Value-Based Reward Shaping Formula

A reinforcement learning engineer wants to add an extra reward signal, denoted as a function f, to an agent's learning process to encourage more efficient exploration. They have access to a function Φ(s) which provides a numerical estimate of a state's value, and a discount factor γ. To guarantee that this additional reward signal does not alter the agent's optimal long-term behavior, which of the following structures must the function f have for a transition from state s_t to s_{t+1}?

Analyzing a Flawed Reward Shaping Implementation

Validating a Potential-Based Shaping Function

An agent is in a state 'S' and must choose between two policies, Policy A and Policy B. The sequence of rewards the agent will receive after starting in state 'S' and following each policy is deterministic and known:

• Policy A Reward Sequence: [+10, +1, +1, +1, ...]
• Policy B Reward Sequence: [+3, +3, +3, +3, ...]

Given the formula for the value of a state, $V(s) = \mathbb{E}[\sum_{t=0}^{\infty} \gamma^t r_t | s_0 = s, \pi]$, which of the following statements correctly analyzes the relationship between the discount factor γ and the value of state 'S' for each policy?

Calculating State Value in a Deterministic Environment

Advantage Function Formula

Temporal Difference (TD) Error as an Advantage Function Estimator

An agent is in a state 'S' and follows a fixed policy. From this state, the environment is stochastic: there is a 50% chance the agent will enter a trajectory with a reward sequence of [+10, 0, 0, ...] and a 50% chance it will enter a different trajectory with a reward sequence of [0, +10, 0, ...]. Given the state-value formula $V(s) = \mathbb{E}[\sum_{t=0}^{\infty} \gamma^t r_t | s_0 = s, \pi]$ and a discount factor (γ) of 0.9, what is the value of state 'S'?

Value-Based Reward Shaping Formula