What role does the discount factor play in the convergence of reinforcement learning algorithms?
Instruction: Discuss the impact of the discount factor on the learning process and its convergence.
Updated
Example Answer
The discount factor controls how strongly future rewards contribute to the return. A smaller gamma places more weight on near-term outcomes; a gamma close to one values consequences further ahead.
It also has a mathematical role. With bounded rewards and gamma below one, discounted returns are bounded, and the usual tabular Bellman operators are contractions. That supports convergence of exact value iteration. It does not guarantee convergence of every reinforcement-learning algorithm, especially with nonlinear approximation and off-policy bootstrapping.
Gamma equal to one is not automatically invalid. A finite-horizon episode can use the undiscounted sum of its rewards. For a continuing task with a positive reward every step, that sum can diverge, so I would use a suitable discounted or average-reward formulation. I would choose the objective first rather than selecting gamma only to make training look stable.
Worked Example
For a three-step episode with rewards [1, 1, 1], the return is 3 at γ = 1 and 1.75 at γ = 0.5. For an endless stream of reward 1, it is 10 at γ = 0.9 and unbounded at γ = 1.