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A significant advantage of the policy gradient method is that the cumulative reward function, R(τ), is not required to be differentiable. This is because the gradient is calculated with respect to the logarithm of the policy's probability, not the reward itself. This property allows for the application of any type of reward function in reinforcement learning, including those that are discontinuous or arbitrarily complex.

Advantage of Policy Gradients: Non-Differentiable Reward Functions

Given the following scenario, evaluate the engineer's claim and justify your reasoning.

Applicability of Policy Gradients with Discrete Rewards

For a policy gradient method to be applicable, the cumulative reward function must be differentiable, as its derivative is required when computing the gradient of the policy performance objective.

In policy gradient methods, the gradient of the performance objective is estimated as an expectation over trajectories. Each trajectory's contribution to this estimate is the product of its cumulative reward and the gradient of its log-probability. Given this structure, why can these methods effectively handle tasks with non-differentiable reward functions, such as a simple binary reward for winning or losing a game?

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