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Sec. 5: Activation Functions

    One of the crucial factors in deep networks is activation function, which brings the non-linearity into networks. Here we will introduce the details and characters of some popular activation functions and give advices later in this section.

1

Sigmoid

 

    The sigmoid non-linearity has the mathematical form 

. 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:
    1. 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 back-propagation, 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.        2. 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 element wise in 

), then the gradient on the weights  w will during back-propagation 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.

2

tanh(x)

 

    The tanh non-linearity 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.

3

Rectified Linear Unit

 

    The Rectified Linear Unit (ReLU) has become very popular in the last few years. It computes the function 

, which is simply thresholded at zero.

    There are several pros and cons to using the ReLUs:    1. (Pros) Compared to sigmoid/tanh neurons that involve expensive operations (exponentials, etc.), the ReLU can be implemented by simply thresholding a matrix of activations at zero. Meanwhile, ReLUs does not suffer from saturating.    2. (Pros) It was found to greatly accelerate (e.g., a factor of 6 in [1]) 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.    3. (Cons) 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.

4

Leaky ReLU

 

    Leaky ReLUs are one attempt to fix the “dying ReLU” problem. Instead of the function being zero when

, a leaky ReLU will instead have a small negative slope (of 0.01, or so). That is, the function computes

  if  

and 

 if

 , where a is a small constant. Some people report success with this form of activation function, but the results are not always consistent.

5

Parametric ReLU

 

    Nowadays, a broader class of activation functions, namely the rectified unit family, were proposed. In the following, we will talk about the variants of ReLU.

    ReLU, Leaky ReLU, PReLU and RReLU. In these figures, for PReLU, ai is learned and for Leaky ReLU ai is fixed. For RReLU, aji is a random variable keeps sampling in a given range, and remains fixed in testing.

    The first variant is called parametric rectified linear unit (PReLU) [4]. In PReLU, the slopes of negative part are learned from data rather than pre-defined. He et al. [4] claimed that PReLU is the key factor of surpassing human-level performance on ImageNet() classification task. The back-propagation and updating process of PReLU is very straightforward and similar to traditional ReLU, which is shown in Page. 43 of the slides.、

6

Randomized ReLU

 

    The second variant is called randomized rectified linear unit (RReLU). In RReLU, the slopes of negative parts are randomized in a given range in the training, and then fixed in the testing. As mentioned in [5], in a recent Kaggle National Data Science Bowl (NDSB) competition(), it is reported that RReLU could reduce overfitting due to its randomized nature. Moreover, suggested by the NDSB competition winner, the random ai in training is sampled from 

 and in test time it is fixed as its expectation, i.e.,

    In [5], the authors evaluated classification performance of two state-of-the-art CNN architectures with different activation functions on theCIFAR-10, CIFAR-100 and NDSB data sets, which are shown in the following tables. Please note that, for these two networks, activation function is followed by each convolutional layer. And the 1/a  in these tables actually indicates , where a is the aforementioned slopes.

    From these tables, we can find the performance of ReLU is not the best for all the three data sets. For Leaky ReLU, a larger slope  will achieve better accuracy rates. PReLU is easy to overfit on small data sets (its training error is the smallest, while testing error is not satisfactory), but still outperforms ReLU. In addition, RReLU is significantly better than other activation functions on NDSB, which shows RReLU can overcome overfitting, because this data set has less training data than that of CIFAR-10/CIFAR-100. In conclusion, three types of ReLU variants all consistently outperform the original ReLU in these three data sets. And PReLU and RReLU seem better choices. Moreover, He et al. also reported similar conclusions in [4].

