How do I derive the gradient with respect to the parameters of the softmax policy?

Question

The gradient of the softmax eligibility trace is given by the following:

\begin{align} \nabla_{\theta} \log(\pi_{\theta}(a|s)) &= \phi(s,a) - \mathbb E[\phi (s, \cdot)]\\ &= \phi(s,a) - \sum_{a'} \pi(a'|s) \phi(s,a') \end{align}

How is this equation derived?

The following relation is true:

\begin{align} \nabla_{\theta} \log(\pi_{\theta}(a|s)) &= \frac{\nabla_{\theta} \pi_{\theta}(a|s)}{\pi_{\theta}(a|s)} \tag{1}\label{1} \end{align}

Thus, the following relation must also be true: \begin{align} \frac{\nabla_{\theta} \pi_{\theta}(a|s)}{\pi_{\theta}(a|s)} &=\phi(s,a) - \sum_{a'} \pi(a'|s) \phi(s,a') \end{align}

Mathematically, why would this be the case? Probably, you just need to answer my question above because \ref{1} is true and it's just the rule to differentiate a logarithm.

Answer 1

Softmax policy $\pi_\theta(s,a)$ is defined as $\frac{\exp{(\phi(s,a)^T \theta})}{\Sigma \exp{(\phi(s,a) ^T \theta) }}$, where the summation is over the action space.
Taking log, this becomes $$ \log \pi_\theta(s,a) = log(e^{\phi(s,a) ^T \theta}) - log({\Sigma e^{\phi(s,a) ^T \theta }}) \\ = \phi(s,a) ^T \theta - log({\Sigma e^{\phi(s,a)^T \theta }}) $$

Taking derivative wrt $\theta$, this becomes $$ \nabla_\theta \log \pi_\theta(s,a) = \phi(s,a) - \nabla_\theta log({\Sigma e^{\phi(s,a) ^T \theta }}) $$

We can rewrite $\nabla_\theta log({\Sigma e^{\phi(s,a)^T \theta }})$ as follows. $$ \nabla_\theta log({\Sigma e^{\phi(s,a)^T \theta }}) = \frac{\nabla_\theta \Sigma e^{\phi(s,a)^T \theta}}{\Sigma e^{\phi(s,a) ^T \theta}} = \frac{\Sigma \phi(s,a) e^{\phi(s,a) ^T \theta}}{\Sigma e^{\phi(s,a) ^T \theta}} = \Sigma \phi(s,a) \pi_\theta(s,a) $$

The final equation then becomes $$ \nabla_\theta \log \pi_\theta(s,a) = \phi(s,a) - \Sigma \phi(s,a) \pi_\theta(s,a) $$