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{{Short description|Reinforcement learning
The '''actor-critic algorithm''' (AC) is a family of [[reinforcement learning]] (RL) algorithms that combine policy-based RL algorithms such as [[
An AC algorithm consists of two main components: an "'''actor'''" that determines which actions to take according to a policy function, and a "'''critic'''" that evaluates those actions according to a value function.<ref>{{Cite journal |
== Overview ==
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The actor-critic methods can be understood as an improvement over pure policy gradient methods like REINFORCE via introducing a baseline.
=== Actor ===
The '''actor''' uses a policy function <math>\pi(a|s)</math>, while the
</math> the advantage function <math>A(s,a)</math>, or any combination thereof.
The actor is a parameterized function <math>\pi_\theta</math>, where <math>\theta</math> are the parameters of the actor. The actor takes as argument the state of the environment <math>s</math> and produces a [[probability distribution]] <math>\pi_\theta(\cdot | s)</math>.
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If the action space is discrete, then <math>\sum_{a} \pi_\theta(a | s) = 1</math>. If the action space is continuous, then <math>\int_{a} \pi_\theta(a | s) da = 1</math>.
The goal of policy optimization is to improve the actor. That is, to find some <math>\theta</math> that maximizes the expected episodic reward <math>J(\theta)</math>:<math display="block">
J(\theta) = \mathbb{E}_{\pi_\theta}\left[\sum_{t=0}^{T} \gamma^t r_t\right]
</math>where <math>
\gamma
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The goal of policy gradient method is to optimize <math>J(\theta)</math> by [[Gradient descent|gradient ascent]] on the policy gradient <math>\nabla J(\theta)</math>.
As detailed on the [[Policy gradient method#Actor-critic methods|policy gradient method]] page, there are many [[
\cdot \Psi_j
\Big|S_0 = s_0 \right]</math>where <math display="inline">\Psi_j</math> is a linear sum of the following:
* <math display="inline">\sum_{0 \leq i\leq T} (\gamma^i R_i)</math>
* <math display="inline">\gamma^j\sum_{j \leq i\leq T} (\gamma^{i-j} R_i)</math>:
* <math display="inline">\gamma^j \sum_{j \leq i\leq T} (\gamma^{i-j} R_i) - b(S_j) </math>:
* <math display="inline">\gamma^j \left(R_j + \gamma V^{\pi_\theta}( S_{j+1}) - V^{\pi_\theta}( S_{j})\right)</math>:
* <math display="inline">\gamma^j Q^{\pi_\theta}(S_j, A_j)</math>.
* <math display="inline">\gamma^j A^{\pi_\theta}(S_j, A_j)</math>: '''Advantage Actor-Critic (A2C)'''.<ref name=":0">{{Citation |
* <math display="inline">\gamma^j \left(R_j + \gamma R_{j+1} + \gamma^2 V^{\pi_\theta}( S_{j+2}) - V^{\pi_\theta}( S_{j})\right)</math>:
* <math display="inline">\gamma^j \left(\sum_{k=0}^{n-1} \gamma^k R_{j+k} + \gamma^n V^{\pi_\theta}( S_{j+n}) - V^{\pi_\theta}( S_{j})\right)</math>:
* <math display="inline">\gamma^j \sum_{n=1}^\infty \frac{\lambda^{n-1}}{1-\lambda}\cdot \left(\sum_{k=0}^{n-1} \gamma^k R_{j+k} + \gamma^n V^{\pi_\theta}( S_{j+n}) - V^{\pi_\theta}( S_{j})\right)</math>: TD(λ) learning, also known as '''GAE (generalized advantage estimate)'''.<ref name="arxiv.org">{{Citation |
=== Critic ===
▲* <math display="inline">\gamma^j \left(R_j + \gamma R_{j+1} + \gamma^2 V^{\pi_\theta}( S_{j+2}) - V^{\pi_\theta}( S_{j})\right)</math>: 2-step TD learning.
In the unbiased estimators given above, certain functions such as <math>V^{\pi_\theta}, Q^{\pi_\theta}, A^{\pi_\theta}</math> appear. These are approximated by the '''critic'''. Since these functions all depend on the actor, the critic must learn alongside the actor. The critic is learned by value-based RL algorithms.
▲* <math display="inline">\gamma^j \left(\sum_{k=0}^{n-1} \gamma^k R_{j+k} + \gamma^n V^{\pi_\theta}( S_{j+n}) - V^{\pi_\theta}( S_{j})\right)</math>: n-step TD learning.
▲* <math display="inline">\gamma^j \sum_{n=1}^\infty \frac{\lambda^{n-1}}{1-\lambda}\cdot \left(\sum_{k=0}^{n-1} \gamma^k R_{j+k} + \gamma^n V^{\pi_\theta}( S_{j+n}) - V^{\pi_\theta}( S_{j})\right)</math>: TD(λ) learning, also known as '''GAE (generalized advantage estimate)'''.<ref>{{Citation |last=Schulman |first=John |title=High-Dimensional Continuous Control Using Generalized Advantage Estimation |date=2018-10-20 |url=https://arxiv.org/abs/1506.02438 |doi=10.48550/arXiv.1506.02438 |last2=Moritz |first2=Philipp |last3=Levine |first3=Sergey |last4=Jordan |first4=Michael |last5=Abbeel |first5=Pieter}}</ref> This is obtained by an exponentially decaying sum of the n-step TD learning ones.
For example, if the critic is estimating the state-value function <math>V^{\pi_\theta}(s)</math>, then it can be learned by any value function approximation method. Let the critic be a function approximator <math>V_\phi(s)</math> with parameters <math>\phi</math>.
