Hardness of Learning Fixed Parities with Neural Networks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912179683328000 |
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| author | Shoshani, Itamar Shamir, Ohad |
| author_facet | Shoshani, Itamar Shamir, Ohad |
| contents | Learning parity functions is a canonical problem in learning theory, which although computationally tractable, is not amenable to standard learning algorithms such as gradient-based methods. This hardness is usually explained via statistical query lower bounds [Kearns, 1998]. However, these bounds only imply that for any given algorithm, there is some worst-case parity function that will be hard to learn. Thus, they do not explain why fixed parities - say, the full parity function over all coordinates - are difficult to learn in practice, at least with standard predictors and gradient-based methods [Abbe and Boix-Adsera, 2022]. In this paper, we address this open problem, by showing that for any fixed parity of some minimal size, using it as a target function to train one-hidden-layer ReLU networks with perturbed gradient descent will fail to produce anything meaningful. To establish this, we prove a new result about the decay of the Fourier coefficients of linear threshold (or weighted majority) functions, which may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_00817 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hardness of Learning Fixed Parities with Neural Networks Shoshani, Itamar Shamir, Ohad Machine Learning Learning parity functions is a canonical problem in learning theory, which although computationally tractable, is not amenable to standard learning algorithms such as gradient-based methods. This hardness is usually explained via statistical query lower bounds [Kearns, 1998]. However, these bounds only imply that for any given algorithm, there is some worst-case parity function that will be hard to learn. Thus, they do not explain why fixed parities - say, the full parity function over all coordinates - are difficult to learn in practice, at least with standard predictors and gradient-based methods [Abbe and Boix-Adsera, 2022]. In this paper, we address this open problem, by showing that for any fixed parity of some minimal size, using it as a target function to train one-hidden-layer ReLU networks with perturbed gradient descent will fail to produce anything meaningful. To establish this, we prove a new result about the decay of the Fourier coefficients of linear threshold (or weighted majority) functions, which may be of independent interest. |
| title | Hardness of Learning Fixed Parities with Neural Networks |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2501.00817 |