Why use tanh for activation function of MLP?

Question

Im personally studying theories of neural network and got some questions.

In many books and references, for activation function of hidden layer, hyper-tangent functions were used.

Books came up with really simple reason that linear combinations of tanh functions can describe nearly all shape of functions with given error.

But, there came a question.

1. Is this a real reason why tanh function is used?
2. If then, is it the only reason why tanh function is used?
3. if then, is tanh function the only function that can do that?
4. if not, what is the real reason?..

I stock here keep thinking... please help me out of this mental(?...) trap!

Answer 1

In deep learning the ReLU has become the activation function of choice because the math is much simpler from sigmoid activation functions such as tanh or logit, especially if you have many layers. To assign weights using backpropagation, you normally calculate the gradient of the loss function and apply the chain rule for hidden layers, meaning you need the derivative of the activation functions. ReLU is a ramp function where you have a flat part where the derivative is 0, and a skewed part where the derivative is 1. This makes the math really easy. If you use the hyperbolic tangent you might run into the fading gradient problem, meaning if x is smaller than -2 or bigger than 2, the derivative gets really small and your network might not converge, or you might end up having a dead neuron that does not fire anymore.

Answer 2

Many of the answers here describe why tanh (i.e. (1 - e^2x) / (1 + e^2x)) is preferable to the sigmoid/logistic function (1 / (1 + e^-x)), but it should noted that there is a good reason why these are the two most common alternatives that should be understood, which is that during training of an MLP using the back propagation algorithm, the algorithm requires the value of the derivative of the activation function at the point of activation of each node in the network. While this could generally be calculated for most plausible activation functions (except those with discontinuities, which is a bit of a problem for those), doing so often requires expensive computations and/or storing additional data (e.g. the value of input to the activation function, which is not otherwise required after the output of each node is calculated). Tanh and the logistic function, however, both have very simple and efficient calculations for their derivatives that can be calculated from the output of the functions; i.e. if the node's weighted sum of inputs is v and its output is u, we need to know du/dv which can be calculated from u rather than the more traditional v: for tanh it is 1 - u^2 and for the logistic function it is u * (1 - u). This fact makes these two functions more efficient to use in a back propagation network than most alternatives, so a compelling reason would usually be required to deviate from them.

Answer 3

Update in attempt to appease commenters: based purely on observation, rather than the theory that is covered above, Tanh and ReLU activation functions are more performant than sigmoid. Sigmoid also seems to be more prone to local optima, or a least extended 'flat line' issues. For example, try limiting the number of features to force logic into network nodes in XOR and sigmoid rarely succeeds whereas Tanh and ReLU have more success.

Tanh seems maybe slower than ReLU for many of the given examples, but produces more natural looking fits for the data using only linear inputs, as you describe. For example a circle vs a square/hexagon thing.

http://playground.tensorflow.org/ <- this site is a fantastic visualisation of activation functions and other parameters to neural network. Not a direct answer to your question but the tool 'provides intuition' as Andrew Ng would say.

Answer 4

Most of time tanh is quickly converge than sigmoid and logistic function, and performs better accuracy [1]. However, recently rectified linear unit (ReLU) is proposed by Hinton [2] which shows ReLU train six times fast than tanh [3] to reach same training error. And you can refer to [4] to see what benefits ReLU provides.

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Accordining to about 2 years machine learning experience. I want to share some stratrgies the most paper used and my experience about computer vision.

Normalizing input is very important

Normalizing well could get better performance and converge quickly. Most of time we will subtract mean value to make input mean to be zero to prevent weights change same directions so that converge slowly [5] .Recently google also points that phenomenon as internal covariate shift out when training deep learning, and they proposed batch normalization [6] so as to normalize each vector having zero mean and unit variance.

More data more accuracy

More training data could generize feature space well and prevent overfitting. In computer vision if training data is not enough, most of used skill to increase training dataset is data argumentation and synthesis training data.

Choosing a good activation function allows training better and efficiently.

ReLU nonlinear acitivation worked better and performed state-of-art results in deep learning and MLP. Moreover, it has some benefits e.g. simple to implementation and cheaper computation in back-propagation to efficiently train more deep neural net. However, ReLU will get zero gradient and do not train when the unit is zero active. Hence some modified ReLUs are proposed e.g. Leaky ReLU, and Noise ReLU, and most popular method is PReLU [7] proposed by Microsoft which generalized the traditional recitifed unit.

Others

• choose large initial learning rate if it will not oscillate or diverge so as to find a better global minimum.
• shuffling data

Answer 5

In theory I in accord with above responses. In my experience, some problems have a preference for sigmoid rather than tanh, probably due to the nature of these problems (since there are non-linear effects, is difficult understand why).

Given a problem, I generally optimize networks using a genetic algorithm. The activation function of each element of the population is choosen randonm between a set of possibilities (sigmoid, tanh, linear, ...). For a 30% of problems of classification, best element found by genetic algorithm has sigmoid as activation function.

Answer 6

To add up to the the already existing answer, the preference for symmetry around 0 isn't just a matter of esthetics. An excellent text by LeCun et al "Efficient BackProp" shows in great details why it is a good idea that the input, output and hidden layers have mean values of 0 and standard deviation of 1.

Answer 7

In truth both tanh and logistic functions can be used. The idea is that you can map any real number ( [-Inf, Inf] ) to a number between [-1 1] or [0 1] for the tanh and logistic respectively. In this way, it can be shown that a combination of such functions can approximate any non-linear function. Now regarding the preference for the tanh over the logistic function is that the first is symmetric regarding the 0 while the second is not. This makes the second one more prone to saturation of the later layers, making training more difficult.