Stochastic Optimization for Non-convex Inf-Projection Problems

Yan Yan, Yi Xu, Lijun Zhang, Wang Xiaoyu, Tianbao Yang
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:10660-10669, 2020.

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.

Cite this Paper

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.} }
Endnote
%0 Conference Paper %T Stochastic Optimization for Non-convex Inf-Projection Problems %A Yan Yan %A Yi Xu %A Lijun Zhang %A Wang Xiaoyu %A Tianbao Yang %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-yan20a %I PMLR %P 10660--10669 %U https://proceedings.mlr.press/v119/yan20a.html %V 119 %X 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.
APA
Yan, Y., Xu, Y., Zhang, L., Xiaoyu, W. & Yang, T.. (2020). Stochastic Optimization for Non-convex Inf-Projection Problems. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:10660-10669 Available from https://proceedings.mlr.press/v119/yan20a.html.

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