具有非线性激活 function（比如 ReLU）的神经网络可以用于线性分类任务吗？

Question

I think the answer would be yes, but I'm unable to reason out a good explanation on this.

Answer 1

Technically, yes .

The reason you could use a non-linear activation function for this task is that you can manually alter the results. Let's say the range the activation function outputs is between 0.0-1.0, then you can round up or down to get a binary 0/1, linear yes/no.

The reason you shouldn't is the same reason that you shouldn't attach an industrial heater to a fan and call it a hair-drier, it's unnecessarily powerful and it could potentially waste resources and time.

I hope this answer helped, have a good day!

Answer 2

The mathematical argument lies in a power to represent linearity, we can use following three lemmas to show that:

Lemma 1

With affine transformations (linear layer) we can map the input hypercube [0,1]^d into arbitrary small box [a,b]^k. Proof is quite simple, we can just make all the biases to be equal to a, and make weights multiply by (ba).

Lemma 2

For sufficiently small scale, many non-linearities are approximately linear. This is actually very much a definition of a derivative, or, taylor expansion. In particular let us take relu(x), for x>0 it is in fact, linear? What about sigmoid, Well if we look at a tiny tiny region [-eps, eps] you can see that it approaches a linear function as eps->0!

Lemma 3

Composition of affine functions is affine. In other words, if I were to make a neural network with multiple linear layers, it is equivalent of having just one. This comes from the matrix composition rules:

W2(W1x + b1) + b2 = W2W1x + W2b1 + b2 = (W2W1)x + (W2b1 + b2)
                                        ------    -----------
                                    New weights   New bias

Combining the above

Composing the three lemmas above we see that with a non-linear layer, there always exists an arbitrarily good approximation of the linear function, We simply use the first layer to map entire input space into the tiny part of the pre-activation spacve where your linearity is approximately linear. and then we "map it back" in the following layer.

General case

This is a very simple proof, now in general you can use Universal Approximation Theorem to show that a non-linear neural network (Sigmoid, Relu, many others) that is sufficiently large, can approximate any smooth target function, which includes linear ones. This proof (originally given by Cybenko) is however much more complex and relies on showing that specific classes of functions are dense in the space of continuous functions.