You've now seen that reinforcement learning problems can be tackled in fundamentally different ways. Chapters 4 and 5 explored value-based methods like Monte Carlo, SARSA, and Q-learning, where the primary goal is to learn accurate estimates of value functions (either state values $V(s)$ or action values $Q(s,a)$ ). Once we have a good estimate of the optimal action-value function $Q^*(s,a)$ , deriving an optimal policy is often straightforward, for instance, by acting greedily with respect to $Q^*$ .

This chapter introduced policy-based methods, exemplified by REINFORCE. Here, the strategy shifts: we directly parameterize the policy itself, $\pi_\theta(a|s)$ , and learn the parameters $\theta$ that maximize the expected return, often using gradient ascent. The value function might still be estimated (for example, as a baseline), but it's not the primary target of learning; the policy is.

Understanding the trade-offs between these two families of algorithms is significant for selecting the right approach for a given problem. Let's compare their characteristics.

Handling Action Spaces

Value-Based Methods: These methods typically work best with discrete action spaces. Finding the best action involves selecting the one with the maximum Q-value, often via an $argmax_a Q(s,a)$ operation. Applying this directly to continuous action spaces is problematic. Finding the maximum of a function over a continuous domain can be computationally expensive or require specific optimization steps within each decision step. While techniques exist to adapt value-based methods (like discretizing the action space or using specific network architectures), it's often less natural than the policy gradient approach.
Policy-Based Methods: Policy gradients handle continuous action spaces quite naturally. Instead of outputting a value for each action, the parameterized policy $\pi_\theta(a|s)$ can directly output the parameters of a probability distribution over actions. For example, in a continuous space, the policy might output the mean $\mu_\theta(s)$ and standard deviation $\sigma_\theta(s)$ of a Gaussian distribution, from which the action $a$ is sampled. Optimizing $\theta$ adjusts the distribution to favor higher-reward actions.

Comparison of information flow in value-based versus policy-based methods for selecting an action. Policy-based methods directly map states to action (probabilities), while value-based methods typically go via learned action values.

Nature of the Learned Policy

Value-Based Methods: Standard value-based methods like Q-learning typically converge towards a deterministic optimal policy (ignoring exploration strategies like $\epsilon$ -greedy, which are used during learning but not part of the converged optimal policy itself). If multiple actions have the same maximal Q-value, ties can be broken arbitrarily or stochastically, but the underlying learned policy derived greedily from $Q^*$ is often deterministic.
Policy-Based Methods: These methods can learn explicitly stochastic policies. The policy network $\pi_\theta(a|s)$ $π_{θ} (a ∣ s)$ outputs probabilities $P(a|s, \theta)$ $P (a ∣ s, θ)$ . This is advantageous in several scenarios:
- Partially Observable Environments: When the agent doesn't have complete information about the state, sometimes the best strategy is inherently random to handle state aliasing (where different underlying world states look identical to the agent).
- Strategic Considerations: In multi-agent settings or games (like Rock-Paper-Scissors), a deterministic policy can be easily exploited, whereas a stochastic policy might be optimal.
- Simplicity: Sometimes, a good stochastic policy is easier to represent and learn than a complex value function that would lead to the same behavior.

Learning Stability and Efficiency

Value-Based Methods: Algorithms like Q-learning and SARSA learn from individual transitions using Temporal Difference (TD) updates. This bootstrapping approach (updating estimates based on other estimates) often leads to higher sample efficiency, meaning they can learn good policies with relatively fewer environment interactions, especially in discrete domains. However, combining TD learning with off-policy training and function approximation (like neural networks in DQN) can sometimes lead to instability during training, requiring techniques like experience replay and target networks to mitigate.
Policy-Based Methods: Basic policy gradient methods like REINFORCE rely on Monte Carlo updates, meaning they only update the policy parameters $\theta$ after observing the full return $G_t$ from a complete episode. This often results in high variance in the gradient estimates because the return depends on a long sequence of actions and state transitions. High variance can slow down learning or make it unstable. Techniques like using baselines (as discussed in the previous section) or moving towards Actor-Critic methods are essential for reducing variance and improving stability and sample efficiency. While policy gradients directly optimize the performance objective which can sometimes lead to smoother convergence, the high variance is a significant practical challenge.

Summary of Comparison

Feature	Value-Based Methods (e.g., Q-Learning)	Policy-Based Methods (e.g., REINFORCE)
Primary Goal	Learn accurate value function ( $Q^*(s,a)$ )	Learn optimal policy parameters ( $\theta$ )
Policy Output	Typically deterministic (derived from values)	Can be stochastic
Action Spaces	Best suited for discrete actions	Handles continuous actions naturally
Sample Efficiency	Often higher (due to TD updates)	Can be lower (esp. Monte Carlo versions)
Gradient Variance	Lower (TD error)	Higher (Monte Carlo return)
Stability	Can be unstable with function approx.	Can be unstable due to high variance
Off-Policy	Q-Learning is naturally off-policy	Off-policy learning can be more complex

Choosing the Right Approach

So, which method should you choose?

If your problem has a discrete action space and sample efficiency is a major concern, a value-based method like Q-learning (or DQN for larger state spaces) might be a good starting point.
If your problem involves a continuous action space or requires an inherently stochastic policy, policy gradient methods are often a more natural fit.
If you face high variance issues with basic policy gradients, consider incorporating baselines or exploring Actor-Critic methods, which represent a hybrid approach attempting to combine the best of both worlds. We'll touch upon Actor-Critic methods next.

Modern reinforcement learning often involves sophisticated algorithms that blend ideas from both value-based and policy-based approaches to leverage their respective strengths and mitigate their weaknesses. Understanding the fundamental differences discussed here provides a solid foundation for navigating these more advanced techniques.