi-gradient algorithm for multi-objective optimization and its application to supervised machine learning
This work addresses multi-objective optimization under uncertainty, which is important for decision-making in real-world applications like machine learning, but it is incremental as it builds on existing stochastic gradient methods.
The paper tackles the problem of stochastic multi-objective optimization (MOO) by extending the stochastic gradient method to handle conflicting objectives, establishing convergence rates similar to single-objective cases for computing points on the Pareto front. It applies this method to supervised machine learning, demonstrating results in logistic binary classification with multiple data groups.
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.