Sec. 6: Regularizations

    There are several ways of controlling the capacity of Neural Networks to prevent overfitting:
    L2 regularization is perhaps the most common form of regularization. It can be implemented by penalizing the squared magnitude of all parameters directly in the objective. That is, for every weight w in the network, we add the term 

 to the objective, where

  is the regularization strength. It is common to see the factor of 1/2 in front because then the gradient of this term with respect to the parameter w is simply 

 instead of 

. The L2 regularization has the intuitive interpretation of heavily penalizing peaky weight vectors and preferring diffuse weight vectors.
    L1 regularization is another relatively common form of regularization, where for each weight w we add the term 

 to the objective. It is possible to combine the L1 regularization with the L2 regularization: 

 (this is called Elastic net regularization). The L1 regularization has the intriguing property that it leads the weight vectors to become sparse during optimization (i.e. very close to exactly zero). In other words, neurons with L1 regularization end up using only a sparse subset of their most important inputs and become nearly invariant to the “noisy” inputs. In comparison, final weight vectors from L2 regularization are usually diffuse, small numbers. In practice, if you are not concerned with explicit feature selection, L2 regularization can be expected to give superior performance over L1.
    Max norm constraints. Another form of regularization is to enforce an absolute upper bound on the magnitude of the weight vector

for every neuron and use projected gradient descent to enforce the constraint. In practice, this corresponds to performing the parameter update as normal, and then enforcing the constraint by clamping the weight vector 

 of every neuron to satisfy

 . Typical values of c are on orders of 3 or 4. Some people report improvements when using this form of regularization. One of its appealing properties is that network cannot “explode” even when the learning rates are set too high because the updates are always bounded.
    Dropout is an extremely effective, simple and recently introduced regularization technique by Srivastava et al. in [6] that complements the other methods (L1, L2, maxnorm). During training, dropout can be interpreted as sampling a Neural Network within the full Neural Network, and only updating the parameters of the sampled network based on the input data. (However, the exponential number of possible sampled networks are not independent because they share the parameters.) During testing there is no dropout applied, with the interpretation of evaluating an averaged prediction across the exponentially-sized ensemble of all sub-networks (more about ensembles in the next section). In practice, the value of dropout ratio 

 is a reasonable default, but this can be tuned on validation data.

    The most popular used regularization techniquedropout [6]. While training, dropout is implemented by only keeping a neuron active with some probability p (a hyper-parameter), or setting it to zero otherwise. In addition, Google applied for a US patent() for dropout in 2014.

Sec. 7: Insights from Figures

    Finally, from the tips above, you can get the satisfactory settings (e.g., data processing, architectures choices and details, etc.) for your own deep networks. During training time, you can draw some figures to indicate your networks’ training effectiveness.
    1. As we have known, the learning rate is very sensitive. From Fig. 1 in the following, a very high learning rate will cause a quite strange loss curve. A low learning rate will make your training loss decrease very slowly even after a large number of epochs. In contrast, a high learning rate will make training loss decrease fast at the beginning, but it will also drop into a local minimum. Thus, your networks might not achieve a satisfactory results in that case. For a good learning rate, as the red line shown in Fig. 1, its loss curve performs smoothly and finally it achieves the best performance.
    2.  Now let’s zoom in the loss curve. The epochs present the number of times for training once on the training data, so there are multiple mini batches in each epoch. If we draw the classification loss every training batch, the curve performs like Fig. 2. Similar to Fig. 1, if the trend of the loss curve looks too linear, that indicates your learning rate is low; if it does not decrease much, it tells you that the learning rate might be too high. Moreover, the “width” of the curve is related to the batch size. If the “width” looks too wide, that is to say the variance between every batch is too large, which points out you should increase the batch size.
    3. Another tip comes from the accuracy curve. As shown in Fig. 3, the red line is the training accuracy, and the green line is the validation one. When the validation accuracy converges, the gap between the red line and the green one will show the effectiveness of your deep networks. If the gap is big, it indicates your network could get good accuracy on the training data, while it only achieve a low accuracy on the validation set. It is obvious that your deep model overfits on the training set. Thus, you should increase the regularization strength of deep networks. However, no gap meanwhile at a low accuracy level is not a good thing, which shows your deep model has low learnability. In that case, it is better to increase the model capacity for better results.

