Learning Gittins Index via Tabular and Deep Reinforcement Learning for Optimal Job Scheduling

Core Concepts

The authors propose tabular (QGI) and deep reinforcement learning (DGN) algorithms to efficiently learn the Gittins index, which is the optimal policy for multi-armed bandit problems. They demonstrate the effectiveness of their algorithms in learning the optimal scheduling policy that minimizes mean flowtime for a batch of jobs with unknown service time distributions.

Abstract

The content discusses the problem of multi-armed bandit (MAB) and the Gittins index, which is the optimal policy for maximizing the expected total discounted reward in MAB problems. However, in most realistic scenarios, the Markovian state transition probabilities are unknown, and reinforcement learning (RL) algorithms are required to learn the Gittins indices.
The authors propose two RL-based algorithms:

Tabular QGI (Q-learning for Gittins Index): This algorithm is based on the retirement formulation of the MAB problem. It updates the Q-values and the Gittins index estimates in a two-timescale manner, leading to lower runtime and better convergence compared to existing RL algorithms.

Deep RL DGN (Deep Gittins Network): This algorithm uses a deep neural network to approximate the state-action value function, along with a separate stochastic approximation update for the Gittins index estimates. The DGN algorithm also leverages a Double DQN architecture for better convergence.

The authors demonstrate the effectiveness of their algorithms on elementary examples and then apply them to the problem of learning the optimal scheduling policy that minimizes the mean flowtime for a batch of jobs with unknown service time distributions. They show that their algorithms outperform the existing RL-based methods in terms of convergence speed, memory requirements, and empirical regret.

Stats

The authors provide the following key figures and metrics:

The time complexity per iteration for QGI is O(N), compared to O(2N + N) for the restart-in-state algorithm and O(NK + N) for QWI.
The space complexity for QGI is O(N × N), compared to O(2 × N × N × K) for the restart-in-state algorithm and QWI.
The authors demonstrate the convergence of Gittins indices for different job size distributions, including Geometric, Binomial, Poisson, Uniform, and Log-normal.
They also show the percentage of optimal actions chosen and the cumulative episodic regret for QGI and the restart-in-state algorithm, with QGI outperforming the existing method.

Quotes

None.

Key Insights Distilled From

Tabular and Deep Reinforcement Learning for Gittins Index

by Harshit Dhan... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.01157.pdf

Deeper Inquiries

How can the proposed algorithms be extended to handle restless multi-armed bandit problems, where the passive arms also undergo Markovian transitions

To extend the proposed algorithms to handle restless multi-armed bandit problems where passive arms also undergo Markovian transitions, we can modify the state transition probabilities and reward structures to account for the passive arms' dynamics. In the context of the Gittins index policy, the Whittle's index policy is often used in restless bandit problems. By incorporating the passive arms' transitions into the state space and updating the Q-values and Gittins indices accordingly, we can adapt the QGI and DGN algorithms to handle this scenario. This adaptation would involve adjusting the update equations to consider the transitions of both active and passive arms, ensuring that the learning process captures the changing dynamics of the system accurately.

Can a policy gradient-based approach be used to learn the Gittins index, and how would it compare to the value function-based methods presented in this work

While the proposed algorithms in this work are based on value function methods for learning the Gittins index, a policy gradient-based approach could offer an alternative perspective. Policy gradient methods directly optimize the policy function without explicitly estimating the value function. By using policy gradients, we can learn the optimal policy for selecting arms in the multi-armed bandit problem without the need for explicit value function estimation. A comparison between policy gradient-based methods and the value function-based methods presented in this work would be interesting to explore. Policy gradient methods may offer advantages in terms of convergence speed, sample efficiency, and robustness to hyperparameters, providing a complementary approach to learning the Gittins index.

Are there any other applications or problem domains where the retirement formulation-based algorithms could be advantageous compared to the existing RL methods for learning the Gittins index

The retirement formulation-based algorithms proposed in this work could be advantageous in various applications and problem domains where the Gittins index policy is applicable. One such domain could be dynamic resource allocation in cloud computing environments. By modeling the resource allocation decisions as a multi-armed bandit problem with Markovian transitions, the proposed algorithms could efficiently learn the optimal allocation strategy to maximize resource utilization and minimize response times. Additionally, in dynamic pricing scenarios where the pricing decisions need to adapt to changing market conditions, the retirement formulation-based algorithms could be beneficial in learning the optimal pricing strategy over time. The ability to handle unknown state transition probabilities and learn the Gittins index in real-time makes these algorithms versatile and suitable for dynamic decision-making problems across various domains.