Home > Articles

This chapter is from the book

6.2 The Critic

Critics are responsible for learning how to evaluate (s, a) pairs and using this to generate Aπ.

In what follows, we first describe the advantage function and why it is a good choice for a reinforcing signal. Then, we present two methods for estimating the advantage function—n-step returns and Generalized Advantage Estimation [123]. Finally, we discuss how they can be learned in practice.

6.2.1 The Advantage Function

Intuitively, the advantage function Aπ(st, at) measures the extent to which an action is better or worse than the policy’s average action in a particular state. The advantage is defined in Equation 6.3.

It has a number of nice properties. First, 𝔼a𝒜[Aπ(st, a)] = 0. This implies that if all actions are essentially equivalent, then Aπ will be 0 for all actions and the probability of taking these actions will remain unchanged when the policy is trained using Aπ. Compare this to a reinforcing signal based on absolute state or state-action values. This signal would have a constant value in the same situation, but it may not be 0. Consequently, it would actively encourage (if positive) or discourage (if negative) the action taken. Since all actions were equivalent, this may not be problematic in practice, although it is unintuitive.

A more problematic example is if the action taken was worse than the average action, but the expected return is still positive. That is, Qπ(st, at) > 0, but Aπ(st, at) < 0. Ideally, the action taken should become less likely since there were better options available. In this case using Aπ yields behavior which matches our intuition more closely since it will discourage the action taken. Using Qπ, or even Qπ with a baseline, may encourage the action.

The advantage is also a relative measure. For a particular state s and action a, it considers the value of the state-action pair, Qπ(s, a), and evaluates whether a will take the policy to a better or worse place, measured relative to Vπ(s). The advantage avoids penalizing an action for the policy currently being in a particularly bad state. Conversely, it does not give credit to an action for the policy being in a good state. This is beneficial because a can only affect the future trajectory, but not how a policy arrived in the current state. We should evaluate the action based on how it changes the value in the future.

Let’s look at an example. In Equation 6.4, the policy is in a good state with Vπ(s) = 100, whereas in Equation 6.5, it is in a bad state with Vπ(s) = −100. In both cases, action a yields a relative improvement of 10, which is captured by each case having the same advantage. However, this would not be clear if we looked at just Qπ(s, a).

Understood this way, the advantage function is able to capture the long-term effects of an action, because it considers all future time steps,1 while ignoring the effects of all the actions to date. Schulman et al. present a similar interpretation in their paper “Generalized Advantage Estimation” [123].

Having seen why the advantage function Aπ(s, a) is a good choice of reinforcing signal to use in an Actor-Critic algorithm, let’s look at two ways of estimating it. Estimating Advantage: n-Step Returns

To calculate the advantage Aπ, we need an estimate for Qπ and Vπ. One idea is that we could learn Qπ and Vπ separately with different neural networks. However, this has two disadvantages. First, care needs to be taken to ensure the two estimates are consistent. Second, it is less efficient to learn. Instead, we typically learn just Vπ and combine it with rewards from a trajectory to estimate Qπ.

Learning Vπ is preferred to learning Qπ for two reasons. First, Qπ is a more complex function and may require more samples to learn a good estimate. This can be particularly problematic in the setting where the actor and the critic are trained jointly. Second, it can be more computationally expensive to estimate Vπ from Qπ. Estimating Vπ(s) from Qπ(s, a) requires computing the values for all possible actions in state s, then taking the action-probability weighted average to obtain Vπ(s). Additionally, this is difficult for environments with continuous actions since estimating Vπ would require a representative sample of actions from a continuous space.

Let’s look at how to estimate Qπ from Vπ.

If we assume for a moment that we have a perfect estimate of Vπ(s), then the Q-function can be rewritten as a mix of the expected rewards for n time steps, followed by Vπ(sn+1) as shown in Equation 6.6. To make this tractable to estimate, we use a single trajectory of rewards (r1, . . . , rn) in place of the expectation, and substitute in f0137-01.jpg learned by the critic. Shown in Equation 6.7, this is known as n-step forward returns.

