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Paper

  • Title: Fast and Accurate Deep Networks Learning By Exponential Linear Units (ELUs)
  • Authors: Djork-Arné Clevert, Thomas Unterthiner, Sepp Hochreiter
  • Link: http://arxiv.org/abs/1511.07289
  • Tags: Neural Network, activation, nonlinearity, ReLU, ELU, PReLU, LeakyReLU
  • Year: 2015

Summary

  • What

    • ELUs are an activation function
    • The are most similar to LeakyReLUs and PReLUs

  • How (formula)

    • f(x):
      • if x >= 0: x
      • else: alpha(exp(x)-1)
    • f'(x) / Derivative:
      • if x >= 0: 1
      • else: f(x) + alpha
    • alpha defines at which negative value the ELU saturates.
    • E. g. alpha=1.0 means that the minimum value that the ELU can reach is -1.0
    • LeakyReLUs however can go to -Infinity, ReLUs can't go below 0.

ELUs vs LeakyReLUs vs ReLUs

Form of ELUs(alpha=1.0) vs LeakyReLUs vs ReLUs.

  • Why

    • They derive from the unit natural gradient that a network learns faster, if the mean activation of each neuron is close to zero.
    • ReLUs can go above 0, but never below. So their mean activation will usually be quite a bit above 0, which should slow down learning.
    • ELUs, LeakyReLUs and PReLUs all have negative slopes, so their mean activations should be closer to 0.
    • In contrast to LeakyReLUs and PReLUs, ELUs saturate at a negative value (usually -1.0).
    • The authors think that is good, because it lets ELUs encode the degree of presence of input concepts, while they do not quantify the degree of absence.
    • So ELUs can measure the presence of concepts quantitatively, but the absence only qualitatively.
    • They think that this makes ELUs more robust to noise.

  • Results

    • In their tests on MNIST, CIFAR-10, CIFAR-100 and ImageNet, ELUs perform (nearly always) better than ReLUs and LeakyReLUs.
    • However, they don't test PReLUs at all and use an alpha of 0.1 for LeakyReLUs (even though 0.33 is afaik standard) and don't test LeakyReLUs on ImageNet (only ReLUs).

CIFAR-100

Comparison of ELUs, LeakyReLUs, ReLUs on CIFAR-100. ELUs ends up with best values, beaten during the early epochs by LeakyReLUs. (Learning rates were optimized for ReLUs.)

Rough chapter-wise notes

  • Introduction

    • Currently popular choice: ReLUs
    • ReLU: max(0, x)
    • ReLUs are sparse and avoid the vanishing gradient problem, because their derivate is 1 when they are active.
    • ReLUs have a mean activation larger than zero.
    • Non-zero mean activation causes a bias shift in the next layer, especially if multiple of them are correlated.
    • The natural gradient (?) corrects for the bias shift by adjusting the weight update.
    • Having less bias shift would bring the standard gradient closer to the natural gradient, which would lead to faster learning.
    • Suggested solutions:
      • Centering activation functions at zero, which would keep the off-diagonal entries of the Fisher information matrix small.
      • Batch Normalization
      • Projected Natural Gradient Descent (implicitly whitens the activations)
    • These solutions have the problem, that they might end up taking away previous learning steps, which would slow down learning unnecessarily.
    • Chosing a good activation function would be a better solution.
    • Previously, tanh was prefered over sigmoid for that reason (pushed mean towards zero).
    • Recent new activation functions:
      • LeakyReLUs: x if x > 0, else alpha*x
      • PReLUs: Like LeakyReLUs, but alpha is learned
      • RReLUs: Slope of part < 0 is sampled randomly
    • Such activation functions with non-zero slopes for negative values seemed to improve results.
    • The deactivation state of such units is not very robust to noise, can get very negative.
    • They suggest an activation function that can return negative values, but quickly saturates (for negative values, not for positive ones).
    • So the model can make a quantitative assessment for positive statements (there is an amount X of A in the image), but only a qualitative negative one (something indicates that B is not in the image).
    • They argue that this makes their activation function more robust to noise.
    • Their activation function still has activations with a mean close to zero.

  • Zero Mean Activations Speed Up Learning

    • Natural Gradient = Update direction which corrects the gradient direction with the Fisher Information Matrix
    • Hessian-Free Optimization techniques use an extended Gauss-Newton approximation of Hessians and therefore can be interpreted as versions of natural gradient descent.
    • Computing the Fisher matrix is too expensive for neural networks.
    • Methods to approximate the Fisher matrix or to perform natural gradient descent have been developed.
    • Natural gradient = inverse(FisherMatrix) * gradientOfWeights
    • Lots of formulas. Apparently first explaining how the natural gradient descent works, then proofing that natural gradient descent can deal well with non-zero-mean activations.
    • Natural gradient descent auto-corrects bias shift (i.e. non-zero-mean activations).
    • If that auto-correction does not exist, oscillations (?) can occur, which slow down learning.
    • Two ways to push means towards zero:
      • Unit zero mean normalization (e.g. Batch Normalization)
      • Activation functions with negative parts

  • Exponential Linear Units (ELUs)

    • Formula
      • f(x):
        • if x >= 0: x
        • else: alpha(exp(x)-1)
      • f'(x) / Derivative:
        • if x >= 0: 1
        • else: f(x) + alpha
      • alpha defines at which negative value the ELU saturates.
      • alpha=0.5 => minimum value is -0.5 (?)
    • ELUs avoid the vanishing gradient problem, because their positive part is the identity function (like e.g. ReLUs)
    • The negative values of ELUs push the mean activation towards zero.
    • Mean activations closer to zero resemble more the natural gradient, therefore they should speed up learning.
    • ELUs are more noise robust than PReLUs and LeakyReLUs, because their negative values saturate and thus should create a small gradient.
    • "ELUs encode the degree of presence of input concepts, while they do not quantify the degree of absence"

  • Experiments Using ELUs

    • They compare ELUs to ReLUs and LeakyReLUs, but not to PReLUs (no explanation why).
    • They seem to use a negative slope of 0.1 for LeakyReLUs, even though 0.33 is standard afaik.
    • They use an alpha of 1.0 for their ELUs (i.e. minimum value is -1.0).
    • MNIST classification:
      • ELUs achieved lower mean activations than ReLU/LeakyReLU
      • ELUs achieved lower cross entropy loss than ReLU/LeakyReLU (and also seemed to learn faster)
      • They used 5 hidden layers of 256 units each (no explanation why so many)
      • (No convolutions)
    • MNIST Autoencoder:
      • ELUs performed consistently best (at different learning rates)
      • Usually ELU > LeakyReLU > ReLU
      • LeakyReLUs not far off, so if they had used a 0.33 value maybe these would have won
    • CIFAR-100 classification:
      • Convolutional network, 11 conv layers
      • LeakyReLUs performed better during the first ~50 epochs, ReLUs mostly on par with ELUs
      • LeakyReLUs about on par for epochs 50-100
      • ELUs win in the end (the learning rates used might not be optimal for ELUs, were designed for ReLUs)
    • CIFER-100, CIFAR-10 (big convnet):
      • 6.55% error on CIFAR-10, 24.28% on CIFAR-100
      • No comparison with ReLUs and LeakyReLUs for same architecture
    • ImageNet
      • Big convnet with spatial pyramid pooling (?) before the fully connected layers
      • Network with ELUs performed better than ReLU network (better score at end, faster learning)
      • Networks were still learning at the end, they didn't run till convergence
      • No comparison to LeakyReLUs