Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization

Xufeng Cai, Chaobing Song, Stephen Wright, Jelena Diakonikolas
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:3469-3494, 2023.

Abstract

Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t. a Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We prove a faster linear convergence result when a Polyak-Łojasiewicz (PŁ) condition holds. To our knowledge, this work is the first to provide non-asymptotic convergence guarantees — variance-reduced or not — for a cyclic block coordinate method in general composite (smooth + nonsmooth) nonconvex settings. Our experimental results demonstrate the efficacy of the proposed cyclic scheme in training deep neural nets.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-cai23e, title = {Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization}, author = {Cai, Xufeng and Song, Chaobing and Wright, Stephen and Diakonikolas, Jelena}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {3469--3494}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/cai23e/cai23e.pdf}, url = {https://proceedings.mlr.press/v202/cai23e.html}, abstract = {Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t. a Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We prove a faster linear convergence result when a Polyak-Łojasiewicz (PŁ) condition holds. To our knowledge, this work is the first to provide non-asymptotic convergence guarantees — variance-reduced or not — for a cyclic block coordinate method in general composite (smooth + nonsmooth) nonconvex settings. Our experimental results demonstrate the efficacy of the proposed cyclic scheme in training deep neural nets.} }
Endnote
%0 Conference Paper %T Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization %A Xufeng Cai %A Chaobing Song %A Stephen Wright %A Jelena Diakonikolas %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-cai23e %I PMLR %P 3469--3494 %U https://proceedings.mlr.press/v202/cai23e.html %V 202 %X Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t. a Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We prove a faster linear convergence result when a Polyak-Łojasiewicz (PŁ) condition holds. To our knowledge, this work is the first to provide non-asymptotic convergence guarantees — variance-reduced or not — for a cyclic block coordinate method in general composite (smooth + nonsmooth) nonconvex settings. Our experimental results demonstrate the efficacy of the proposed cyclic scheme in training deep neural nets.
APA
Cai, X., Song, C., Wright, S. & Diakonikolas, J.. (2023). Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:3469-3494 Available from https://proceedings.mlr.press/v202/cai23e.html.

Related Material