Home > Articles

This chapter is from the book

Different Activation Functions to Avoid Vanishing Gradient in Hidden Layers

The previous section showed how we can solve the problem of saturated neurons in the output layer by choosing a different loss function. However, this does not help for the hidden layers. The hidden neurons can still be saturated, resulting in derivatives close to 0 and vanishing gradients. At this point, you may wonder if we are solving the problem or just fighting symptoms. We have modified (standardized) the input data, used elaborate techniques to initialize the weights based on the number of inputs and outputs, and changed our loss function to accommodate the behavior of our activation function. Could it be that the activation function itself is the cause of the problem?

How did we end up with the tanh and logistic sigmoid functions as activation functions anyway? We started with early neuron models from McCulloch and Pitts (1943) and Rosenblatt (1958) that were both binary in nature. Then Rumelhart, Hinton, and Williams (1986) added the constraint that the activation function needs to be differentiable, and we switched to the tanh and logistic sigmoid functions. These functions kind of look like the sign function yet are still differentiable, but what good is a differentiable function in our algorithm if its derivative is 0 anyway?

Based on this discussion, it makes sense to explore alternative activation functions. One such attempt is shown in Figure 5-8, where we have complicated the activation function further by adding a linear term 0.2*x to the output to prevent the derivative from approaching 0.


FIGURE 5-8 Modified tanh function with an added linear term

Although this function might well do the trick, it turns out that there is no good reason to overcomplicate things, so we do not need to use this function. We remember from the charts in the previous section that a derivative of 0 was a problem only in one direction because, in the other direction, the output value already matched the ground truth anyway. In other words, it is fine with a derivative of 0 on one side of the chart. Based on this reasoning, we can consider the rectified linear unit (ReLU) activation function in Figure 5-9, which has been shown to work for neural networks (Glorot, Bordes, and Bengio, 2011).


FIGURE 5-9 Rectified linear unit (ReLU) activation function

Now, a fair question is how this function can possibly be used after our entire obsession with differentiable functions. The function in Figure 5-9 is not differentiable at x = 0. However, this does not present a big problem. It is true that from a mathematical point of view, the function is not differentiable in that one point, but nothing prevents us from just defining the derivative as 1 in that point and then trivially using it in our backpropagation algorithm implementation. The key issue to avoid is a function with a discontinuity, like the sign function. Can we simply remove the kink in the line altogether and use y = x as an activation function? The answer is that this does not work. If you do the calculations, you will discover that this will let you collapse the entire network into a linear function and, as we saw in Chapter 1, “The Rosenblatt Perceptron,” a linear function (like the perceptron) has severe limitations. It is even common to refer to the activation function as a nonlinearity, which stresses how important it is to not pick a linear function as an activation function.

An obvious benefit with the ReLU function is that it is cheap to compute. The implementation involves testing only whether the input value is less than 0, and if so, it is set to 0. A potential problem with the ReLU function is when a neuron starts off as being saturated in one direction due to a combination of how the weights and inputs happen to interact. Then that neuron will not participate in the network at all because its derivative is 0. In this situation, the neuron is said to be dead. One way to look at this is that using ReLUs gives the network the ability to remove certain connections altogether, and it thereby builds its own network topology, but it could also be that it accidentally kills neurons that could be useful if they had not happened to die. Figure 5-10 shows a variation of the ReLU function known as leaky ReLU, which is defined so its derivative is never 0.


FIGURE 5-10 Leaky rectified linear unit (ReLU) activation function

All in all, the number of activation functions we can think of is close to unlimited, and many of them work equally well. Figure 5-11 shows a number of important activation functions that we should add to our toolbox. We have already seen tanh, ReLU, and leaky ReLU (Xu, Wang, et al., 2015). We now add the softplus function (Dugas et al., 2001), the exponential linear unit also known as elu (Shah et al., 2016), and the maxout function (Goodfellow et al., 2013). The maxout function is a generalization of the ReLU function in which, instead of taking the max value of just two lines (a horizontal line and a line with positive slope), it takes the max value of an arbitrary number of lines. In our example, we use three lines, one with a negative slope, one that is horizontal, and one with a positive slope.

All of these activation functions except for tanh should be effective at fighting vanishing gradients when used as hidden units. There are also some alternatives to the logistic sigmoid function for the output units, but we save that for Chapter 6.


FIGURE 5-11 Important activation functions for hidden neurons. Top row: tanh, ReLU. Middle row: leaky ReLU, softplut. Bottom row: elu, maxout.

We saw previously how we can choose tanh as an activation function for the neurons in a layer in TensorFlow, also shown in Code Snippet 5-10.

Code Snippet 5-10 Setting the Activation Function for a Layer

keras.layers.Dense(25, activation='tanh',

If we want a different activation function, we simply replace 'tanh' with one of the other supported functions (e.g., 'sigmoid', 'relu', or 'elu'). We can also omit the activation argument altogether, which results in a layer without an activation function; that is, it will just output the weighted sum of the inputs. We will see an example of this in Chapter 6.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020