Learning a Single Neuron for Non-monotonic Activation Functions
This addresses a gap in theoretical understanding for practical non-monotonic activations like SiLU/Swish and GELU, which outperform monotonic ones in many applications, though it is incremental as it extends existing positive results beyond monotonicity.
The paper tackles the problem of learning a single neuron with non-monotonic activation functions using gradient descent, establishing that under mild conditions (e.g., a dominating linear part), polynomial-time and sample learnability is guaranteed for Gaussian inputs, with extensions to non-degenerate distributions under stronger assumptions.
We study the problem of learning a single neuron $\mathbf{x}\mapsto σ(\mathbf{w}^T\mathbf{x})$ with gradient descent (GD). All the existing positive results are limited to the case where $σ$ is monotonic. However, it is recently observed that non-monotonic activation functions outperform the traditional monotonic ones in many applications. To fill this gap, we establish learnability without assuming monotonicity. Specifically, when the input distribution is the standard Gaussian, we show that mild conditions on $σ$ (e.g., $σ$ has a dominating linear part) are sufficient to guarantee the learnability in polynomial time and polynomial samples. Moreover, with a stronger assumption on the activation function, the condition of input distribution can be relaxed to a non-degeneracy of the marginal distribution. We remark that our conditions on $σ$ are satisfied by practical non-monotonic activation functions, such as SiLU/Swish and GELU. We also discuss how our positive results are related to existing negative results on training two-layer neural networks.