Rollout Clustering in Reinforcement Learning

• Rollout clustering is a reinforcement learning method that adaptively groups simulation trajectories based on statistically significant outcome differences.
• It employs Hoeffding’s inequality and multi-armed bandit strategies to efficiently allocate sampling efforts and reduce computational resources.
• Integration with classifier-based policy iteration demonstrates its effectiveness on tasks like the inverted pendulum and mountain-car with substantial sample savings.

Rollout clustering is a methodological innovation in reinforcement learning and sequential decision-making, whereby simulation resources or trajectories are adaptively grouped according to statistically significant outcome separations. It serves to reduce sample complexity and computational effort by partitioning state, action, or trajectory pools into resolved and unresolved clusters based on empirical value differences. This approach enables intelligent allocation of sampling and learning effort, achieving strong numerical efficiency while maintaining policy performance. The concept is formalized and evaluated in standard RL domains, notably in the inverted pendulum and mountain-car tasks, as described in "Rollout Sampling Approximate Policy Iteration" (0805.2027).

1. Rollout Sampling Framework

The foundational element of rollout clustering is the unbiased estimation of state–action value functions via multiple simulated trajectories. For a given state–action pair $(s, a)$, the algorithm executes $K$ rollouts where the first action is fixed to $a$, and subsequent steps are governed by the current policy $\pi$. Each rollout yields a cumulative discounted reward:

$$\tilde{Q}^{\pi, T}_i(s,a) = \sum_{t=0}^{T} \gamma^t r_t^{(i)}$$

The empirical Q-value estimate is:

$$\hat{Q}^\pi_K(s,a) = \frac{1}{K}\sum_{i=1}^{K}\tilde{Q}^{\pi, T}_i(s,a)$$

Rather than treating each state independently, rollout clustering reframes the set of rollout states as arms in a multi-armed bandit problem. Adaptive interleaved sampling exploits observed differences between estimated Q-values, focusing effort on states/actions whose optimality is ambiguous and quickly terminating sampling for states showing a clear dominating action.

2. Statistical Tests and State Clustering

A critical component is the use of Hoeffding’s inequality to test when a state's action ordering is reliably determined. The action gap estimate for a state is:

$$\hat{\Delta}^\pi(s) = \hat{Q}^\pi_K(s,\hat{a}^*_s) - \max_{a \neq \hat{a}^*_s} \hat{Q}^\pi_K(s,a)$$

Sampling is terminated when:

$$\hat{\Delta}^\pi(s) \ge \sqrt{\frac{(b_2-b_1)^2}{2c(s)} \ln\left(\frac{|\mathcal{A}|-1}{\delta}\right)}$$

where $c(s)$ is the number of samples at state $s$, $[b_1, b_2]$ the support of possible returns, and $\delta$ the error tolerance.

States resolved by this test are clustered as having a "dominant action," so further rollouts are unnecessary. Unresolved states—those not meeting the statistical threshold—form the alternative cluster demanding further sampling. This partitioning enables a dynamic and efficient sampling schedule.

3. Integration with Classifier-Based Policy Iteration

Rollout clustering is employed in an approximate policy iteration (API) architecture that eschews explicit value function learning. Instead, resolved states provide labeled data for classifier-based policy representation:

• For each state $s$ in pool $S_R$, rollout estimates are generated for all actions.
• States where the dominant action is statistically validated form positive training examples for that action, and negative for all others.
• Classifier training (on the aggregated data) outputs the improved policy for the next API iteration.
• The pool $S_R$ is updated to cover additional regions of the state space, ensuring comprehensive data coverage.

Utilizing supervised learning in lieu of value function approximation combines statistical efficiency (through clustering) with robust classifier generalization, leveraging statistical theory underlying both bandit sampling and supervised classification.

4. Computational Efficiency and Scaling

Empirical evidence—quantitative comparisons with Relational Classification Policy Iteration (RCPI)—demonstrates order-of-magnitude reductions in sample requirements when employing rollout clustering. In the inverted pendulum domain, UCB-A–based selective sampling achieves high success rates (e.g., balancing >1,000 steps) with substantially fewer rollouts than uniform RCPI sampling. In the mountain-car task, policies reaching the goal in <75 steps are found with sparse simulation effort. The mechanism's efficiency derives from resource-sensitive clustering that reallocates sampling to informative or ambiguous states, avoiding unnecessary computation in resolved regions of the state space.

5. Experimental Domains and Robustness

The approach is validated in continuous, stochastic domains:

• Inverted Pendulum: Clustering allows rapid identification of states with clear action preference, yielding robust, low-variance classifier training sets and a high rate of successful policy iterations.
• Mountain-Car: Rollout clustering selectively invests sampling effort in ambiguous states, producing near-optimal policies with dramatically reduced sample counts.

The method proves robust to stochasticity (action noise) and demonstrates consistent performance gains across heterogeneous RL tasks. Clustering acts as an implicit regularization, isolating high-confidence decisions and minimizing error propagation due to oversampling.

6. Algorithmic Innovations and Extensions

Rollout clustering leverages multi-armed bandit allocation strategies (including UCB, counting, and successive elimination variants) to reinforce efficient state selection. The use of Hoeffding-based stopping criteria systematically clusters rollout states by sampling utility. Representing policies as classifiers—rather than through explicit value functions—clarifies the link between RL and supervised learning, facilitating adoption of advanced classification models within API loops.

This suggests that rollout clustering frameworks can be extended to richer action/state spaces, and may benefit from integration with more sophisticated classifiers or exploration mechanisms. A plausible implication is the generalization of rollout clustering to large-scale, high-dimensional RL domains, wherever adaptive resource allocation and label-efficient learning are essential.

7. Significance and Future Directions

Rollout clustering mitigates the core sampling bottleneck endemic to rollout-based policy improvement schemes, making classifier-based API practical for complex real-world domains. By partitioning states according to statistical resolution of action gaps, computational resources are deployed adaptively, yielding superior efficiency and policy quality indistinguishable from sampling-intensive alternatives. Extension to other forms of decision problem where state, action, or trajectory sets have resolvable ambiguity—such as multiagent coordination, causal inference, or planning with combinatorial structure—is anticipated.

The method introduces a technically rigorous foundation for adaptive sampling and clustering in simulation-based RL, establishing fertile ground for future research in scalable, label-efficient, and classifier-driven API frameworks.