Content deleted Content added
m →Critic |
→Critic: advantage |
||
Line 52:
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.
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>
== Variants ==
| |||