[Paper Review] Target Transfer Q-Learning and Its Convergence Analysis

Yue Wang, Qi Meng+4 more|arXiv (Cornell University)|Sep 21, 2018

Distributed Sensor Networks and Detection Algorithms4 citations

TL;DR

This paper proposes Target Transfer Q-Learning (TTQL), a novel reinforcement learning method that transfers the Q-function from a source task to accelerate learning in a target task when specific safe conditions are met. The method ensures faster convergence than standard Q-learning by leveraging the Bellman error of the transferred Q-function as a safety criterion, with theoretical guarantees on convergence rate under similarity between source and target MDPs.

ABSTRACT

Q-learning is one of the most popular methods in Reinforcement Learning (RL). Transfer Learning aims to utilize the learned knowledge from source tasks to help new tasks to improve the sample complexity of the new tasks. Considering that data collection in RL is both more time and cost consuming and Q-learning converges slowly comparing to supervised learning, different kinds of transfer RL algorithms are designed. However, most of them are heuristic with no theoretical guarantee of the convergence rate. Therefore, it is important for us to clearly understand when and how will transfer learning help RL method and provide the theoretical guarantee for the improvement of the sample complexity. In this paper, we propose to transfer the Q-function learned in the source task to the target of the Q-learning in the new task when certain safe conditions are satisfied. We call this new transfer Q-learning method target transfer Q-Learning. The safe conditions are necessary to avoid the harm to the new tasks and thus ensure the convergence of the algorithm. We study the convergence rate of the target transfer Q-learning. We prove that if the two tasks are similar with respect to the MDPs, the optimal Q-functions in the source and new RL tasks are similar which means the error of the transferred target Q-function in new MDP is small. Also, the convergence rate analysis shows that the target transfer Q-Learning will converge faster than Q-learning if the error of the transferred target Q-function is smaller than the current Q-function in the new task. Based on our theoretical results, we design the safe condition as the Bellman error of the transferred target Q-function is less than the current Q-function. Our experiments are consistent with our theoretical founding and verified the effectiveness of our proposed target transfer Q-learning method.

Motivation & Objective

• To address the slow convergence of Q-learning in reinforcement learning due to high sample complexity.
• To develop a transfer learning method for RL that provides theoretical convergence guarantees, unlike most existing heuristic approaches.
• To identify safe conditions under which transferring a pre-trained Q-function from a source task improves learning in a new target task.
• To establish a theoretical framework linking the similarity of MDPs to the convergence speed of transfer Q-learning.
• To design a practical, verifiable safety condition based on Bellman error to ensure algorithmic stability and convergence.

Proposed method

• Proposes Target Transfer Q-Learning (TTQL), which uses the Q-function from a source task as the target in the temporal difference update of the target task.
• Introduces a safe condition based on the Bellman error: the transferred Q-function's maximum Bellman error must be less than or equal to that of the current Q-function in the target task.
• Uses the discounted relative Q-function error ratio $\gamma\beta$ as a key metric to analyze convergence rate, where $\beta$ is the relative error of the transferred Q-function.
• Derives theoretical bounds showing that convergence is guaranteed only if $\gamma\beta < 1$, and faster convergence occurs when $\gamma\beta$ is small.
• Establishes a mathematical relationship between the maximum norm error (MNE) and maximum Bellman error (MNBE): $\mathbf{MNE}(Q) \leq \frac{\mathbf{MNBE}(Q)}{1 - \gamma}$, enabling practical estimation.
• Employs a two-stage analysis: the error in TTQL consists of initialization error and sampling error, both dependent on $\gamma\beta$.

Experimental results

Research questions

• RQ1 Under what conditions does transferring a Q-function from a source task to a target task improve learning speed in Q-learning?
• RQ2 How can we theoretically guarantee the convergence of transfer Q-learning when transferring knowledge across tasks?
• RQ3 What metric can serve as a safe, verifiable condition to prevent harmful transfer and ensure convergence in transfer Q-learning?
• RQ4 How does the similarity between source and target MDPs affect the convergence rate of TTQL?
• RQ5 Can the Bellman error of the transferred Q-function be used as a practical proxy for the relative error in the Q-function, enabling real-time safety checks?

Key findings

• TTQL converges faster than standard Q-learning when the optimal Q-functions of the source and target tasks are similar, as measured by the discounted relative error ratio $\gamma\beta$.
• The convergence of TTQL is guaranteed only if the discounted relative error ratio $\gamma\beta < 1$, and convergence slows as $\gamma\beta$ approaches 1.
• The safe condition based on Bellman error ($\mathbf{MNBE}(Q_1^*) \leq \mathbf{MNBE}(Q_n)$) ensures that the transferred Q-function is not worse than the current estimate, preventing degradation.
• Experiments confirm that TTQL outperforms standard Q-learning across all MDP variations (in $\gamma$, $r$, $P$), with faster convergence on more similar source-target pairs.
• The safe condition is essential: TTQL without the condition fails to converge in some cases, validating its necessity for stability.
• The theoretical relationship $\mathbf{MNE}(Q) \leq \frac{\mathbf{MNBE}(Q)}{1 - \gamma}$ enables practical estimation of Q-function error using Bellman error, which is computationally feasible.