Answers.Body: <p>Note that the documentation states that the <code>weights</code> parameter is set to <span class="math-container">$R(\tau)$</span>.
  Therefore, the <span class="math-container">$\nabla_\theta \log \pi_\theta (a_t|s_t)R(\tau)$</span> terms are the same whether calculating the policy gradient over trajectories (the equation) or over state-action pairs (the pseudocode). Also, note that each state-action pair is associated with exactly one trajectory; therefore,<br />
  <span class="math-container">$$\hat{h} = \sum_{\tau \in D} \sum_{t=0}^T \nabla_\theta \log \pi_\theta (a_t|s_t)R(\tau)$$</span>
  calculates both the unnormalized policy gradient over the trajectories and also the unnormalized policy gradient over the state-action pairs. The policy gradient over the trajectories is given as <span class="math-container">$\hat{g} = \frac{1}{|D|}\hat{h}$</span>. Assuming there are <span class="math-container">$|K|$</span> state-action pairs, the policy gradient instead averaged over the state-action pairs is <span class="math-container">$\frac{1}{|K|}\hat{h} = \frac{|D|}{|K|}\left(\frac{1}{|D|}\hat{h}\right) = \frac{|D|}{|K|}\hat{g}$</span>. Thus, the policy gradient averaged over state-action pairs is proportional by factor <span class="math-container">$\frac{|D|}{|K|}$</span> to the policy gradient over trajectories. Since multiplying a gradient by a constant (generally) does not change the underlying optimization problem, these two forms are practically equivalent.</p>
  
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Answers.CreationDate: 2024-08-23T18:41:41.023Z
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Answers.UserDisplayName: DeepQZero
Body: <p>In <a href="https://spinningup.openai.com/en/latest/spinningup/rl_intro3.html" rel="nofollow noreferrer">Part 3: Intro to Policy Optimization</a> from spinningup documentation, there is a formula to compute the estimate of the policy gradient:</p>
  <p>'''
  This is an expectation, which means that we can estimate it with a sample mean. If we collect a set of trajectories <span class="math-container">$ D = \{\tau_i\}_{i=1,...,N} $</span> where each trajectory is obtained by letting the agent act in the environment using the policy <span class="math-container">$\pi_{\theta}$</span>, the policy gradient can be estimated with</p>
  <p><span class="math-container">$$
  \hat{g} = \frac{1}{|D|} \sum_{\tau \in D} \sum_{t=0}^{T} \nabla_{\theta} \log \pi_{\theta}(a_t |s_t) R(\tau),
  $$</span></p>
  <p>where <span class="math-container">$|D|$</span> is the number of trajectories in <span class="math-container">$D$</span> (here, <span class="math-container">$N$</span>).
  '''</p>
  <p>Also in the code on the same page to compute loss:</p>
  <pre><code># make loss function whose gradient, for the right data, is policy gradient 
  def compute_loss(obs, act, weights):
      logp = get_policy(obs).log_prob(act)
      return -(logp * weights).mean()
  </code></pre>
  <p>Here the mean() is calculated over all the (state, action) pairs. However, the formula above calculates the mean over trajectories. There seems to be a disconnect. Are these two forms equivalant?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-08-23T18:42:46.607Z
Comments.Text: Hi @Charles Ju and welcome to AI Stack Exchange. I wanted to commend you on writing a very clear and well-researched question. Thank you for posting, and we look forward to more of your questions here soon!
Comments.UserDisplayName: DeepQZero
Tags: <policy-gradients>
Title: question on Implementing the Simplest Policy Gradient from spinningup
UserDisplayName: Charles Ju
ThreadText: # question on Implementing the Simplest Policy Gradient from spinningup
  
  (Asked by: username_0 on 2024-08-21)
  
  In [Part 3: Intro to Policy Optimization](https://spinningup.openai.com/en/latest/spinningup/rl_intro3.html) from spinningup documentation, there is a formula to compute the estimate of the policy gradient:
  
  '''
  This is an expectation, which means that we can estimate it with a sample mean. If we collect a set of trajectories $ D = \{\tau\_i\}\_{i=1,...,N} $ where each trajectory is obtained by letting the agent act in the environment using the policy $\pi\_{\theta}$, the policy gradient can be estimated with
  
  $$
  \hat{g} = \frac{1}{|D|} \sum\_{\tau \in D} \sum\_{t=0}^{T} \nabla\_{\theta} \log \pi\_{\theta}(a\_t |s\_t) R(\tau),
  $$
  
  where $|D|$ is the number of trajectories in $D$ (here, $N$).
  '''
  
  Also in the code on the same page to compute loss:
  
  ```
  # make loss function whose gradient, for the right data, is policy gradient 
  def compute_loss(obs, act, weights):
      logp = get_policy(obs).log_prob(act)
      return -(logp * weights).mean()
  
  ```
  
  Here the mean() is calculated over all the (state, action) pairs. However, the formula above calculates the mean over trajectories. There seems to be a disconnect. Are these two forms equivalant?
  
  **Tags:** policy-gradients
  
  > **Comments:**
  > **username_1**: Hi @username_0 and welcome to AI Stack Exchange. I wanted to commend you on writing a very clear and well-researched question. Thank you for posting, and we look forward to more of your questions here soon!  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2024-08-23. Score: 1
  
  Note that the documentation states that the `weights` parameter is set to $R(\tau)$.
  Therefore, the $\nabla\_\theta \log \pi\_\theta (a\_t|s\_t)R(\tau)$ terms are the same whether calculating the policy gradient over trajectories (the equation) or over state-action pairs (the pseudocode). Also, note that each state-action pair is associated with exactly one trajectory; therefore,  
  $$\hat{h} = \sum\_{\tau \in D} \sum\_{t=0}^T \nabla\_\theta \log \pi\_\theta (a\_t|s\_t)R(\tau)$$
  calculates both the unnormalized policy gradient over the trajectories and also the unnormalized policy gradient over the state-action pairs. The policy gradient over the trajectories is given as $\hat{g} = \frac{1}{|D|}\hat{h}$. Assuming there are $|K|$ state-action pairs, the policy gradient instead averaged over the state-action pairs is $\frac{1}{|K|}\hat{h} = \frac{|D|}{|K|}\left(\frac{1}{|D|}\hat{h}\right) = \frac{|D|}{|K|}\hat{g}$. Thus, the policy gradient averaged over state-action pairs is proportional by factor $\frac{|D|}{|K|}$ to the policy gradient over trajectories. Since multiplying a gradient by a constant (generally) does not change the underlying optimization problem, these two forms are practically equivalent.
  
  
  ---
  
  