Q-Values Vs. Reward Functions

A gentle introduction to the fundamentals of reinforcement learning

In the context of Markov Decision Processes (MDPs), both Q-values and reward functions are fundamental concepts, but they represent different things:

Reward Functions


Serious Stuff


A reward function \(r(s, a, s')\) specifies the immediate reward an agent receives for taking action \(a\) in state \(s\) and transitioning to state \(s'\).

\[r=r(s, a, s')\]



In tabular settings, it’s often presented as a table or matrix, and in more complex scenarios (like continuous states and actions), it might be approximated or represented using function approximators.

Story Walk Through for Reward Functions:



Serious Stuff


A Q-value, denoted \(Q(s, a)\), indicates the expected cumulative reward of taking action \(a\) in state \(s\) and then acting optimally thereafter.

It is defined as:

\[Q(s,a) = \sum_{s' \in S} P_a(s' \mid s)\ [r(s,a,s') + \gamma\ V(s') ]\]

which follows directly from the Bellman Equation of state values:

\[V(s) = \max_{a \in A(s)} Q(s,a)\]



Like reward functions, Q-values can be tabular (in simpler environments) or approximated using neural networks or other function approximators in more complex scenarios (Deep Q Networks or DQN is an example where neural networks are used to approximate Q-values).

Story Walk through for Q-values:

Key Ideas

Q-Values and Reward functions are foundational blocks in understanding model-based methods in reinforcement learning. This section summarizes key ideas from these building blocks.

  1. Q-Values vs. Reward Functions

    In an MDP, there are two key functions that guide an agent’s actions:

    • Reward Function: Provides immediate feedback for actions taken in specific states. It essentially tells the agent how “good” or “bad” its last action was. This feedback is based on the state transition and action.
    • Q-Values (Action-Value Function): Denotes the expected cumulative future reward of taking a specific action in a particular state and then following an optimal policy thereafter.
  2. Illustrating Q-Values

    Imagine a robot in a simple 3x3 grid world. This robot must navigate obstacles to reach a goal. While the reward function might give the robot immediate feedback (e.g., -1 for every move, +10 for reaching the goal), the Q-value helps the robot decide the optimal path by accounting for both immediate and future rewards.

  3. Dynamics of Reward Functions

    Rewards can be deterministic (fixed for specific actions in states) or stochastic (varying even for the same action-state pair). For instance, a robot trying to pick up an object might mostly succeed and receive a positive reward. Still, occasionally, due to external factors, it might fail and receive a negative reward.

  4. Iterative Evaluation of the Value Function

    It’s crucial to differentiate between the reward and value functions when applying iterative algorithms like policy iteration or value iteration. While the reward function provides immediate feedback, the value function gets updated using the Bellman expectation equation, which integrates both immediate rewards and the anticipated value of future states.

  5. Designing Reward Functions

    Designing a reward function is a critical task. While one can, in principle, select an arbitrary reward function, the choice can drastically influence an agent’s behavior. A poorly chosen reward might even lead to unintended consequences. The challenge is to align the reward structure with desired outcomes.

  6. Are Q-Values Goal-Dependent?

    Indeed, Q-values can change if the goal of the task changes, leading to a different reward structure. For example, in our robot grid world, if the goal shifts to a different location, the Q-values for various actions will need to be recalculated or relearned.

  7. Range of Q-Values

    The range of Q-values depends on multiple factors: the reward function, discount factor, and task structure. While no fixed universal range applies to all scenarios, understanding these factors can help predict Q-value dynamics.