Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and ReLU Regression under Gaussian Marginals

Ilias Diakonikolas, Daniel Kane, Lisheng Ren
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:7922-7938, 2023.

Abstract

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\\mathbf{x},y)$ from an unknown distribution on $\\mathbb{R}^n \\times \\{\pm 1 \\}$, whose marginal distribution on $\\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\\mathrm{OPT}+\\epsilon$, where $\\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-diakonikolas23b, title = {Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and {R}e{LU} Regression under {G}aussian Marginals}, author = {Diakonikolas, Ilias and Kane, Daniel and Ren, Lisheng}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {7922--7938}, 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/diakonikolas23b/diakonikolas23b.pdf}, url = {https://proceedings.mlr.press/v202/diakonikolas23b.html}, abstract = {We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\\mathbf{x},y)$ from an unknown distribution on $\\mathbb{R}^n \\times \\{\pm 1 \\}$, whose marginal distribution on $\\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\\mathrm{OPT}+\\epsilon$, where $\\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.} }
Endnote
%0 Conference Paper %T Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and ReLU Regression under Gaussian Marginals %A Ilias Diakonikolas %A Daniel Kane %A Lisheng Ren %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-diakonikolas23b %I PMLR %P 7922--7938 %U https://proceedings.mlr.press/v202/diakonikolas23b.html %V 202 %X We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\\mathbf{x},y)$ from an unknown distribution on $\\mathbb{R}^n \\times \\{\pm 1 \\}$, whose marginal distribution on $\\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\\mathrm{OPT}+\\epsilon$, where $\\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.
APA
Diakonikolas, I., Kane, D. & Ren, L.. (2023). Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and ReLU Regression under Gaussian Marginals. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:7922-7938 Available from https://proceedings.mlr.press/v202/diakonikolas23b.html.

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