If Rectified Linear Units Are Linear, How Do They Add Nonlinearity?

One may be inclined to point out that ReLUs cannot extrapolate; that is, a series of ReLUs fitted to resemble a sine wave from -4 < x < 4 will not be able to continue the sine wave for values of x outside of those bounds. It’s important to remember, however, that it’s not the goal of a neural network to extrapolate, the goal is to generalize. Consider, for instance, a model fitted to predict house price based on number of bathrooms and number of bedrooms. It doesn’t matter if the model struggles to carry the pattern to negative values of number of bathrooms or values of number of bedrooms exceeding five hundred, because it’s not the objective of the model. (You can read more about generalization vs extrapolation here .)

The strength of the ReLU function lies not in itself, but in an entire army of ReLUs. This is why using a few ReLUs in a neural network does not yield satisfactory results; instead, there must be an abundance of ReLU activations to allow the network to construct an entire map of points. In multi-dimensional space, rectified linear units combine to form complex polyhedra along the class boundaries.

Here lies the reason why ReLU works so well: when there are enough of them, they can approximate any function just as well as other activation functions like sigmoid or tanh, much like stacking hundreds of Legos, without the downsides. There are several issues with smooth-curve functions that do not occur with ReLU — one being that computing the derivative, or the rate of change, the driving force behind gradient descent, is much cheaper with ReLU than with any other smooth-curve function.

Another is that sigmoid and other curves have an issue with the vanishing gradient problem; because the derivative of the sigmoid function gradually slopes off for larger absolute values of x . Because the distributions of inputs may shift around heavily earlier during training away from 0, the derivative will be so small that no useful information can be backpropagated to update the weights. This is often a major problem in neural network training.

On the other hand, the derivative of the ReLU function is simple; it’s the slope of whatever line the input is on. It will reliably return a useful gradient, and while the fact that x = 0 { x < 0} may sometimes lead to a ‘dead neuron problem’, ReLU has still shown to be, in general, more powerful than not only curved functions (sigmoid, tanh) but also ReLU variants attempting to solve the dead neuron problem, like Leaky ReLU.

ReLU is designed to work in abundance; with heavy volume it approximates well, and with good approximation it performs just as well as any other activation function, without the downsides.

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