Timezone: »
We study the implicit regularization of optimization methods for linear models interpolating the training data in the under-parametrized and over-parametrized regimes. For over-parameterized linear regression, where there are infinitely many interpolating solutions, different optimization methods can converge to solutions with varying generalization performance. In this setting, we show that projections onto linear spans can be used to move between solutions. Furthermore, via a simple reparameterization, we can ensure that an arbitrary optimizer converges to the minimum l2-norm solution with favourable generalization properties. For under-parameterized linear classification, optimizers can converge to different decision boundaries separating the data. We prove that for any such classifier, there exists a family of quadratic norms ||.||_P such that the classifier's direction is the same as that of the maximum P-margin solution. We argue that analyzing convergence to the standard maximum l2-margin is arbitrary and show that minimizing the norm induced by the data can result in better generalization. We validate our theoretical results via experiments on synthetic and real datasets.
Author Information
Sharan Vaswani (Mila, Université de Montréal)
More from the Same Authors
-
2021 : Towards Noise-adaptive, Problem-adaptive Stochastic Gradient Descent »
Sharan Vaswani · Benjamin Dubois-Taine · Reza Babanezhad Harikandeh -
2022 : Target-based Surrogates for Stochastic Optimization »
Jonathan Lavington · Sharan Vaswani · Reza Babanezhad Harikandeh · Mark Schmidt · Nicolas Le Roux -
2022 : Poster Session 1 »
Andrew Lowy · Thomas Bonnier · Yiling Xie · Guy Kornowski · Simon Schug · Seungyub Han · Nicolas Loizou · xinwei zhang · Laurent Condat · Tabea E. Röber · Si Yi Meng · Marco Mondelli · Runlong Zhou · Eshaan Nichani · Adrian Goldwaser · Rudrajit Das · Kayhan Behdin · Atish Agarwala · Mukul Gagrani · Gary Cheng · Tian Li · Haoran Sun · Hossein Taheri · Allen Liu · Siqi Zhang · Dmitrii Avdiukhin · Bradley Brown · Miaolan Xie · Junhyung Lyle Kim · Sharan Vaswani · Xinmeng Huang · Ganesh Ramachandra Kini · Angela Yuan · Weiqiang Zheng · Jiajin Li -
2022 Poster: Near-Optimal Sample Complexity Bounds for Constrained MDPs »
Sharan Vaswani · Lin Yang · Csaba Szepesvari -
2021 : Poster Session 1 (gather.town) »
Hamed Jalali · Robert Hönig · Maximus Mutschler · Manuel Madeira · Abdurakhmon Sadiev · Egor Shulgin · Alasdair Paren · Pascal Esser · Simon Roburin · Julius Kunze · Agnieszka Słowik · Frederik Benzing · Futong Liu · Hongyi Li · Ryotaro Mitsuboshi · Grigory Malinovsky · Jayadev Naram · Zhize Li · Igor Sokolov · Sharan Vaswani -
2020 : Poster Session 2 (gather.town) »
Sharan Vaswani · Nicolas Loizou · Wenjie Li · Preetum Nakkiran · Zhan Gao · Sina Baghal · Jingfeng Wu · Roozbeh Yousefzadeh · Jinyi Wang · Jing Wang · Cong Xie · Anastasia Borovykh · Stanislaw Jastrzebski · Soham Dan · Yiliang Zhang · Mark Tuddenham · Sarath Pattathil · Ievgen Redko · Jeremy Cohen · Yasaman Esfandiari · Zhanhong Jiang · Mostafa ElAraby · Chulhee Yun · Michael Psenka · Robert Gower · Xiaoyu Wang -
2020 : Contributed talks in Session 2 (Zoom) »
Martin Takac · Samuel Horváth · Guan-Horng Liu · Nicolas Loizou · Sharan Vaswani -
2020 : Contributed Video: Adaptive Gradient Methods Converge Faster with Over-Parameterization (and you can do a line-search), Sharan Vaswani »
Sharan Vaswani -
2019 : Poster Session »
Eduard Gorbunov · Alexandre d'Aspremont · Lingxiao Wang · Liwei Wang · Boris Ginsburg · Alessio Quaglino · Camille Castera · Saurabh Adya · Diego Granziol · Rudrajit Das · Raghu Bollapragada · Fabian Pedregosa · Martin Takac · Majid Jahani · Sai Praneeth Karimireddy · Hilal Asi · Balint Daroczy · Leonard Adolphs · Aditya Rawal · Nicolas Brandt · Minhan Li · Giuseppe Ughi · Orlando Romero · Ivan Skorokhodov · Damien Scieur · Kiwook Bae · Konstantin Mishchenko · Rohan Anil · Vatsal Sharan · Aditya Balu · Chao Chen · Zhewei Yao · Tolga Ergen · Paul Grigas · Chris Junchi Li · Jimmy Ba · Stephen J Roberts · Sharan Vaswani · Armin Eftekhari · Chhavi Sharma -
2019 Poster: Painless Stochastic Gradient: Interpolation, Line-Search, and Convergence Rates »
Sharan Vaswani · Aaron Mishkin · Issam Laradji · Mark Schmidt · Gauthier Gidel · Simon Lacoste-Julien