Provable Benefit of Mixup for Finding Optimal Decision Boundaries

Junsoo Oh, Chulhee Yun
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:26403-26450, 2023.

Abstract

We investigate how pair-wise data augmentation techniques like Mixup affect the sample complexity of finding optimal decision boundaries in a binary linear classification problem. For a family of data distributions with a separability constant $\kappa$, we analyze how well the optimal classifier in terms of training loss aligns with the optimal one in test accuracy (i.e., Bayes optimal classifier). For vanilla training without augmentation, we uncover an interesting phenomenon named the curse of separability. As we increase $\kappa$ to make the data distribution more separable, the sample complexity of vanilla training increases exponentially in $\kappa$; perhaps surprisingly, the task of finding optimal decision boundaries becomes harder for more separable distributions. For Mixup training, we show that Mixup mitigates this problem by significantly reducing the sample complexity. To this end, we develop new concentration results applicable to $n^2$ pair-wise augmented data points constructed from $n$ independent data, by carefully dealing with dependencies between overlapping pairs. Lastly, we study other masking-based Mixup-style techniques and show that they can distort the training loss and make its minimizer converge to a suboptimal classifier in terms of test accuracy.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-oh23a, title = {Provable Benefit of Mixup for Finding Optimal Decision Boundaries}, author = {Oh, Junsoo and Yun, Chulhee}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {26403--26450}, 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/oh23a/oh23a.pdf}, url = {https://proceedings.mlr.press/v202/oh23a.html}, abstract = {We investigate how pair-wise data augmentation techniques like Mixup affect the sample complexity of finding optimal decision boundaries in a binary linear classification problem. For a family of data distributions with a separability constant $\kappa$, we analyze how well the optimal classifier in terms of training loss aligns with the optimal one in test accuracy (i.e., Bayes optimal classifier). For vanilla training without augmentation, we uncover an interesting phenomenon named the curse of separability. As we increase $\kappa$ to make the data distribution more separable, the sample complexity of vanilla training increases exponentially in $\kappa$; perhaps surprisingly, the task of finding optimal decision boundaries becomes harder for more separable distributions. For Mixup training, we show that Mixup mitigates this problem by significantly reducing the sample complexity. To this end, we develop new concentration results applicable to $n^2$ pair-wise augmented data points constructed from $n$ independent data, by carefully dealing with dependencies between overlapping pairs. Lastly, we study other masking-based Mixup-style techniques and show that they can distort the training loss and make its minimizer converge to a suboptimal classifier in terms of test accuracy.} }
Endnote
%0 Conference Paper %T Provable Benefit of Mixup for Finding Optimal Decision Boundaries %A Junsoo Oh %A Chulhee Yun %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-oh23a %I PMLR %P 26403--26450 %U https://proceedings.mlr.press/v202/oh23a.html %V 202 %X We investigate how pair-wise data augmentation techniques like Mixup affect the sample complexity of finding optimal decision boundaries in a binary linear classification problem. For a family of data distributions with a separability constant $\kappa$, we analyze how well the optimal classifier in terms of training loss aligns with the optimal one in test accuracy (i.e., Bayes optimal classifier). For vanilla training without augmentation, we uncover an interesting phenomenon named the curse of separability. As we increase $\kappa$ to make the data distribution more separable, the sample complexity of vanilla training increases exponentially in $\kappa$; perhaps surprisingly, the task of finding optimal decision boundaries becomes harder for more separable distributions. For Mixup training, we show that Mixup mitigates this problem by significantly reducing the sample complexity. To this end, we develop new concentration results applicable to $n^2$ pair-wise augmented data points constructed from $n$ independent data, by carefully dealing with dependencies between overlapping pairs. Lastly, we study other masking-based Mixup-style techniques and show that they can distort the training loss and make its minimizer converge to a suboptimal classifier in terms of test accuracy.
APA
Oh, J. & Yun, C.. (2023). Provable Benefit of Mixup for Finding Optimal Decision Boundaries. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:26403-26450 Available from https://proceedings.mlr.press/v202/oh23a.html.

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