Environment-related difference of Deep Q-Learning and Double Deep Q-Learning

This project was conducted for the Reinforcement Learning course of the Master Artificial Intelligence at the University of Amsterdam during the Winter term 2018/2019.

Samples of the performance of Double Deep Q Networks on different environments. While it is able to solve most of the tasks at hand, it performed visibly worse for Pendulum, where the goal is to balance it on top.

Introduction

The Q-learning algorithm is known to overestimate state-action values under certain conditions. A positive bias is introduced, as Q-learning uses the same function to select actions and evaluate a state-action pair. Hence, overoptimistic values are more likely to be used in updates. Hasselt (2016) showed that this can indeed be harmful for performance using Deep Neural Networks and proposed the Double Deep Q Networks algorithm. We apply both algorithms and compare their performance on different environments provided by Open AI Gym (Brockman et al., 2016).

Experiments

We used the environments CartPole-v1, Acrobot-v1, MountainCar-v0, Pendulum-v0 from Open AI Gym and performed joined grid search over the hyperparameters on random seeds for both models with 10 models each , the best configuration was selected based on the highest overall rewards achieved. The hyperparameters can be found in hyperparameters.py.

Afterwards, we trained 15 models on each environment and used a two-tailed Mann-Whitney U (Nachar et al., 2008) test to test whether the significance in Q-value estimates is statistically significant (p=0.05). The test was chosen because after performing Shapiro-Wilk test for normality, most of the distributions could not be shown to be gaussian (p=0.05).

Results

The figures below show the average of Q-Value estimates during training for DQN and DDQN on four different environments over 15 different models. Intervals are determined by averaging the two most extreme values during every time step. The true values are determined by applying Monte Carlo rollout where the whole episode is played out to compute the total rewards following the policy found after training terminated. These true returns are indicated by the dashed lines. Markers (bottom of x-axis) indicate time steps with statistically significant differences between DQN and DDQN (p = 0.05).

We observe the following things:

• Both algorithms perform well on CartPole-v1; environment less challenging due to easy credit assignment (immediate, positive and constant rewards)
• Pendulum-v0: Both get similar Q-value estimates, but DQN performs better than DDQN. The reason for these observations might be due to the complex reward function requiring careful actions
• We can confirm the findings of Hasselt et al. (2016) for Acrobot-v1: DDQN performs better on this task while also having Q-estimates that are closer to the true values
• Although both algorithms learn to solve the environment MountainCar-v0 , we suspect that using Deep Neural Networks to parameterize the Q-function leads to unstable and unrealistic Q-values (non-negative), likely influenced by "deadly triad" (Hasselt et al., 2018).

Conclusion

• Only Acrobot-v1 shows significant performance improvement when using DDQN
• Whether DDQN improves performance depends on the reward structure of the environment, in our environments the reward distributions are very narrow, so there is not much room for maximization bias to occur
• Due to parameterization with Deep Neural Networks, Q-values can be unstable while still achieving the objective (the "deathly triad" of Reinforcement Learning: Bootstrapping, off-policy learning and function approximation)
• Unrealistic q-value estimates don't necessarily translate into bad performance (e.g. most MountainCar models were still able to solve the environment although using wildly unrealistic estimates)
• Function approximation (Neural Network), bootstrapping and off-policy learning (“deadly triad”) can lead to unstable Q-values while still achieving the objective

Usage

To run the experiments yourself, first install all the necessary requirements

cd quirky-quokka
pip3 install -r requirements.txt

Then run the training script:

python3 train.py

Models will be saved to the models/ folder, plots and the data they are based on into img/ and data/ respectively. Lastly, trained models can also be tested by rendering their environments using

cd models
python3 ../test.py <model_name>.pt

where the environment name has to be included in <model_name>, e.g.

python3 ../test.py Acrobot-v1_ddqn0.pt

References

Hado V Hasselt. Double q-learning. In Advances in Neural Information Processing Systems, pages 2613–2621, 2010.

Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym. arXiv preprint arXiv:1606.01540, 2016.

Nadim Nachar et al. The mann-whitney u: A test for assessing whether two independent samples come from the same distribution. Tutorials in Quantitative Methods for Psychology, 4(1):13–20, 2008.

Hado van Hasselt, Arthur Guez, and David Silver. Deep reinforcement learning with double q-learning. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, AAAI’16, pages 2094–2100. AAAI Press, 2016.

Hado van Hasselt, Yotam Doron, Florian Strub, Matteo Hessel, Nicolas Sonnerat, and Joseph Modayil. Deep reinforcement learning and the deadly triad. arXiv preprint arXiv:1812.02648, 2018.

Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature 518(7540):529, 2015.