Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing

Jikai Jin, Zhiyuan Li, Kaifeng Lyu, Simon Shaolei Du, Jason D. Lee
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:15200-15238, 2023.

Abstract

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics and follows an incremental learning procedure: GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-jin23a, title = {Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing}, author = {Jin, Jikai and Li, Zhiyuan and Lyu, Kaifeng and Du, Simon Shaolei and Lee, Jason D.}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {15200--15238}, 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/jin23a/jin23a.pdf}, url = {https://proceedings.mlr.press/v202/jin23a.html}, abstract = {It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics and follows an incremental learning procedure: GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.} }
Endnote
%0 Conference Paper %T Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing %A Jikai Jin %A Zhiyuan Li %A Kaifeng Lyu %A Simon Shaolei Du %A Jason D. Lee %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-jin23a %I PMLR %P 15200--15238 %U https://proceedings.mlr.press/v202/jin23a.html %V 202 %X It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics and follows an incremental learning procedure: GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.
APA
Jin, J., Li, Z., Lyu, K., Du, S.S. & Lee, J.D.. (2023). Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:15200-15238 Available from https://proceedings.mlr.press/v202/jin23a.html.

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