Why is it that logistic regression is considered a linear classifier, but a neural network with a sigmoid activation function and sigmoid output layer is considered non-linear? I am aware that logistic regression is considered linear because its inputs are a linear combination of the original inputs, but couldn't the same be said of the neural network example?
A neural network with a single sigmoid neuron is also a linear classifier when the output is thresholded for classification. But, more than one layer produces a non-linear one because the decision rule can't be written in the following form: $$\sum w_i x_i+b>\tau$$ where $\tau$ is threshold, $w_i$ are learnable weights and $x_i$ are features. This is why logistic regression is referred as a linear classifier.