Sec. 8: Ensemble

    In machine learning, ensemble methods [8] that train multiple learners and then combine them for use are a kind of state-of-the-art learning approach. It is well known that an ensemble is usually significantly more accurate than a single learner, and ensemble methods have already achieved great success in many real-world tasks. In practical applications, especially challenges or competitions, almost all the first-place and second-place winners used ensemble methods.
    Here we introduce several skills for ensemble in the deep learning scenario.    Same model, different initialization. Use cross-validation to determine the best hyperparameters, then train multiple models with the best set of hyperparameters but with different random initialization. The danger with this approach is that the variety is only due to initialization.    Top models discovered during cross-validation. Use cross-validation to determine the best hyperparameters, then pick the top few (e.g., 10) models to form the ensemble. This improves the variety of the ensemble but has the danger of including suboptimal models. In practice, this can be easier to perform since it does not require additional retraining of models after cross-validation. Actually, you could directly select several state-of-the-art deep models from Caffe Model Zoo () to perform ensemble.    Different checkpoints of a single model. If training is very expensive, some people have had limited success in taking different checkpoints of a single network over time (for example after every epoch) and using those to form an ensemble. Clearly, this suffers from some lack of variety, but can still work reasonably well in practice. The advantage of this approach is that is very cheap.    Some practical examples. If your vision tasks are related to high-level image semantic, e.g., event recognition from still images, a better ensemble method is to employ multiple deep models trained on different data sources to extract different and complementary deep representations. For example in the Cultural Event Recognition () challenge in associated with ICCV’15 () , we utilized five different deep models trained on images of ImageNet (), Place Database () and the cultural images supplied by the competition organizers (). After that, we extracted five complementary deep features and treat them as multi-view data. Combining “early fusion” and “late fusion” strategies described in [7], we achieved one of the best performance and ranked the 2nd place in that challenge. Similar to our work, [9] presented the Stacked NN framework to fuse more deep networks at the same time.

References & Source Links

[1] A. Krizhevsky, I. Sutskever, and G. E. Hinton. ImageNet Classification with Deep Convolutional Neural Networks.() In NIPS, 2012
[2] A Brief Overview of Deep Learning () , which is a guest post by Ilya Sutskever.
[3] CS231n: Convolutional Neural Networks for Visual Recognition of Stanford University, held by Prof. Fei-Fei Li and Andrej Karpathy.
[4] K. He, X. Zhang, S. Ren, and J. Sun. Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification.（）InICCV, 2015.
[5] B. Xu, N. Wang, T. Chen, and M. Li. Empirical Evaluation of Rectified Activations in Convolution Network（）. In ICML Deep Learning Workshop, 2015.
[6] N. Srivastava, G. E. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov. Dropout: A Simple Way to Prevent Neural Networks from Overfitting. （）JMLR, 15(Jun):1929−1958, 2014.
[7] X.-S. Wei, B.-B. Gao, and J. Wu. Deep Spatial Pyramid Ensemble for Cultural Event Recognition. （http://lamda.nju.edu.cn/weixs/publication/iccvw15_CER.pdf）In ICCV ChaLearn Looking at People Workshop, 2015.
[8] Z.-H. Zhou. Ensemble Methods: Foundations and Algorithms（）. Boca Raton, FL: Chapman & HallCRC/, 2012. (ISBN 978-1-439-830031)
[9] M. Mohammadi, and S. Das. S-NN: Stacked Neural Networks. Project in Stanford CS231n Winter Quarter, 2015.（）
[10] P. Hensman, and D. Masko. The Impact of Imbalanced Training Data for Convolutional Neural Networks.（） Degree Project in Computer Science, DD143X, 2015.

作者简介魏秀参
南京大学计算机系机器学习与数据挖掘所（LAMDA）博士生，研究方向为计算机视觉和机器学习，特别是深度学习和弱监督学习。曾在国际顶级期刊和会议发表学术论文。个人主页：，微博ID：Wilson_NJUer

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