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Rectifier (neural networks)

  (Redirected from ReLU)
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 the positive part of its argument:


where x is the input to a neuron. This is also known as a ramp function and is analogous to half-wave rectification in electrical engineering. This activation function was first introduced to a dynamical network by Hahnloser et al. in a 2000 paper in Nature[1] with strong biological motivations and mathematical justifications.[2]. It has been demonstrated for the first time in 2011 to enable better training of deeper networks [3], compared to the widely used activation functions prior to 2011, i.e., the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more practical[4] counterpart, the hyperbolic tangent. The rectifier is, as of 2018, the most popular activation function for deep neural networks.[5][6]

A unit employing the rectifier is also called a rectified linear unit (ReLU).[7]

A smooth approximation to the rectifier is the analytic function

which is called the softplus function.[8] The derivative of softplus is , i.e. the logistic function.

Rectified linear units find applications in computer vision[3] and speech recognition[9][10] using deep neural nets.



Noisy ReLUsEdit

Rectified linear units can be extended to include Gaussian noise, making them noisy ReLUs, giving[7]

 , with  

Noisy ReLUs have been used with some success in restricted Boltzmann machines for computer vision tasks.[7]

Leaky ReLUsEdit

Leaky ReLUs allow a small, non-zero gradient when the unit is not active.[10]


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.[11]


Note that for  , this is equivalent to


and thus has a relation to "maxout" networks.[11]


Exponential linear units try to make the mean activations closer to zero which speeds up learning. It has been shown that ELUs can obtain higher classification accuracy than ReLUs.[12]


  is a hyper-parameter to be tuned and   is a constraint.


  • 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 or exploding gradient problems.
  • Efficient computation: Only comparison, addition and multiplication.
  • Scale-invariant:  .

Rectifying activation functions were used to separate specific excitation and unspecific inhibition in the Neural Abstraction Pyramid, which was trained in a supervised way to learn several computer vision tasks[13]. In 2011,[3] 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 problemsEdit

  • Non-differentiable at zero: however it is differentiable anywhere else, including points arbitrarily close to (but not equal to) zero.
  • Non-zero centered
  • Unbounded
  • Dying ReLU problem: ReLU neurons can sometimes be pushed into states in which they become inactive for essentially all inputs. In this state, no gradients flow backward through the neuron, and so the neuron becomes stuck in a perpetually inactive state and "dies." In some cases, large numbers of neurons in a network can become stuck in dead states, effectively decreasing the model capacity. This problem typically arises when the learning rate is set too high. It may be mitigated by using Leaky ReLUs instead.

See alsoEdit


  1. ^ R Hahnloser, R. Sarpeshkar, M A Mahowald, R. J. Douglas, H.S. Seung (2000). Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit. Nature. 405. pp. 947–951. 
  2. ^ R Hahnloser, H.S. Seung (2001). Permitted and Forbidden Sets in Symmetric Threshold-Linear Networks. NIPS 2001. 
  3. ^ a b c Xavier Glorot, Antoine Bordes and Yoshua Bengio (2011). Deep sparse rectifier neural networks (PDF). AISTATS. 
  4. ^ 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. 
  5. ^ LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey (2015). "Deep learning". Nature. 521 (7553): 436–444. Bibcode:2015Natur.521..436L. doi:10.1038/nature14539. PMID 26017442. 
  6. ^ Ramachandran, Prajit; Barret, Zoph; Quoc, V. Le (October 16, 2017). "Searching for Activation Functions". arXiv:1710.05941  [cs.NE]. 
  7. ^ a b c Vinod Nair and Geoffrey Hinton (2010). Rectified linear units improve restricted Boltzmann machines (PDF). ICML. 
  8. ^ C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, R. Garcia, NIPS'2000, (2001),Incorporating Second-Order Functional Knowledge for Better Option Pricing.
  9. ^ László Tóth (2013). Phone Recognition with Deep Sparse Rectifier Neural Networks (PDF). ICASSP. 
  10. ^ a b Andrew L. Maas, Awni Y. Hannun, Andrew Y. Ng (2014). Rectifier Nonlinearities Improve Neural Network Acoustic Models
  11. ^ a b He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2015). "Delving Deep into Rectifiers: Surpassing Human-Level Performance on Image Net Classification". arXiv:1502.01852  [cs.CV]. 
  12. ^ Clevert, Djork-Arné; Unterthiner, Thomas; Hochreiter, Sepp (2015). "Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)". arXiv:1511.07289  [cs.LG]. 
  13. ^ Behnke, Sven (2003). Hierarchical Neural Networks for Image Interpretation. Lecture Notes in Computer Science. 2766. Springer. doi:10.1007/b11963.