Rectifier (neural networks)

Plot of the rectifier (blue) and softplus (green) functions near x = 0.

In the context of artificial neural networks, the rectifier is an activation function defined as

$f(x) = \max(0, x)$

where x is the input to a neuron. This is also known as a ramp function, and it is analogous to half-wave rectification in electrical engineering. This activation function has been argued to be more biologically plausible^[1] than the widely used logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more practical^[2] counterpart, the hyperbolic tangent. The rectifier is, as of 2015, the most popular activation function for deep neural networks.^[3]

A unit employing the rectifier is also called a rectified linear unit (ReLU).^[4]

A smooth approximation to the rectifier is the analytic function

$f(x) = \ln(1 + e^x)$

which is called the softplus function.^[5] The derivative of softplus is $f'(x) = e^x / (e^x+1) = 1 / (1 + e^{-x})$, i.e. the logistic function.

Rectified linear units find applications in computer vision,^[1] and speech recognition^{[6] [7]} using deep neural nets.

Variants

Noisy ReLUs

Rectified linear units can be extended to include Gaussian noise, making them noisy ReLUs, giving^[4]

$f(x) = \max(0, x + Y)$, with $Y \sim \mathcal{N}(0, \sigma(x))$

Noisy ReLUs have been used with some success in restricted Boltzmann machines for computer vision tasks.^[4]

Leaky ReLUs

Leaky ReLUs allow a small, non-zero gradient when the unit is not active.^[7]

$f(x) = \begin{cases} x & \mbox{if } x > 0 \\ 0.01x & \mbox{otherwise} \end{cases}$

Parametric ReLUs take this idea further by making the coefficient of leakage into a parameter that is learned along with the other neural network parameters.^[8]

$f(x) = \begin{cases} x & \mbox{if } x > 0 \\ a x & \mbox{otherwise} \end{cases}$

Note that for $a<1$, this is equivalent to

$f(x) = \max(x, ax)$

and thus has a relation to "maxout" networks.^[8]

Advantages

• Biological plausibility: One-sided, compared to the antisymmetry of tanh.
• Sparse activation: For example, in a randomly initialized network, only about 50% of hidden units are activated (having a non-zero output).
• Efficient gradient propagation: No vanishing gradient problem or exploding effect.
• Efficient computation: Only comparison, addition and multiplication.

For the first time in 2011,^[1] the use of the rectifier as a non-linearity has been shown to enable training deep supervised neural networks without requiring unsupervised pre-training. Rectified linear units, compared to sigmoid function or similar activation functions, allow for faster and effective training of deep neural architectures on large and complex datasets.

Potential problems

• Non-differentiable at zero: however it is differentiable anywhere else, including points arbitrarily close to (but not equal to) zero.

See also

• Softmax function
• Sigmoid function
• Tobit model

References

1. Xavier Glorot, Antoine Bordes and Yoshua Bengio (2011). Deep sparse rectifier neural networks (PDF). AISTATS.
2. Yann LeCun, Leon Bottou, Genevieve B. Orr and Klaus-Robert Müller (1998). "Efficient BackProp" (PDF). In G. Orr and K. Müller. Neural Networks: Tricks of the Trade. Springer.
3. LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey (2015). "Deep learning". Nature 521: 436–444. doi:10.1038/nature14539.
4. Vinod Nair and Geoffrey Hinton (2010). Rectified linear units improve restricted Boltzmann machines (PDF). ICML.
5. C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, R. Garcia, NIPS'2000, (2001),Incorporating Second-Order Functional Knowledge for Better Option Pricing
6. László Tóth (2013). Phone Recognition with Deep Sparse Rectifier Neural Networks (PDF). ICASSP.
7. Andrew L. Maas, Awni Y. Hannun, Andrew Y. Ng (2014). Rectifier Nonlinearities Improve Neural Network Acoustic Models
8. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun (2015) Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification