Benign Overfitting in Two-Layer ReLU Convolutional Neural Networks for XOR Data

Xuran Meng, Difan Zou, Yuan Cao
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:35404-35469, 2024.

Abstract

Modern deep learning models are usually highly over-parameterized so that they can overfit the training data. Surprisingly, such overfitting neural networks can usually still achieve high prediction accuracy. To study this “benign overfitting” phenomenon, a line of recent works has theoretically studied the learning of linear models and two-layer neural networks. However, most of these analyses are still limited to the very simple learning problems where the Bayes-optimal classifier is linear. In this work, we investigate a class of XOR-type classification tasks with label-flipping noises. We show that, under a certain condition on the sample complexity and signal-to-noise ratio, an over-parameterized ReLU CNN trained by gradient descent can achieve near Bayes-optimal accuracy. Moreover, we also establish a matching lower bound result showing that when the previous condition is not satisfied, the prediction accuracy of the obtained CNN is an absolute constant away from the Bayes-optimal rate. Our result demonstrates that CNNs have a remarkable capacity to efficiently learn XOR problems, even in the presence of highly correlated features.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-meng24c, title = {Benign Overfitting in Two-Layer {R}e{LU} Convolutional Neural Networks for {XOR} Data}, author = {Meng, Xuran and Zou, Difan and Cao, Yuan}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {35404--35469}, 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/meng24c/meng24c.pdf}, url = {https://proceedings.mlr.press/v235/meng24c.html}, abstract = {Modern deep learning models are usually highly over-parameterized so that they can overfit the training data. Surprisingly, such overfitting neural networks can usually still achieve high prediction accuracy. To study this “benign overfitting” phenomenon, a line of recent works has theoretically studied the learning of linear models and two-layer neural networks. However, most of these analyses are still limited to the very simple learning problems where the Bayes-optimal classifier is linear. In this work, we investigate a class of XOR-type classification tasks with label-flipping noises. We show that, under a certain condition on the sample complexity and signal-to-noise ratio, an over-parameterized ReLU CNN trained by gradient descent can achieve near Bayes-optimal accuracy. Moreover, we also establish a matching lower bound result showing that when the previous condition is not satisfied, the prediction accuracy of the obtained CNN is an absolute constant away from the Bayes-optimal rate. Our result demonstrates that CNNs have a remarkable capacity to efficiently learn XOR problems, even in the presence of highly correlated features.} }
Endnote
%0 Conference Paper %T Benign Overfitting in Two-Layer ReLU Convolutional Neural Networks for XOR Data %A Xuran Meng %A Difan Zou %A Yuan Cao %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-meng24c %I PMLR %P 35404--35469 %U https://proceedings.mlr.press/v235/meng24c.html %V 235 %X Modern deep learning models are usually highly over-parameterized so that they can overfit the training data. Surprisingly, such overfitting neural networks can usually still achieve high prediction accuracy. To study this “benign overfitting” phenomenon, a line of recent works has theoretically studied the learning of linear models and two-layer neural networks. However, most of these analyses are still limited to the very simple learning problems where the Bayes-optimal classifier is linear. In this work, we investigate a class of XOR-type classification tasks with label-flipping noises. We show that, under a certain condition on the sample complexity and signal-to-noise ratio, an over-parameterized ReLU CNN trained by gradient descent can achieve near Bayes-optimal accuracy. Moreover, we also establish a matching lower bound result showing that when the previous condition is not satisfied, the prediction accuracy of the obtained CNN is an absolute constant away from the Bayes-optimal rate. Our result demonstrates that CNNs have a remarkable capacity to efficiently learn XOR problems, even in the presence of highly correlated features.
APA
Meng, X., Zou, D. & Cao, Y.. (2024). Benign Overfitting in Two-Layer ReLU Convolutional Neural Networks for XOR Data. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:35404-35469 Available from https://proceedings.mlr.press/v235/meng24c.html.

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