Stochastic Optimization for Non-convex Inf-Projection Problems
Yan Yan, Yi Xu, Lijun Zhang, Wang Xiaoyu, Tianbao Yang
Abstract
In this paper, we study a family of non-convex and possibly non-smooth inf-projection minimization problems, where the target objective function is equal to minimization of a joint function over another variable. This problem include difference of convex (DC) functions and a family of bi-convex functions as special cases. We develop stochastic algorithms and establish their first-order convergence for finding a (nearly) stationary solution of the target non-convex function under different conditions of the component functions. To the best of our knowledge, this is the first work that comprehensively studies stochastic optimization of non-convex inf-projection minimization problems with provable convergence guarantee. Our algorithms enable efficient stochastic optimization of a family of non-decomposable DC functions and a family of bi-convex functions. To demonstrate the power of the proposed algorithms we consider an important application in variance-based regularization. Experiments verify the effectiveness of our inf-projection based formulation and the proposed stochastic algorithm in comparison with previous stochastic algorithms based on the min-max formulation for achieving the same effect.
BibTeX
@InProceedings{pmlr-v119-yan20a,
title = {Stochastic Optimization for Non-convex Inf-Projection Problems},
author = {Yan, Yan and Xu, Yi and Zhang, Lijun and Xiaoyu, Wang and Yang, Tianbao},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {10660--10669},
year = {2020},
editor = {III, Hal Daumé and Singh, Aarti},
volume = {119},
series = {Proceedings of Machine Learning Research},
month = {13--18 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v119/yan20a/yan20a.pdf},
url = {https://proceedings.mlr.press/v119/yan20a.html},
abstract = {In this paper, we study a family of non-convex and possibly non-smooth inf-projection minimization problems, where the target objective function is equal to minimization of a joint function over another variable. This problem include difference of convex (DC) functions and a family of bi-convex functions as special cases. We develop stochastic algorithms and establish their first-order convergence for finding a (nearly) stationary solution of the target non-convex function under different conditions of the component functions. To the best of our knowledge, this is the first work that comprehensively studies stochastic optimization of non-convex inf-projection minimization problems with provable convergence guarantee. Our algorithms enable efficient stochastic optimization of a family of non-decomposable DC functions and a family of bi-convex functions. To demonstrate the power of the proposed algorithms we consider an important application in variance-based regularization. Experiments verify the effectiveness of our inf-projection based formulation and the proposed stochastic algorithm in comparison with previous stochastic algorithms based on the min-max formulation for achieving the same effect.}
}