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After decomposing the trajectory log-probability gradient, it is typical in reinforcement learning settings to assume the environment's dynamics are not directly influenced by the policy parameters $$\theta$$. Consequently, the derivative of the dynamics gradient, $$\frac{\partial}{\partial \theta} \sum_{t=1}^{T} \log \Pr(s_{t+1}|s_t,a_t)$$, is usually zero. We can therefore simplify the overall gradient to focus entirely on optimizing the policy component:
$$ \frac{\partial \log \mathrm{Pr}_{\theta}(\tau)}{\partial \theta} = \frac{\partial}{\partial \theta} \sum_{t=1}^{T} \log \pi_{\theta}(a_t|s_t) $$
This simplification allows for concentrating solely on policy updates without the need to understand or model the underlying environmental dynamics.

Simplification of the Trajectory Log-Probability Gradient

The general form of the policy gradient is an expectation over trajectories sampled from the policy. By treating every trajectory $$\tau$$ in a sampled dataset $$\mathcal{D}$$ (with size $$m$$ or $$|\mathcal{D}|$$) as equally probable, we use the practical estimator:
$$ \frac{\partial J(\theta)}{\partial \theta} = \frac{1}{m} \sum_{\tau \in \mathcal{D}} \frac{\partial \log \mathrm{Pr}_{\theta}(\tau)}{\partial \theta} R(\tau) $$
By decomposing the sequence probability and dropping the dynamics gradient (which does not depend on $$\theta$$), and knowing that the cumulative reward is $$R(\tau) = \sum_{t=1}^{T} r_t$$, this objective expands to:
$$ \frac{\partial J(\theta)}{\partial \theta} = \frac{1}{|\mathcal{D}|} \sum_{\tau \in \mathcal{D}} \frac{\partial}{\partial \theta} \Big( \sum_{t=1}^{T} \log \pi_{\theta}(a_t|s_t) \sum_{t=1}^{T} r_t \Big) $$
This formulation highlights that the reward function $$R(\tau)$$ does not need to be differentiable for optimization.

Policy Gradient Estimate under Uniform Trajectory Probability

Based on the standard derivation of the policy gradient, can the engineer still optimize the agent's policy using this method? Justify your answer by explaining which component of the full trajectory log-probability gradient is affected by this lack of a model and why it does not prevent the optimization.

Applicability of Policy Gradient Methods

In the derivation of the policy gradient, the full expression for the gradient of a trajectory's log-probability is simplified by dropping the term related to the environment's transition probabilities. Analyze the primary practical advantage of this simplification. What does this imply about the types of environments or problems where this family of algorithms can be successfully applied, compared to methods that might require a model of the environment?

Practical Implications of the Policy Gradient Simplification

In policy gradient methods, the gradient of the log-probability of a trajectory is initially expressed as the sum of two components: one related to the agent's actions and another related to the environment's transitions. The expression is then simplified by removing the environment's component before optimization. Given the initial expression: $$ \frac{\partial}{\partial \theta} \left[ \sum_{t=1}^{T} \log \pi_{\theta}(a_t|s_t) + \sum_{t=1}^{T} \log \text{Pr}(s_{t+1}|s_t, a_t) \right] $$ What is

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