Criteria and Bias of Parameterized Linear Regression under Edge of Stability Regime
Keywords: Edge of Stability, gradient descent, implicit bias
Abstract: Classical optimization theory requires a small step-size for gradient-based methods to converge. Nevertheless, recent findings (Cohen et al., 2021) challenge the traditional idea by empirically demonstrating Gradient Descent (GD) converges even when the step-size $\eta$ exceeds the threshold of $2/L$, where $L$ is the global smooth constant. This is usually known as the \emph{Edge of Stability} (EoS) phenomenon. A widely held belief suggests that an objective function with subquadratic growth plays an important role in incurring EoS. In this paper, we provide a more comprehensive answer by considering the task of finding linear interpolator $\beta \in \mathbb{R}^{d}$ for regression with loss function $l(\cdot)$, where $\beta$ admits parameterization as $\beta = w^2_{+} - w^2_{-}$. Contrary to the previous work that suggests a subquadratic $l$ is necessary for EoS, our novel finding reveals that EoS occurs even when $l$ is quadratic under proper conditions. This argument is made rigorous by both empirical and theoretical evidence, demonstrating the GD trajectory converges to a linear interpolator in a non-asymptotic way. Moreover, the model under quadratic $l$, also known as a depth-$2$ \emph{diagonal linear network}, remains largely unexplored under the EoS regime. Our analysis then sheds some new light on the implicit bias of diagonal linear networks when a larger step-size is employed, enriching the understanding of EoS on more practical models.
Primary Area: optimization
Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.
Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.
Reciprocal Reviewing: I understand the reciprocal reviewing requirement as described on https://iclr.cc/Conferences/2025/CallForPapers. If none of the authors are registered as a reviewer, it may result in a desk rejection at the discretion of the program chairs. To request an exception, please complete this form at https://forms.gle/Huojr6VjkFxiQsUp6.
Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.
No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.
Submission Number: 1154
Loading