### Web: http://arxiv.org/abs/2206.08918

June 20, 2022, 1:11 a.m. | Ilias Diakonikolas, Vasilis Kontonis, Christos Tzamos, Nikos Zarifis

We study the fundamental problem of learning a single neuron, i.e., a
function of the form $\mathbf{x}\mapsto\sigma(\mathbf{w}\cdot\mathbf{x})$ for
monotone activations $\sigma:\mathbb{R}\mapsto\mathbb{R}$, with respect to the
$L_2^2$-loss in the presence of adversarial label noise. Specifically, we are
given labeled examples from a distribution $D$ on $(\mathbf{x}, y)\in\mathbb{R}^d \times \mathbb{R}$ such that there exists
$\mathbf{w}^\ast\in\mathbb{R}^d$ achieving $F(\mathbf{w}^\ast)=\epsilon$, where
$F(\mathbf{w})=\mathbf{E}_{(\mathbf{x},y)\sim D}[(\sigma(\mathbf{w}\cdot \mathbf{x})-y)^2]$. The goal of the learner is to output a hypothesis vector
$\mathbf{w}$ such that $F(\mathbb{w})=C\, \epsilon$ with high probability,
where $C>1$ …

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