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As discussed in this question, the policy gradient algorithms given in Reinforcement Learning: An Introduction use the gradient \begin{align*} \gamma^t \hat A_t \nabla_{\theta} \log \pi(a_t \, | \, s_t, \theta) \end{align*} where $\hat A_t$ is the advantage estimate for step $t$. For example, $\hat A_t = r_t + \gamma V(s_{t+1}) - V(s_t)$ in the one-step actor-critic algorithm given in section 13.5.

In the answers to the linked question, it is claimed that the extra discounting is "correct", which implies that it should be included.

If I look in the literature to a seminal paper such as Proximal Policy Optimization Algorithms by OpenAI, they do not include the extra discounting factor, i.e. they use a gradient defined as \begin{align*} \hat A_t \dfrac{\nabla_{\theta}\pi(a_t \, | \, s_t, \theta)}{\pi(a_t \, | \,s_t, \theta_{\rm old})} \end{align*} which does not include the discounting factor (of course, it's dealing with the off-policy case, but I don't see how that would make a difference in terms of the discounting). OpenAI's implementation of PPO also does not include the extra discounting factor.

So, how am I supposed to interpret this discrepancy? I agree that the extra discounting factor should be present, from a theoretical standpoint. Then, why is it not in the OpenAI code or paper?

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If you want to maximize the expected reward \begin{align*} \mathbb{E}\bigg[\sum_{t=1}^nr_t \bigg] \end{align*} and are using a score function based gradient estimator (as opposed to a SAC/DDPG style update), you have the unbiased gradient estimator \begin{align*} \sum_{i=1}^n \sum_{k=i}^n r_k\nabla_{\theta}\log\pi(a_t) \tag{1} \end{align*} Then, you can add discounting as a variance reduction technique; the gradient estimator \begin{align*} \sum_{i=1}^n \sum_{k=i}^n \gamma^{k-i}r_k\nabla_{\theta}\log\pi(a_t) \tag{2} \end{align*} will have a lower variance than Eq. (1) (see this answer).

If you want to maximize the discounted expected reward \begin{align*} \mathbb{E}\bigg[\sum_{t=1}^n \gamma^{t-1}r_t \bigg] \end{align*} you get the unbiased gradient estimator \begin{align*} \sum_{i=1}^n \sum_{k=i}^n \gamma^{k-1}r_k\nabla_{\theta}\log\pi(a_t) \tag{3} \end{align*} So in Sutton and Barto they are essentially presenting the formulation Eq. (3). The difference between (2) and (3) is the factor of $\gamma^{i-1}$ which is what I was confused about in the question.

Thus in summary, the formulation (2) is an biased estimator of the expected reward; whereas (3) is an unbiased estimator of the expected discounted reward.

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I believe you will find the answer in the paper High-Dimensional Continuous Control Using Generalized Advantage Estimation, which is the basis for the advantage function used in the PPO paper that you referenced.

From the paper, the estimate of the advantage function is defined as: \begin{align*} \hat{A}_{t}^{GAE(\gamma,\lambda)} = \sum_{l=0}^{\infty}(\gamma\lambda)^{l}\delta_{t+1}^{V} \end{align*} where $\delta_{t}^{V}$, the TD residual of $V$, is defined as: \begin{align*} \delta_{t}^{V} = r_{t}+\gamma V(s_{t+1})-V(s_{t}) \end{align*} where $V$ is an approximate of the value function.

If you look closely at these two equations you will see that the discount $\gamma$ is applied twice.

I never went through the code of the whole OpenAI implementation of PPO, but if I am not mistaken the implementation of the above equations can be found here in ppo2/runner.py.

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  • $\begingroup$ This does not answer the question. I am familiar with GAE and the equations you give are consistent with the PPO paper. The question is whether or not it is supposed to be $\gamma^{t-1}\hat A_t$ (which is what Sutton and Barto give and what I would expect accordingly to theory); or whether it should be just $\hat A_t$, which is what is present in the PPO paper, PPO code, and GAE paper. $\endgroup$
    – Taw
    Jan 25 at 15:32
  • $\begingroup$ Apologies for not being clear in my answer. From my understanding of the GAE and PPO implementation, the discount factor is there twice like in S&B, it is just not applied in the same order. $\endgroup$
    – Lars
    Jan 25 at 17:32
  • $\begingroup$ You have given the definition of $\hat A_t$. I agree with the definition. The question is whether the gradient of the (log probability) should use the coefficient $\hat A_t$ or $\gamma^{t-1} \hat A_t$. $\endgroup$
    – Taw
    Jan 26 at 7:47
  • $\begingroup$ No, I don't believe that $\gamma$ should be applied to the gradient in $\hat{A}_{t}\frac{\nabla_{\theta}\pi(a_t \, | \, s_t, \theta)}{\pi(a_t \, | \,s_t, \theta_{\rm old})}$ for the reason that it is applied in the calculation of the estimate of the advantage and in the calculation of $\delta^{V}_{t}$ . $\endgroup$
    – Lars
    Jan 26 at 10:59

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