We consider the fundamental problem of learning a single neuron $x \mapsto\sigma(w^\top x)$ using standard gradient methods. As opposed to previous works, which considered specific (and not always realistic) input distributions and activation functions $\sigma(\cdot)$, we ask whether a more general result is attainable, under milder assumptions. On the one hand, we show that some assumptions on the distribution and the activation function are necessary. On the other hand, we prove positive guarantees under mild assumptions, which go beyond those studied in the literature so far. We also point out and study the challenges in further strengthening and generalizing our results.

中文翻译：

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使用梯度方法学习单个神经元

我们考虑使用标准梯度方法学习单个神经元 $x \mapsto\sigma(w^\top x)$ 的基本问题。与之前的工作不同，这些工作考虑了特定的（并不总是现实的）输入分布和激活函数 $\sigma(\cdot)$，我们询问在更温和的假设下是否可以获得更一般的结果。一方面，我们表明对分布和激活函数的一些假设是必要的。另一方面，我们在温和的假设下证明了积极的保证，这超出了迄今为止文献中的研究。我们还指出并研究了进一步加强和推广我们的结果的挑战。