Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs) Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)
Paper summary * 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](https://raw.githubusercontent.com/aleju/papers/master/neural-nets/images/ELUs__slopes.png?raw=true "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](https://raw.githubusercontent.com/aleju/papers/master/neural-nets/images/ELUs__cifar100.png?raw=true "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
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Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)
Djork-Arné Clevert and Thomas Unterthiner and Sepp Hochreiter
arXiv e-Print archive - 2015 via Local arXiv
Keywords: cs.LG

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