A similar question was asked on CV: Comprehensive list of activation functions in neural networks with pros/cons.
I copy below one of the answers:
One such a list, though not much exhaustive:
http://cs231n.github.io/neural-networks-1/
Commonly used activation functions
Every activation function (or non-linearity) takes a single number
and performs a certain fixed mathematical operation on it. There are
several activation functions you may encounter in practice:
Left: Sigmoid non-linearity
squashes real numbers to range between [0,1] Right: The tanh
non-linearity squashes real numbers to range between [-1,1].
Sigmoid. The sigmoid non-linearity has the mathematical form $\sigma(x) = 1 / (1 + e^{-x})$ and is shown in the image above on
the left. As alluded to in the previous section, it takes a
real-valued number and "squashes" it into range between 0 and 1. In
particular, large negative numbers become 0 and large positive numbers
become 1. The sigmoid function has seen frequent use historically
since it has a nice interpretation as the firing rate of a neuron:
from not firing at all (0) to fully-saturated firing at an assumed
maximum frequency (1). In practice, the sigmoid non-linearity has
recently fallen out of favor and it is rarely ever used. It has two
major drawbacks:
- Sigmoids saturate and kill gradients. A very undesirable property of the sigmoid neuron is that when the neuron's activation
saturates at either tail of 0 or 1, the gradient at these regions is
almost zero. Recall that during backpropagation, this (local) gradient
will be multiplied to the gradient of this gate's output for the whole
objective. Therefore, if the local gradient is very small, it will
effectively "kill" the gradient and almost no signal will flow through
the neuron to its weights and recursively to its data. Additionally,
one must pay extra caution when initializing the weights of sigmoid
neurons to prevent saturation. For example, if the initial weights are
too large then most neurons would become saturated and the network
will barely learn.
- Sigmoid outputs are not zero-centered. This is undesirable since neurons in later layers of processing in a Neural Network (more on
this soon) would be receiving data that is not zero-centered. This has
implications on the dynamics during gradient descent, because if the
data coming into a neuron is always positive (e.g. $x > 0$
elementwise in $f = w^Tx + b$)), then the gradient on the weights
$w$ will during backpropagation become either all be positive, or
all negative (depending on the gradient of the whole expression
$f$). This could introduce undesirable zig-zagging dynamics in the
gradient updates for the weights. However, notice that once these
gradients are added up across a batch of data the final update for the
weights can have variable signs, somewhat mitigating this issue.
Therefore, this is an inconvenience but it has less severe
consequences compared to the saturated activation problem above.
Tanh. The tanh non-linearity is shown on the image above on the right. It squashes a real-valued number to the range [-1, 1]. Like the
sigmoid neuron, its activations saturate, but unlike the sigmoid
neuron its output is zero-centered. Therefore, in practice the tanh
non-linearity is always preferred to the sigmoid nonlinearity. Also
note that the tanh neuron is simply a scaled sigmoid neuron, in
particular the following holds: $ \tanh(x) = 2 \sigma(2x) -1 $.
Left: Rectified Linear
Unit (ReLU) activation function, which is zero when x < 0 and then
linear with slope 1 when x > 0. Right: A plot from Krizhevsky
et al. (pdf) paper indicating the 6x improvement in convergence
with the ReLU unit compared to the tanh unit.
ReLU. The Rectified Linear Unit has become very popular in the last few years. It computes the function $f(x) = \max(0, x)$. In
other words, the activation is simply thresholded at zero (see image
above on the left). There are several pros and cons to using the
ReLUs:
- (+) It was found to greatly accelerate (e.g. a factor of 6 in Krizhevsky et
al.) the
convergence of stochastic gradient descent compared to the
sigmoid/tanh functions. It is argued that this is due to its linear,
non-saturating form.
- (+) Compared to tanh/sigmoid neurons that involve expensive operations (exponentials, etc.), the ReLU can be implemented by simply
thresholding a matrix of activations at zero.
- (-) Unfortunately, ReLU units can be fragile during training and can "die". For example, a large gradient flowing through a ReLU neuron
could cause the weights to update in such a way that the neuron will
never activate on any datapoint again. If this happens, then the
gradient flowing through the unit will forever be zero from that point
on. That is, the ReLU units can irreversibly die during training since
they can get knocked off the data manifold. For example, you may find
that as much as 40% of your network can be "dead" (i.e. neurons that
never activate across the entire training dataset) if the learning
rate is set too high. With a proper setting of the learning rate this
is less frequently an issue.
Leaky ReLU. Leaky ReLUs are one attempt to fix the "dying ReLU" problem. Instead of the function being zero when x < 0, a leaky ReLU will instead have a small negative slope (of 0.01, or so). That is, the function computes $f(x) = \mathbb{1}(x < 0) (\alpha x) + \mathbb{1}(x>=0) (x) $ where $\alpha$ is a small constant. Some people report success with this form of activation function, but the results are not always consistent. The slope in the negative region can also be made into a parameter of each neuron, as seen in PReLU neurons, introduced in Delving Deep into Rectifiers, by Kaiming He et al., 2015. However, the consistency of the benefit across tasks is presently unclear.
Maxout. Other types of units have been proposed that do not have the functional form $f(w^Tx + b)$ where a non-linearity is applied
on the dot product between the weights and the data. One relatively
popular choice is the Maxout neuron (introduced recently by
Goodfellow et
al.) that
generalizes the ReLU and its leaky version. The Maxout neuron computes
the function $\max(w_1^Tx+b_1, w_2^Tx + b_2)$. Notice that both
ReLU and Leaky ReLU are a special case of this form (for example, for
ReLU we have $w_1, b_1 = 0$). The Maxout neuron therefore enjoys
all the benefits of a ReLU unit (linear regime of operation, no
saturation) and does not have its drawbacks (dying ReLU). However,
unlike the ReLU neurons it doubles the number of parameters for every
single neuron, leading to a high total number of parameters.
This concludes our discussion of the most common types of neurons and
their activation functions. As a last comment, it is very rare to mix
and match different types of neurons in the same network, even though
there is no fundamental problem with doing so.
TLDR: "What neuron type should I use?" Use the ReLU non-linearity, be careful with your learning rates and possibly
monitor the fraction of "dead" units in a network. If this concerns
you, give Leaky ReLU or Maxout a try. Never use sigmoid. Try tanh, but
expect it to work worse than ReLU/Maxout.
License:
The MIT License (MIT)
Copyright (c) 2015 Andrej Karpathy
Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:
The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.*