== Variants ==▼
The simplest example is TD(1) learning, which trains the critic to minimize the TD(1) error:<math display="block">\delta_i = R_i + \gamma V_\phi(S_{i+1}) - V_\phi(S_i)</math>The critic parameters are updated by gradient descent on the squared TD error:<math display="block">\phi \leftarrow \phi - \alpha \nabla_\phi (\delta_i)^2 = \phi + \alpha \delta_i \nabla_\phi V_\phi(S_i)</math>where <math>\alpha</math> is the learning rate. Note that the gradient is taken with respect to the <math>\phi</math> in <math>V_\phi(S_i)</math> only, since the <math>\phi</math> in <math>\gamma V_\phi(S_{i+1})</math> constitutes a moving target, and the gradient is not taken with respect to that. This is a common source of error in implementations that use [[automatic differentiation]], and requires "stopping the gradient" at that point.
* '''Asynchronous Advantage Actor-Critic (A3C)''': [[Parallel computing|Parallel and asynchronous]] version of A2C.<ref name=":0" />▼
* '''Soft Actor-Critic (SAC)''': Incorporates entropy maximization for improved exploration.<ref>{{Citation |last=Haarnoja |first=Tuomas |title=Soft Actor-Critic Algorithms and Applications |date=2019-01-29 |url=https://arxiv.org/abs/1812.05905 |doi=10.48550/arXiv.1812.05905 |last2=Zhou |first2=Aurick |last3=Hartikainen |first3=Kristian |last4=Tucker |first4=George |last5=Ha |first5=Sehoon |last6=Tan |first6=Jie |last7=Kumar |first7=Vikash |last8=Zhu |first8=Henry |last9=Gupta |first9=Abhishek}}</ref>▼
Similarly, if the critic is estimating the action-value function <math>Q^{\pi_\theta}</math>, then it can be learned by [[Q-learning]] or [[State–action–reward–state–action|SARSA]]. In SARSA, the critic maintains an estimate of the Q-function, parameterized by <math>\phi</math>, denoted as <math>Q_\phi(s, a)</math>. The temporal difference error is then calculated as <math>\delta_i = R_i + \gamma Q_\theta(S_{i+1}, A_{i+1}) - Q_\theta(S_i,A_i)</math>. The critic is then updated by<math display="block">\theta \leftarrow \theta + \alpha \delta_i \nabla_\theta Q_\theta(S_i, A_i)</math>The advantage critic can be trained by training both a Q-function <math>Q_\phi(s,a)</math> and a state-value function <math>V_\phi(s)</math>, then let <math>A_\phi(s,a) = Q_\phi(s,a) - V_\phi(s)</math>. Although, it is more common to train just a state-value function <math>V_\phi(s)</math>, then estimate the advantage by<ref name=":0" /><math display="block">A_\phi(S_i,A_i) \approx \sum_{j\in 0:n-1} \gamma^{j}R_{i+j} + \gamma^{n}V_\phi(S_{i+n}) - V_\phi(S_i)</math>Here, <math>n</math> is a positive integer. The higher <math>n</math> is, the more lower is the bias in the advantage estimation, but at the price of higher variance.
* '''Deep Deterministic Policy Gradient (DDPG)''': Specialized for continuous action spaces.<ref>{{Citation |last=Lillicrap |first=Timothy P. |title=Continuous control with deep reinforcement learning |date=2019-07-05 |url=https://arxiv.org/abs/1509.02971 |doi=10.48550/arXiv.1509.02971 |last2=Hunt |first2=Jonathan J. |last3=Pritzel |first3=Alexander |last4=Heess |first4=Nicolas |last5=Erez |first5=Tom |last6=Tassa |first6=Yuval |last7=Silver |first7=David |last8=Wierstra |first8=Daan}}</ref>▼
\lambda
</math> that smoothly interpolates between Monte Carlo returns (<math>
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</math>, low variance, high bias). This hyperparameter can be adjusted to pick the optimal bias-variance trade-off in advantage estimation. It uses an exponentially decaying average of n-step returns with <math>
\lambda
</math> being the decay strength.<ref name="arxiv.org"/>
▲== Variants ==
▲* '''Asynchronous Advantage Actor-Critic (A3C)''': [[Parallel computing|Parallel and asynchronous]] version of A2C.<ref name=":0" />
▲* '''Soft Actor-Critic (SAC)''': Incorporates entropy maximization for improved exploration.<ref>{{Citation |
▲* '''Deep Deterministic Policy Gradient (DDPG)''': Specialized for continuous action spaces.<ref>{{Citation |
== See also ==
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== References ==
{{Reflist|30em}}
* {{Cite journal |
* {{Cite book |
* {{Cite book |last=Bertsekas |first=Dimitri P. |title=Reinforcement learning and optimal control |date=2019 |publisher=Athena Scientific |isbn=978-1-886529-39-7 |edition=2 |___location=Belmont, Massachusetts}}
* {{Cite book |last=Grossi |first=Csaba |title=Algorithms for Reinforcement Learning |date=2010 |publisher=Springer International Publishing |isbn=978-3-031-00423-0 |edition=1 |series=Synthesis Lectures on Artificial Intelligence and Machine Learning |___location=Cham}}
* {{Cite journal |last1=Grondman |first1=Ivo |last2=Busoniu |first2=Lucian |last3=Lopes |first3=Gabriel A. D. |last4=Babuska |first4=Robert |date=November 2012 |title=A Survey of Actor-Critic Reinforcement Learning: Standard and Natural Policy Gradients |journal=IEEE Transactions on Systems, Man, and Cybernetics - Part C: Applications and Reviews |volume=42 |issue=6 |pages=1291–1307 |doi=10.1109/TSMCC.2012.2218595 |bibcode=2012ITHMS..42.1291G |issn=1094-6977 |url=https://hal.science/hal-00756747 }}
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