Blockwise Stochastic Variance-Reduced Methods with Parallel Speedup for Multi-Block Bilevel Optimization

Quanqi Hu, Zi-Hao Qiu, Zhishuai Guo, Lijun Zhang, Tianbao Yang
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:13550-13583, 2023.

Abstract

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve $m\gg 1$ lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling $I$ blocks and sampling $B$ samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of $O(\frac{m\epsilon^{-3}\mathbb{I}(I \textless m)}{I\sqrt{I}}+\frac{m\epsilon^{-3}}{I\sqrt{B}})$ for finding an $\epsilon$-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-hu23d, title = {Blockwise Stochastic Variance-Reduced Methods with Parallel Speedup for Multi-Block Bilevel Optimization}, author = {Hu, Quanqi and Qiu, Zi-Hao and Guo, Zhishuai and Zhang, Lijun and Yang, Tianbao}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {13550--13583}, 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/hu23d/hu23d.pdf}, url = {https://proceedings.mlr.press/v202/hu23d.html}, abstract = {In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve $m\gg 1$ lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling $I$ blocks and sampling $B$ samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of $O(\frac{m\epsilon^{-3}\mathbb{I}(I \textless m)}{I\sqrt{I}}+\frac{m\epsilon^{-3}}{I\sqrt{B}})$ for finding an $\epsilon$-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.} }
Endnote
%0 Conference Paper %T Blockwise Stochastic Variance-Reduced Methods with Parallel Speedup for Multi-Block Bilevel Optimization %A Quanqi Hu %A Zi-Hao Qiu %A Zhishuai Guo %A Lijun Zhang %A Tianbao Yang %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-hu23d %I PMLR %P 13550--13583 %U https://proceedings.mlr.press/v202/hu23d.html %V 202 %X In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve $m\gg 1$ lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling $I$ blocks and sampling $B$ samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of $O(\frac{m\epsilon^{-3}\mathbb{I}(I \textless m)}{I\sqrt{I}}+\frac{m\epsilon^{-3}}{I\sqrt{B}})$ for finding an $\epsilon$-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.
APA
Hu, Q., Qiu, Z., Guo, Z., Zhang, L. & Yang, T.. (2023). Blockwise Stochastic Variance-Reduced Methods with Parallel Speedup for Multi-Block Bilevel Optimization. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:13550-13583 Available from https://proceedings.mlr.press/v202/hu23d.html.

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