[Paper Review] Robust Bayesian reinforcement learning through tight lower bounds

Christos Dimitrakakis|arXiv (Cornell University)|Jun 18, 2011

Reinforcement Learning in Robotics26 references4 citations

TL;DR

This paper proposes a novel method for robust Bayesian reinforcement learning by computing tight lower bounds on the value function using a backwards induction algorithm (MMBI) and its stochastic variant (MSBI), which identify near-optimal memoryless policies under belief distributions over MDPs. The approach significantly outperforms standard Bayesian RL methods like MCBRL and the $ς$-optimal policy, especially as sample size increases, due to tighter bounds and better exploration strategies.

ABSTRACT

In the Bayesian approach to sequential decision making, exact calculation of the (subjective) utility is intractable. This extends to most special cases of interest, such as reinforcement learning problems. While utility bounds are known to exist for this problem, so far none of them were particularly tight. In this paper, we show how to efficiently calculate a lower bound, which corresponds to the utility of a near-optimal memoryless policy for the decision problem, which is generally different from both the Bayes-optimal policy and the policy which is optimal for the expected MDP under the current belief. We then show how these can be applied to obtain robust exploration policies in a Bayesian reinforcement learning setting.

Motivation & Objective

• To address the intractability of computing Bayes-optimal policies in Bayesian reinforcement learning under belief distributions over MDPs.
• To develop a computationally efficient method that provides tight lower bounds on the value function for near-optimal memoryless policies.
• To enable robust exploration in Bayesian RL by leveraging these bounds to guide policy selection.
• To generalize existing sampling-based Bayesian RL approaches by using multiple MDP samples and tighter policy evaluation.
• To demonstrate that the proposed method outperforms both the expected MDP heuristic and baseline Bayesian RL algorithms on benchmark tasks.

Proposed method

• Introduces MMBI (Memoryless Maximum Belief Induction), a backwards induction procedure to compute a lower bound on the value function for a memoryless policy under a belief distribution over MDPs.
• Develops MSBI (Stochastic MMBI), a randomized variant of MMBI that provides a lower bound in expectation with a polynomially bounded gap to MMBI.
• Uses the MMBI/MSBI policy as a robust exploration strategy in the MCBRL algorithm, which samples multiple MDPs at regular intervals from the belief distribution.
• Applies the lower bound to guide action selection, ensuring the policy is near-optimal among memoryless policies rather than relying on the expected MDP.
• Employs a backward induction process over a finite set of MDPs to compute value functions and policy estimates efficiently.
• Integrates the method into a Bayesian RL framework where the policy is updated based on tight lower bounds rather than point estimates of the MDP.

Experimental results

Research questions

• RQ1 Can we compute a tight lower bound on the value function for a memoryless policy in Bayesian reinforcement learning under belief distributions over MDPs?
• RQ2 How does the performance of the MMBI/MSBI policy compare to the Bayes-optimal policy and the policy optimal for the expected MDP?
• RQ3 Can the proposed method improve exploration and overall learning performance in Bayesian RL compared to existing sampling-based approaches?
• RQ4 What is the computational and statistical gap between the MMBI policy and the true Bayes-optimal policy?
• RQ5 Does increasing the number of sampled MDPs in the belief distribution lead to tighter bounds and better performance?

Key findings

• The MMBI policy provides a significantly tighter lower bound on the value function than the policy optimal for the expected MDP ($\hat{\mu}_{\xi}$), especially in uncertain environments.
• For the chain task, the proposed method (Alg. 4) achieved a mean total reward of 3287 over 1000 steps, outperforming MCBRL with $n=1$ (3166) and $n=8$ (3358), and approaching the performance of more complex algorithms like Beetle and AMP-EM.
• As the number of MDP samples ($n$) increased from 1 to 16, the total reward improved from 3166 to 3376, demonstrating the benefit of tighter bounds with more samples.
• The method reduced the number of runs where the agent got stuck in a suboptimal regime to less than 1% of runs, indicating robustness and reliable performance.
• The MMBI policy was shown to be closer to the Bayes-optimal policy than the $\hat{\mu}_{\xi}$-optimal policy, with a theoretically bounded gap that depends polynomially on the number of samples.
• The MSBI algorithm provides a lower bound in expectation that is close to MMBI, with a gap that grows polynomially in the number of samples, making it scalable and practical.