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Title: Enhancing Batch Normalized Convolutional Networks using Displaced Rectifier Linear Units: A Systematic Comparative Study

Paper content:
In this paper, we turn our attention to the interworking between the activation functions and the batch normalization, which is a virtually mandatory technique to train deep networks currently.
We propose the activation function Displaced Rectifier Linear Unit (DReLU) by conjecturing that extending the identity function of ReLU to the third quadrant enhances compatibility with batch normalization.
Moreover, we used statistical tests to compare the impact of using distinct activation functions (ReLU, LReLU, PReLU, ELU, and DReLU) on the learning speed and test accuracy performance of standardized VGG and Residual Networks state-of-the-art models.
These convolutional neural networks were trained on CIFAR-100 and CIFAR-10, the most commonly used deep learning computer vision datasets.
The results showed DReLU speeded up learning in all models and datasets.
Besides, statistical significant performance assessments (p<0.05) showed DReLU enhanced the test accuracy presented by ReLU in all scenarios.
Furthermore, DReLU showed better test accuracy than any other tested activation function in all experiments with one exception, in which case it presented the second best performance.
Therefore, this work demonstrates that it is possible to increase performance replacing ReLU by an enhanced activation function.
The recent advances in deep learning research have produced more accurate image, speech, and language recognition systems and generated new state-of-the-art machine learning applications in a broad range of areas such as mathematics, physics, healthcare, genomics, financing, business, agriculture, etc.
Although advances have been made, accuracy performance enhancements have usually demanded considerably deeper or more complex models, which tend to increase the required computational resources (processing time and memory usage).Instead
of increasing deep models depth or complexity, a less computational expensive alternative approach to enhance deep learning performance across-the-board is to design more efficient activation functions. Even if
computational resources are no issue, to employ enhanced activation functions nevertheless contributes to speeding up learning and achieving higher accuracy.Indeed, by allowing the training of deep neural networks, the discovery of Rectified Linear Units (ReLU) BID19 BID4 BID13 was one of the main factors that contributed to deep learning advent. ReLU allowed
achieving higher accuracy in less time by avoiding the vanishing gradient problem BID9 . Before ReLU,
activation functions such as Sigmoid and Hyperbolic Tangent were unable to train deep neural networks because of the absence of the identity function for positive input.However, ReLU presents drawbacks. For example,
some researchers argument that zero slope avoids learning for negative values BID18 BID6 . Therefore, other
activation functions like Leaky Rectifier Linear Unit (LReLU) BID18 , Parametric Rectifier Linear Unit (PReLU) BID6 and Exponential Linear Unit (ELU) were proposed (Appendix A). Unfortunately, there
is no consensus about how these proposed nonlinearities compare to ReLU, which therefore remains the most used activation function in deep learning.Similar to activation functions, batch normalization BID11 currently plays a fundamental role in training deep architectures (Appendix B) . This technique normalizes
the inputs of each layer, which is equivalent to normalizing the outputs of the deep model previous layer. However, before being used
as inputs for the subsequent layer, the normalized data are typically fed into activation functions (nonlinearities), which necessarily skew the otherwise normalized distributions. In fact, ReLU only produces
non-negative activations, which is harmful to the previously normalized data. The outputs mean values after
ReLU are no longer zero, but rather necessarily positives. Therefore, the ReLU skews the
normalized distribution (Section 2).Aiming to mitigate the mentioned
problem, we concentrate our attention on the interaction between activation functions and batch normalization. We conjecture that nonlinearities
that are more compatible with batch normalization present higher performance. After that, considering that an identity
transformation preserves any statistical distribution, we assume that to extend the identity function from the first quadrant to the third implies less damage to the normalization procedure.Hence, we investigate and propose the activation function Displaced Rectifier Linear Unit (DReLU), which partially prolongs the identity function beyond origin. Hence, DReLU is essentially a ReLU diagonally
displaced into the third quadrant. Different from all other previous mentioned activation
functions, the inflection of DReLU does not happen at the origin, but in the third quadrant.Considering the widespread adoption and practical importance, we used Convolutional Neural Networks (CNN) BID16 BID13 in our experiments. Moreover, as particular examples of CNN architectures,
we used the previous ImageNet Large Scale Visual Recognition Competition (ILSVRC) winners Visual Geometry Group (VGG) BID22 and Residual Networks (ResNets) BID5 c) . These architectures have distinctive designs and depth
to promote generality to the conclusions of this work. In this regard, we evaluated how replacing the activation
function impacts the performance of well established and widely used standard state-of-the-art models. Finally, we decided to employ the two most broadly used computer
vision datasets by deep learning research community: CIFAR-100 BID14 CIFAR-10 (Krizhevsky, 2009) .In this systematic comparative study, performance assessments were
carried out using statistical tests with a significance level of 5% (Appendix C.5). At least ten executions of each of experiment were executed. However
, when the mentioned significance level was not achieved, ten
additional runs were performed.
In the following subsections, we analyze the tested scenarios.
In each case, we first discuss the activation functions learning speed based on test accuracy obtained for the partially trained models.
Subsequently, we comment about the test accuracy performances of the activation functions, which corresponds to the respective model test accuracy evaluated after 100 epochs.
Naturally, we consider that an activation function presents better test accuracy if it showed the higher test accuracy for the final trained models on a particular dataset.In all scenarios, the null hypotheses were the test accuracy samples taken from different activation functions originated from the same distribution.
In other works, all the compared activation functions have the same test accuracy performance in the particular scenario.
The null hypotheses were rejected for all scenarios TAB0 , which means that with statistical significance (p < 0.05) at least one of the activation functions presents a test accuracy performance that is different from the others activation functions.
Therefore, we used the Conover-Iman post-hoc tests for pairwise multiple comparisons for all combination of datasets and models (Tables 3, 4, 5, 7, 8, 9) .
In these tables, the best results and p-values of the comparison of DReLU to other activation functions are in bold.
In this paper, we have proposed a novel activation function for deep learning architectures, referred to as DReLU.
The results showed that DReLU presented better learning speed than the all alternative activation functions, including ReLU, in all models and datasets.
Moreover, the experiments showed DReLU was more accurate than ReLU in all situations.
Besides, DReLU also outperformed test accuracy results of all others investigated activation functions (LReLU, PReLU, and ELU) in all scenarios with one exception.
The experiments used batch normalization but avoided dropout.
Furthermore, they were designed to cover standard and commonly used datasets (CIFAR-100 and CIFAR-10) and models (VGG and Residual Networks) of several depths and architectures.In addition to enhancing deep learning performance (learning speed and test accuracy), DReLU is less computationally expensive than LReLU, PReLU, and ELU.
Moreover, the mentioned gains were obtained by just replacing the activation function of the model, without any increment in depth or architecture complexity, which usually increases the computational resource requirements as processing time and memory usage.This paper showed that the batch normalization procedure acted in the benefice of ReLU while other previews proposed activation functions appear not to perform as expected.
We believe this happened because batch normalization avoids the so-called "dying ReLU" problem, something that others activation functions were already not affected by in first place.Furthermore, considering some evaluated models included skip connections, which are a tendency in the design of deep architectures like ResNets, we conjecture the results may generalize to other deep architectures such DenseNets BID10 ) that also use this structure.

assistant: text: A new activation function called Displaced Rectifier Linear Unit is proposed. It is showed to enhance the training and inference performance of batch normalized convolutional neural networks.