Learning a Single Neuron Robustly to Distributional Shifts and Adversarial Label Noise

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial distribution shifts, where the labels can be arbitrary, and the goal is to find a best-fit function.More precisely, given training samples from a reference distribution $p_0$, the goal is to approximate the vector $\mathbf{w}^*$which minimizes the squared loss with respect to the worst-case distribution that is close in $\chi^2$-divergence to $p_{0}$.We design a computationally efficient algorithm that recovers a vector $ \hat{\mathbf{w}}$satisfying $\mathbb{E}\_{p^*} (\sigma(\hat{\mathbf{w}} \cdot \mathbf{x}) - y)^2 \leq C \hspace{0.2em}