Nearly Minimax Optimal Reinforcement Learning for Linear Mixture Markov Decision Processes (2012.08507v2)

Abstract: We study reinforcement learning (RL) with linear function approximation where the underlying transition probability kernel of the Markov decision process (MDP) is a linear mixture model (Jia et al., 2020; Ayoub et al., 2020; Zhou et al., 2020) and the learning agent has access to either an integration or a sampling oracle of the individual basis kernels. We propose a new Bernstein-type concentration inequality for self-normalized martingales for linear bandit problems with bounded noise. Based on the new inequality, we propose a new, computationally efficient algorithm with linear function approximation named $\text{UCRL-VTR}^{+}$ for the aforementioned linear mixture MDPs in the episodic undiscounted setting. We show that $\text{UCRL-VTR}^{+}$ attains an $\tilde O(dH\sqrt{T})$ regret where $d$ is the dimension of feature mapping, $H$ is the length of the episode and $T$ is the number of interactions with the MDP. We also prove a matching lower bound $\Omega(dH\sqrt{T})$ for this setting, which shows that $\text{UCRL-VTR}^{+}$ is minimax optimal up to logarithmic factors. In addition, we propose the $\text{UCLK}^{+}$ algorithm for the same family of MDPs under discounting and show that it attains an $\tilde O(d\sqrt{T}/(1-\gamma)^{1.5})$ regret, where $\gamma\in [0,1)$ is the discount factor. Our upper bound matches the lower bound $\Omega(d\sqrt{T}/(1-\gamma)^{1.5})$ proved by Zhou et al. (2020) up to logarithmic factors, suggesting that $\text{UCLK}^{+}$ is nearly minimax optimal. To the best of our knowledge, these are the first computationally efficient, nearly minimax optimal algorithms for RL with linear function approximation.

• The paper introduces a new Bernstein-type concentration inequality for vector-valued martingales that refines analysis in linear bandit problems.
• The paper presents UCRL-VTR+ and UCLK+ algorithms that achieve regret bounds of Õ(dH√T) and Õ(d√T/(1-γ)^(1.5)), nearly matching theoretical lower limits.
• The paper establishes nearly minimax optimal performance, demonstrating robust theoretical guarantees for RL in large or structured domains.

Nearly Minimax Optimal Reinforcement Learning for Linear Mixture Markov Decision Processes

This paper presents significant advancements in Reinforcement Learning (RL) by focusing on linear mixture Markov Decision Processes (MDPs). The authors propose novel algorithms that approach near minimax optimality, specifically targeting situations where transition probability kernels are modeled as linear mixtures of basis kernels. These findings stand at the intersection of enhancing computational efficiency and minimizing regret in both episodic and discounted RL settings.

Main Contributions

1. Bernstein-type Concentration Inequality: The paper introduces a new concentration inequality for vector-valued martingales pivotal for linear bandit problems with bounded noise. This inequality provides a refined tool for handling issues associated with self-normalization, aiding in maintaining optimal regret bounds in RL with linear function approximation.
2. Algorithmic Innovation: The authors propose two algorithms:
   • UCRL-VTR$^+$ for episodic undiscounted settings, which achieves a regret bound of $\tilde O(dH\sqrt{T})$. This work marks the first computationally efficient, nearly minimax optimal bound for the linear mixture MDP setting, where $d$ is the dimension of the feature mapping, $H$ the episode length, and $T$ is the number of interactions.
   • UCLK$^+$ for discounted MDPs that achieves an $\tilde O(d\sqrt{T}/(1-\gamma)^{1.5})$ regret, showing strong bounds relative to a known lower bound of $\Omega(d\sqrt{T}/(1-\gamma)^{1.5})$.
3. Theoretical Bounds: Both algorithms are claimed to be near-optimal as their upper bounds match theoretically proven lower bounds up to logarithmic factors. This positions the proposed methods as robustly designed solutions when dealing with RL in large or structured domains.

Implications and Future Directions

The results presented have practical implications for designing RL algorithms that can more effectively handle large state spaces by using linear approximations. This extends robustness beyond tabular settings to more dynamic and real-world scenarios where linearity assumptions can still be applied to good effect.

Moreover, this research lays a foundation for future work in reducing the complexity of RL methods without compromising on theoretical guarantees. Future research could explore extending these approaches to nonlinear function approximations or integrating them into systems with constrained resources where dynamic adaptation is critical.

Speculation on future developments in AI includes using these findings to enhance algorithmic efficiency in high-dimensional and complex environments, such as robotics and real-time decision-making systems. This line of inquiry could potentially bridge the gap between theoretical guarantees and their applicability in tasks requiring real-time operation.