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The score function in policy gradient methods is the gradient of the log-probability of a trajectory $\tau$ with respect to the policy parameters $\theta$. It is expressed as $\nabla_{\theta} \log \text{Pr}_{\theta}(\tau)$ or, using partial derivative notation, $\frac{\partial \log \text{Pr}_{\theta}(\tau)}{\partial \theta}$. This term is fundamental to deriving the policy gradient objective, as it indicates the direction in the parameter space that increases the probability of a given trajectory.

Score Function in Policy Gradients

In a policy gradient method, an agent executes a specific trajectory τ. The score function, defined as the gradient of the log-probability of this trajectory with respect to the policy parameters (∇θ log Prθ(τ)), is calculated. What is the fundamental interpretation of this score function vector?

Consider a reinforcement learning agent using a policy gradient method. The agent completes a trajectory that results in a very high positive reward. Explain the role of the score function for this specific trajectory in the subsequent policy update. How does the high reward influence this update process?

Applying the Score Function in Policy Updates

An agent is being trained using a policy gradient method. After an episode, the agent has sampled two distinct trajectories. Analyze how the policy update mechanism would treat each trajectory, focusing specifically on the role of the score function, ∇θ log Prθ(τ).

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