Minimax Regret Bounds for Reinforcement Learning (1703.05449v2)

Abstract: We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of $\tilde{O}( \sqrt{HSAT} + H^2S^2A+H\sqrt{T})$ where $H$ is the time horizon, $S$ the number of states, $A$ the number of actions and $T$ the number of time-steps. This result improves over the best previous known bound $\tilde{O}(HS \sqrt{AT})$ achieved by the UCRL2 algorithm of Jaksch et al., 2010. The key significance of our new results is that when $T\geq H^3S^3A$ and $SA\geq H$, it leads to a regret of $\tilde{O}(\sqrt{HSAT})$ that matches the established lower bound of $\Omega(\sqrt{HSAT})$ up to a logarithmic factor. Our analysis contains two key insights. We use careful application of concentration inequalities to the optimal value function as a whole, rather than to the transitions probabilities (to improve scaling in $S$), and we define Bernstein-based "exploration bonuses" that use the empirical variance of the estimated values at the next states (to improve scaling in $H$).

• The paper presents a novel UCBVI algorithm that achieves regret bounds of O(√(HSAT) + H²S²A + H√T) for finite-horizon MDPs, outperforming previous methods.
• It leverages concentration inequalities on the optimal value function and Bernstein-based exploration bonuses to reduce dependencies on state and time horizon parameters.
• The improved regret bounds have significant implications for efficient RL in complex applications such as robotics, autonomous systems, and adaptive control.

Minimax Regret Bounds for Reinforcement Learning

The paper "Minimax Regret Bounds for Reinforcement Learning" by Mohammad Gheshlaghi Azar, Ian Osband, and Remi Munos addresses the complex problem of achieving provably optimal exploration in reinforcement learning (RL) for finite-horizon Markov Decision Processes (MDPs). The authors introduce an optimistic modification to value iteration and present a significant improvement in the regret bounds over previous state-of-the-art algorithms.

Key Contributions

The algorithm proposed in the paper, named Upper Confidence Bound Value Iteration (UCBVI), achieves a regret bound of $O(\sqrt{HSAT} + H^2S^2A + H\sqrt{T})$, where $H$ is the time horizon, $S$ is the number of states, $A$ is the number of actions, and $T$ is the number of time-steps. This result notably improves over the best previously known bound $O(HS \sqrt{AT})$ from the UCRL2 algorithm.

Two major insights underlie this enhanced performance:

1. Concentration Inequalities: The use of concentration inequalities directly on the optimal value function rather than on the transition probabilities, which improves the scaling with respect to $S$.
2. Bernstein-Based Exploration Bonuses: The implementation of exploration bonuses based on the empirical variance of the estimated values at the next states, inspired by Bernstein's inequality, which improves the scaling in $H$.

The theoretical significance of the result is that for sufficiently large $T$ (specifically when $T \geq H^3 S^3 A$ and $SA \geq H$), the regret bound $O(\sqrt{HSAT})$ matches the established lower bound of $\Omega(\sqrt{HSAT})$, up to logarithmic factors.

Methodological Insights

The methodological advances can be distilled as follows:

• The authors formulate two variants of the UCBVI algorithm. The first, UCBVI-CH, employs Chernoff-Hoeffding bounds and achieves a regret bound of $O(H\sqrt{SAT})$, improving the dependency on $S$ over previous algorithms.
• The second variant, UCBVI-BF, leverages Bernstein-Freedman inequalities, enabling the regret to depend on $\sqrt{H}$ rather than $H$, hence achieving the $O(\sqrt{HSAT})$ bound.

These approaches defer from traditional algorithms by not constructing confidence sets for transition probabilities and rewards. Instead, they directly address the concentration of the optimal value function.

Practical and Theoretical Implications

The practical implications of this work are profound for designing RL algorithms that can operate efficiently in large state and action spaces. This is particularly useful in applications where premature exploitation can lead to suboptimal policies. These scenarios are common in robotics, autonomous systems, and adaptive control settings.

Theoretically, the results provide a clearer answer to the ongoing question about the fundamental lower bounds for regret in finite-horizon MDPs. The established methods could serve as a foundation for future research, improving bounds in more complex MDP settings such as weakly communicating MDPs.

Future Directions

The authors acknowledge certain limitations, opening avenues for future research. Significant areas include:

• Extending the analysis to the broader class of weakly communicating MDPs.
• Investigating whether the higher-order term $H^2S^2A$ can be improved to $HS^2A$ or even further.
• Refining the term $H\sqrt{T}$ to $\sqrt{HT}$.

Moreover, exploring the practical computational performance of the proposed algorithms in various real-world problems would further cement their applicability.

In summary, this paper presents a significant step forward in the exploration-exploitation trade-off in reinforcement learning, providing both computational efficiency and tighter theoretical bounds on regret. The insights and methods developed here have the potential to influence a broad class of future RL research and applications.