Equation 6.7 makes the tradeoff between bias and variance of the estimator explicit. The n steps of actual rewards are unbiased but have high variance since they come from only a single trajectory. f0138-01.jpg has lower variance since it reflects an expectation over all of the trajectories seen so far, but is biased because it is calculated using a function approximator. The intuition behind mixing these two types of estimates is that the variance of the actual rewards typically increases the more steps away from t you take. Close to t, the benefits of using an unbiased estimate may outweigh the variance introduced. As n increases, the variance in the estimates will likely start to become problematic, and switching to a lower-variance but biased estimate is better. The number of steps of actual rewards, n, controls the tradeoff between the two.

Combining the n-step estimate for Qπ with f0142-01.jpg we get an formula for estimating the advantage function, shown in Equation 6.8.

The number of steps of actual rewards, n, controls the amount of variance in the advantage estimator, and is a hyperparameter that needs to be tuned. Small n results in an estimator with lower variance but higher bias, large n results in an estimator with higher variance but lower bias. Estimating Advantage: Generalized Advantage Estimation (GAE)

Generalized Advantage Estimation (GAE) [123] was proposed by Schulman et al. as an improvement over the n-step returns estimate for the advantage function. It addresses the problem of having to explicitly choose the number of steps of returns, n. The main idea behind GAE is that instead of picking one value of n, we mix multiple values of n. That is, we calculate the advantage using a weighted average of individual advantages calculated with n = 1, 2, 3, . . . , k. The purpose of GAE is to significantly reduce the variance of the estimator while keeping the bias introduced as low as possible.

GAE is defined as an exponentially weighted average of all of the n-step forward return advantages. It is shown in Equation 6.9 and the full derivation for GAE is given in Box 6.1.

Intuitively, GAE is taking a weighted average of a number of advantage estimators with different bias and variance. GAE weights the high-bias, low-variance 1-step advantage the most, but also includes contributions from lower-bias, higher-variance estimators using 2, 3, . . . , n steps. The contribution decays at an exponential rate as the number of steps increases. The decay rate is controlled by the coefficient λ. Therefore, the larger λ, the higher the variance.

Box 6.1 Generalized Advantage Estimation Derivation

Both GAE and the n-step advantage function estimates include the discount factor γ which controls how much an algorithm “cares” about future rewards compared to the current reward. Additionally, they both have a parameter that controls the bias-variance tradeoff: n for the advantage function and λ for GAE. So what have we gained with GAE?

Even though n and λ both control the bias-variance tradeoff, they do so in different ways. n represents a hard choice, since it precisely determines the point at which the high-variance rewards are switched for the V -function estimate. In contrast, λ represents a soft choice: smaller values of λ will more heavily weight the V -function estimate, whilst larger values will weight the actual rewards more. However, unless λ = 02 or λ = 1,3 using λ still allows higher or lower variance estimates to contribute—hence the soft choice.

6.2.2 Learning the Advantage Function

We have seen two ways to estimate the advantage function. Both these methods assume we have access to an estimate for Vπ, as shown below.

We learn Vπ using TD learning in the same way we used it to learn Qπ for DQN. In brief, learning proceeds as follows. Parametrize Vπ with θ, generate f0141-01.jpg for each of the experiences an agent gathers, and minimize the difference between f0141-03.jpg and f0141-01.jpg using a regression loss such as MSE. Repeat this process for many steps.

f0141-01.jpg can be generated using any appropriate estimate. The simplest method is to set f0141-02.jpg. This naturally generalizes to an n-step estimate, as shown in Equation 6.18.

Alternatively, we can use a Monte Carlo estimate for f0141-01.jpg shown in Equation 6.19.

Or, we can set

Practically, to avoid additional computation, the choice of f0141-01.jpg is often related to the method used to estimate the advantage. For example, we can use Equation 6.18 when estimating advantages using n-step returns, or Equation 6.20 when estimating advantages using GAE.

It is also possible to use a more advanced optimization procedure when learning f0141-04.jpg is learned using a trust-region method.

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