Must Know Tips/Tricks in Deep Neural Networks (by Xiu-Shen Wei)

tricks 

Deep Neural Networks, especially Convolutional Neural Networks (CNN), allows computational models that are composed of multiple processing layers to learn representations of data with multiple levels of abstraction. These methods have dramatically improved the state-of-the-arts in visual object recognition, object detection, text recognition and many other domains such as drug discovery and genomics.

In addition, many solid papers have been published in this topic, and some high quality open source CNN software packages have been made available. There are also well-written CNN tutorials or CNN software manuals. However, it might lack a recent and comprehensive summary about the details of how to implement an excellent deep convolutional neural networks from scratch. Thus, we collected and concluded many implementation details for DCNNs. Here we will introduce these extensive implementation details, i.e., tricks or tips, for building and training your own deep networks.

Introduction

We assume you already know the basic knowledge of deep learning, and here we will present the implementation details (tricks or tips) in Deep Neural Networks, especially CNN for image-related tasks, mainly in eight aspects: 1) data augmentation; 2) pre-processing on images; 3) initializations of Networks; 4) some tips during training; 5) selections of activation functions; 6) diverse regularizations; 7) some insights found from figures and finally 8) methods of ensemble multiple deep networks.

Additionally, the corresponding slides are available at [slide]. If there are any problems/mistakes in these materials and slides, or there are something important/interesting you consider that should be added, just feel free to contact me.

Sec. 1: Data Augmentation

Since deep networks need to be trained on a huge number of training images to achieve satisfactory performance, if the original image data set contains limited training images, it is better to do data augmentation to boost the performance. Also, data augmentation becomes the thing must to do when training a deep network.

Sec. 2: Pre-Processing

Now we have obtained a large number of training samples (images/crops), but please do not hurry! Actually, it is necessary to do pre-processing on these images/crops. In this section, we will introduce several approaches for pre-processing.

The first and simple pre-processing approach is zero-center the data, and then normalize them, which is presented as two lines Python codes as follows:

>>> X -= np.mean(X, axis = 0) # zero-center
>>> X /= np.std(X, axis = 0) # normalize

where, X is the input data (NumIns×NumDim). Another form of this pre-processing normalizes each dimension so that the min and max along the dimension is -1 and 1 respectively. It only makes sense to apply this pre-processing if you have a reason to believe that different input features have different scales (or units), but they should be of approximately equal importance to the learning algorithm. In case of images, the relative scales of pixels are already approximately equal (and in range from 0 to 255), so it is not strictly necessary to perform this additional pre-processing step.

Another pre-processing approach similar to the first one is PCA Whitening. In this process, the data is first centered as described above. Then, you can compute the covariance matrix that tells us about the correlation structure in the data:

>>> X -= np.mean(X, axis = 0) # zero-center
>>> cov = np.dot(X.T, X) / X.shape[0] # compute the covariance matrix

After that, you decorrelate the data by projecting the original (but zero-centered) data into the eigenbasis:

>>> U,S,V = np.linalg.svd(cov) # compute the SVD factorization of the data covariance matrix
>>> Xrot = np.dot(X, U) # decorrelate the data

The last transformation is whitening, which takes the data in the eigenbasis and divides every dimension by the eigenvalue to normalize the scale:

>>> Xwhite = Xrot / np.sqrt(S + 1e-5) # divide by the eigenvalues (which are square roots of the singular values)

Note that here it adds 1e-5 (or a small constant) to prevent division by zero. One weakness of this transformation is that it can greatly exaggerate the noise in the data, since it stretches all dimensions (including the irrelevant dimensions of tiny variance that are mostly noise) to be of equal size in the input. This can in practice be mitigated by stronger smoothing (i.e., increasing 1e-5 to be a larger number).

Please note that, we describe these pre-processing here just for completeness. In practice, these transformations are not used with Convolutional Neural Networks. However, it is also very important to zero-center the data, and it is common to see normalization of every pixel as well.

Sec. 3: Initializations

Now the data is ready. However, before you are beginning to train the network, you have to initialize its parameters.

All Zero Initialization

In the ideal situation, with proper data normalization it is reasonable to assume that approximately half of the weights will be positive and half of them will be negative. A reasonable-sounding idea then might be to set all the initial weights to zero, which you expect to be the “best guess” in expectation. But, this turns out to be a mistake, because if every neuron in the network computes the same output, then they will also all compute the same gradients during back-propagation and undergo the exact same parameter updates. In other words, there is no source of asymmetry between neurons if their weights are initialized to be the same.

