Implicit Bias of Policy Gradient in Linear Quadratic Control: Extrapolation to Unseen Initial States

Noam Razin, Yotam Alexander, Edo Cohen-Karlik, Raja Giryes, Amir Globerson, Nadav Cohen
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:42275-42331, 2024.

Abstract

In modern machine learning, models can often fit training data in numerous ways, some of which perform well on unseen (test) data, while others do not. Remarkably, in such cases gradient descent frequently exhibits an implicit bias that leads to excellent performance on unseen data. This implicit bias was extensively studied in supervised learning, but is far less understood in optimal control (reinforcement learning). There, learning a controller applied to a system via gradient descent is known as policy gradient, and a question of prime importance is the extent to which a learned controller extrapolates to unseen initial states. This paper theoretically studies the implicit bias of policy gradient in terms of extrapolation to unseen initial states. Focusing on the fundamental Linear Quadratic Regulator (LQR) problem, we establish that the extent of extrapolation depends on the degree of exploration induced by the system when commencing from initial states included in training. Experiments corroborate our theory, and demonstrate its conclusions on problems beyond LQR, where systems are non-linear and controllers are neural networks. We hypothesize that real-world optimal control may be greatly improved by developing methods for informed selection of initial states to train on.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-razin24a, title = {Implicit Bias of Policy Gradient in Linear Quadratic Control: Extrapolation to Unseen Initial States}, author = {Razin, Noam and Alexander, Yotam and Cohen-Karlik, Edo and Giryes, Raja and Globerson, Amir and Cohen, Nadav}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {42275--42331}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/razin24a/razin24a.pdf}, url = {https://proceedings.mlr.press/v235/razin24a.html}, abstract = {In modern machine learning, models can often fit training data in numerous ways, some of which perform well on unseen (test) data, while others do not. Remarkably, in such cases gradient descent frequently exhibits an implicit bias that leads to excellent performance on unseen data. This implicit bias was extensively studied in supervised learning, but is far less understood in optimal control (reinforcement learning). There, learning a controller applied to a system via gradient descent is known as policy gradient, and a question of prime importance is the extent to which a learned controller extrapolates to unseen initial states. This paper theoretically studies the implicit bias of policy gradient in terms of extrapolation to unseen initial states. Focusing on the fundamental Linear Quadratic Regulator (LQR) problem, we establish that the extent of extrapolation depends on the degree of exploration induced by the system when commencing from initial states included in training. Experiments corroborate our theory, and demonstrate its conclusions on problems beyond LQR, where systems are non-linear and controllers are neural networks. We hypothesize that real-world optimal control may be greatly improved by developing methods for informed selection of initial states to train on.} }
Endnote
%0 Conference Paper %T Implicit Bias of Policy Gradient in Linear Quadratic Control: Extrapolation to Unseen Initial States %A Noam Razin %A Yotam Alexander %A Edo Cohen-Karlik %A Raja Giryes %A Amir Globerson %A Nadav Cohen %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-razin24a %I PMLR %P 42275--42331 %U https://proceedings.mlr.press/v235/razin24a.html %V 235 %X In modern machine learning, models can often fit training data in numerous ways, some of which perform well on unseen (test) data, while others do not. Remarkably, in such cases gradient descent frequently exhibits an implicit bias that leads to excellent performance on unseen data. This implicit bias was extensively studied in supervised learning, but is far less understood in optimal control (reinforcement learning). There, learning a controller applied to a system via gradient descent is known as policy gradient, and a question of prime importance is the extent to which a learned controller extrapolates to unseen initial states. This paper theoretically studies the implicit bias of policy gradient in terms of extrapolation to unseen initial states. Focusing on the fundamental Linear Quadratic Regulator (LQR) problem, we establish that the extent of extrapolation depends on the degree of exploration induced by the system when commencing from initial states included in training. Experiments corroborate our theory, and demonstrate its conclusions on problems beyond LQR, where systems are non-linear and controllers are neural networks. We hypothesize that real-world optimal control may be greatly improved by developing methods for informed selection of initial states to train on.
APA
Razin, N., Alexander, Y., Cohen-Karlik, E., Giryes, R., Globerson, A. & Cohen, N.. (2024). Implicit Bias of Policy Gradient in Linear Quadratic Control: Extrapolation to Unseen Initial States. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:42275-42331 Available from https://proceedings.mlr.press/v235/razin24a.html.

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