Google

The temporal difference (TD) error is a common estimator for the advantage function, $$A(s_t, a_t)$$. This value, denoted $$\hat{A}(s_t, a_t)$$, is calculated as the difference between the immediate reward plus the discounted expected value of the next state and the expected value of the current state. The formula for the TD error is: $$ \hat{A}(s_t, a_t) = r_t + \gamma V(s_{t+1}) - V(s_t) $$ This formulation, also known as the one-step advantage estimate, is a foundational component in many actor-critic algorithms. By substituting the action-value function with the return and next state value, it allows the advantage to be efficiently computed using a single critic network $$V(s_t)$$.

Temporal Difference (TD) Error as an Advantage Function Estimator

Explain why the expression `r_t + γV(s_{t+1}) - V(s_t)` is considered a valid single-sample estimate for the advantage of taking action `a_t` in state `s_t`. Your explanation should break down the expression and relate its components to the formal definitions of the action-value and state-value functions.

Derivation of the Advantage Function Estimator

Based on the scenario below, calculate the one-step advantage estimate for the action taken. Then, explain what the sign of this estimate implies about both the action itself and the agent's original valuation of its starting state.

Evaluating an Agent's Action Choice

The value-based shaped reward, defined as $$r' = r + \gamma V(s_{t+1}) - V(s_t)$$, is mathematically equivalent to the Temporal Difference (TD) error, which is a common estimator for the advantage function. This equivalence establishes a direct relationship between advantage-based methods, such as PPO, and reward shaping, demonstrating that the advantage function can be interpreted as a specific instance of a shaped reward.

Advantage Function as a Form of Shaped Reward

An autonomous agent is navigating a maze. At a particular state, the agent's value function estimates the value of its current state to be 10. The agent decides to move to an adjacent state, receiving an immediate reward of -1 for the move. The value function estimates the value of the new state to be 15. Assuming a discount factor of 0.9, calculate the one-step advantage estimate for the action taken and determine its implication for future action selection.

Learn Before

Related