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In policy-based RL, the policy gradient theorem is provides a Monte Carlo gradient estimator that can be used for directly learning a policy. The policy gradient theorem states that under a policy with parameters \(\theta\), the gradient of the expected return \(R(\tau)\) (where \(\tau := s_1, a_1, r_1, s_2, ...\) is a trjaectory) with respect to \(\theta\) is:

\[\nabla_{\theta} \mathbb{E}_{\tau \sim p_{\theta}}[R(\tau)] = \mathbb{E}_{\tau \sim p_{\theta}} \Bigg[ R(\tau) \nabla_{\theta} \sum_t \nabla_{\theta} \log p_{\theta}(a_t | s_t) \Bigg]\]In practice, because the distribution over trajectories is almost never known, we approximate the right-hand side expectation with a Monte Carlo estimate:

\[\nabla_{\theta} \mathbb{E}_{\tau \sim p_{\theta}}[R(\tau)] \approx \frac{1}{N}\sum_{n} R(\tau^{(n)}) \sum_t \nabla_{\theta} \log p_{\theta}(a_t^{(n)}| s_t^{(n)})\]As with an estimator, we want to minimize its variance if possible. Consequently, many techniques have emerged for creating better gradient estimators.

The return \(R(\tau)\) is defined as the (optionally discounted) sum of future rewards:

\[R(\tau) := \sum_{t=1}^{T} r_t\]Plugging this into the estimate of the policy gradient yields:

\[\frac{1}{N}\sum_{n} \Big(\sum_{t'=1}^{t'=T} r_{t'}^{(n)} \Big) \sum_t \nabla_{\theta} \log p_{\theta}(a_t^{(n)}| s_t^{(n)})\]If we consider the causal structure of the agent interacting with its environment, actions cannot affect rewards from previous timesteps. We can leverage this knowledge to rearrange and drop certain terms:

\[\begin{equation} \frac{1}{N}\sum_{n} \sum_t \nabla_{\theta} \log p_{\theta}(a_t^{(n)}| s_t^{(n)}) \Big(\sum_{t'=t}^{t'=T} r_{t'}^{(n)} \Big) \end{equation}\]This policy gradient estimator will have lower variance because fewer random variables are included in its sum.

One well-known family of approaches for improving an estimator is control variates,
more commonly known as *baselines* in RL. The idea is that by subtracting a particular function from
the return (or return to go), we can further reduce the variance in the policy gradient estimator.
Let \(b: \mathcal{S} \rightarrow \mathbb{R}\) be a function called the *baseline*, and suppose we subtract
that baseline at each time step:

To understand why we do this, I claim two facts:

- Subtracting the baseline introduces no bias i.e. the expectation with and without the baseline is equal
- Subtracting the baseline reduces the variance in the policy gradient estimator

Claim 1: Subtracting the baseline introduces no bias. To show this, we could show that the expectation of the estimate with and without the baseline is equal, but it’s easier (and equivalent) to show the expectation of the baseline is 0:

\[\begin{align*} \frac{1}{N}\sum_{n} \sum_t \nabla_{\theta} \log p_{\theta}(a_t^{(n)}| s_t^{(n)}) \Big(b(s_t^{(n)}) \Big) \end{align*}\]