[Paper Review] Towards Minimax Optimal Reinforcement Learning in Factored Markov Decision Processes

Yi Tian, Jian Qian+1 more|arXiv (Cornell University)|Jun 24, 2020

Reinforcement Learning in Robotics38 references4 citations

TL;DR

This paper proposes two model-based reinforcement learning algorithms—F-UCBVI and F-EULER—for episodic factored Markov decision processes (FMDPs) with minimax optimal regret. By introducing an inverse telescoping-based bonus term, the algorithms achieve tight regret bounds that close the gap to the $ ilde{ m O}( olimits\sqrt{HSAT})$ lower bound, with F-EULER achieving problem-dependent regret matching the theoretical limit.

ABSTRACT

We study minimax optimal reinforcement learning in episodic factored Markov decision processes (FMDPs), which are MDPs with conditionally independent transition components. Assuming the factorization is known, we propose two model-based algorithms. The first one achieves minimax optimal regret guarantees for a rich class of factored structures, while the second one enjoys better computational complexity with a slightly worse regret. A key new ingredient of our algorithms is the design of a bonus term to guide exploration. We complement our algorithms by presenting several structure-dependent lower bounds on regret for FMDPs that reveal the difficulty hiding in the intricacy of the structures.

Motivation & Objective

• Address the challenge of achieving minimax optimal regret in episodic factored MDPs (FMDPs), where state transitions exhibit conditional independence.
• Overcome the difficulty of ensuring optimism in exploration when using scalar concentration instead of $L_1$-norm concentration in component-wise transition estimation.
• Design algorithms that exploit structural sparsity in FMDPs to reduce regret beyond standard MDP bounds, particularly by eliminating suboptimal $\sqrt{S_i}$ factors.
• Establish structure-dependent regret lower bounds to characterize the inherent difficulty of different FMDP structures.
• Balance regret optimality with computational efficiency by proposing two algorithms with different trade-offs: one with tighter regret and simpler bonus, the other with better computational complexity.

Proposed method

• Propose F-UCBVI, a model-based algorithm using Hoeffding-style bonuses derived from scalar concentration to ensure optimism in exploration.
• Introduce an inverse telescoping technique to construct a bonus term that maintains optimism across component transitions, enabling tighter regret bounds.
• Design F-EULER, a model-based algorithm using Bernstein-style bonuses for problem-dependent regret, improving over F-UCBVI in worst-case regret scaling.
• Leverage the known factorization of the FMDP to estimate transition probabilities component-wise and combine them with the novel bonus structure.
• Apply concentration inequalities (Hoeffding, Bernstein, and empirical Bernstein) to bound estimation errors and derive high-probability confidence intervals.
• Use the inverse telescoping bonus to ensure that the optimistic value function remains a valid upper bound on the true value function, even when component-wise bonuses are summed.

Experimental results

Research questions

• RQ1 Can we achieve minimax optimal regret in episodic FMDPs by overcoming the limitations of naive component-wise bonus summation?
• RQ2 What structural properties of FMDPs allow for regret bounds that match the $\tilde{\mathcal{O}}(\sqrt{HSAT})$ lower bound for nonfactored MDPs?
• RQ3 How can we design a bonus term that ensures optimism in FMDPs when using scalar concentration instead of $L_1$-norm concentration?
• RQ4 What are the fundamental limits of regret in FMDPs, and how do they depend on the underlying factorization structure?
• RQ5 Can we achieve both regret optimality and computational efficiency in FMDP learning through algorithmic design?

Key findings

• F-UCBVI achieves a regret bound of $\tilde{\mathcal{O}}\left(\sum_{i=1}^{m}\sqrt{H^2 X[I_i] T}\right)$, matching the minimax optimal rate up to logarithmic factors.
• F-EULER achieves a problem-dependent regret bound that reduces to $\tilde{\mathcal{O}}\left(\sum_{i=1}^{m}\sqrt{H X[I_i] T}\right)$ in the worst case, closing the gap to the $\Omega(\sqrt{HSAT})$ lower bound.
• The proposed inverse telescoping bonus technique enables optimism in the presence of scalar concentration, overcoming a key obstacle in prior approaches.
• Structure-dependent regret lower bounds are established for FMDPs, revealing that the difficulty depends on the specific factorization structure.
• The F-EULER algorithm achieves tighter regret than prior work by eliminating the $\sqrt{S_i}$ factor in component-wise regret, improving over $\tilde{\mathcal{O}}(\sum \sqrt{D^2 S_i X[I_i] T})$ bounds.
• Despite a $\sqrt{H}$-factor overhead, F-UCBVI offers significantly lower computational complexity due to its simpler bonus structure.