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Hardness of Learning a Single Neuron with Adversarial Label Noise

Ilias Diakonikolas · Daniel Kane · Pasin Manurangsi · Lisheng Ren


We study the problem of distribution-free learning of a single neuronunder adversarial label noise with respect to the squared loss.For a wide range of activation functions, including ReLUs and sigmoids,we prove hardness of learning results in the Statistical Query model andunder a well-studied assumption on the complexity of refuting XOR formulas.Specifically, we establish that no polynomial-time learning algorithm, even improper,can approximate the optimal loss value within any constant factor.

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