Initialization with Small Random Numbers

Thus, you still want the weights to be very close to zero, but not identically zero. In this way, you can random these neurons to small numbers which are very close to zero, and it is treated as symmetry breaking. The idea is that the neurons are all random and unique in the beginning, so they will compute distinct updates and integrate themselves as diverse parts of the full network. The implementation for weights might simply look like weightssim 0.001times N(0,1), where N(0,1) is a zero mean, unit standard deviation gaussian. It is also possible to use small numbers drawn from a uniform distribution, but this seems to have relatively little impact on the final performance in practice.

Calibrating the Variances

One problem with the above suggestion is that the distribution of the outputs from a randomly initialized neuron has a variance that grows with the number of inputs. It turns out that you can normalize the variance of each neuron's output to 1 by scaling its weight vector by the square root of its fan-in (i.e., its number of inputs), which is as follows:

>>> w = np.random.randn(n) / sqrt(n) # calibrating the variances with 1/sqrt(n)

where “randn” is the aforementioned Gaussian and “n” is the number of its inputs. This ensures that all neurons in the network initially have approximately the same output distribution and empirically improves the rate of convergence. The detailed derivations can be found from Page. 18 to 23 of the slides. Please note that, in the derivations, it does not consider the influence of ReLU neurons.

Current Recommendation

As aforementioned, the previous initialization by calibrating the variances of neurons is without considering ReLUs. A more recent paper on this topic by He et al. [4] derives an initialization specifically for ReLUs, reaching the conclusion that the variance of neurons in the network should be 2.0/n as:

>>> w = np.random.randn(n) * sqrt(2.0/n) # current recommendation

which is the current recommendation for use in practice, as discussed in [4].

Sec. 4: During Training

Now, everything is ready. Let’s start to train deep networks!

table 

Fine-tune your data on pre-trained models. Different strategies of fine-tuning are utilized in different situations. For data sets, Caltech-101 is similar to ImageNet, where both two are object-centric image data sets; while Place Database is different from ImageNet, where one is scene-centric and the other is object-centric.

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.

neuron 

Figures courtesy of Stanford CS231n.

Sigmoid

sigmod 

The sigmoid non-linearity has the mathematical form sigma(x)=1/(1+e^{-x}). 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 f=w^Tx+b), 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.

tanh(x)

tanh 

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.

Rectified Linear Unit

relu 

The Rectified Linear Unit (ReLU) has become very popular in the last few years. It computes the function f(x)=max(0,x), 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.

Leaky ReLU

lrelu 

Leaky ReLUs are one attempt to fix the “dying ReLU” problem. Instead of the function being zero when x<0, a leaky ReLU will instead have a small negative slope (of 0.01, or so). That is, the function computes f(x)=alpha x if x<0 and f(x)=x if xgeq 0, where alpha is a small constant. Some people report success with this form of activation function, but the results are not always consistent.

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.

relufamily 

ReLU, Leaky ReLU, PReLU and RReLU. In these figures, for PReLU, alpha_i is learned and for Leaky ReLU alpha_i is fixed. For RReLU, alpha_{ji} 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.

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 alpha_i in training is sampled from 1/U(3,8) and in test time it is fixed as its expectation, i.e., 2/(l+u)=2/11.

In [5], the authors evaluated classification performance of two state-of-the-art CNN architectures with different activation functions on the CIFAR-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 a in these tables actually indicates 1/alpha, where alpha is the aforementioned slopes.

relures 

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 alpha 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:

dropout 

The most popular used regularization technique dropout [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.

trainfigs 

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.

Miscellaneous

In real world applications, the data is usually class-imbalanced: some classes have a large number of images/training instances, while some have very limited number of images. As discussed in a recent technique report [10], when deep CNNs are trained on these imbalanced training sets, the results show that imbalanced training data can potentially have a severely negative impact on overall performance in deep networks. For this issue, the simplest method is to balance the training data by directly up-sampling and down-sampling the imbalanced data, which is shown in [10]. Another interesting solution is one kind of special crops processing in our challenge solution [7]. Because the original cultural event images are imbalanced, we merely extract crops from the classes which have a small number of training images, which on one hand can supply diverse data sources, and on the other hand can solve the class-imbalanced problem. In addition, you can adjust the fine-tuning strategy for overcoming class-imbalance. For example, you can divide your own data set into two parts: one contains the classes which have a large number of training samples (images/crops); the other contains the classes of limited number of samples. In each part, the class-imbalanced problem will be not very serious. At the beginning of fine-tuning on your data set, you firstly fine-tune on the classes which have a large number of training samples (images/crops), and secondly, continue to fine-tune but on the classes with limited number samples.

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. In ICCV, 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. 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.