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Title: Temporal Difference Weighted Ensemble For Reinforcement Learning

Paper content:
Combining multiple function approximators in machine learning models typically leads to better performance and robustness compared with a single function.
In reinforcement learning, ensemble algorithms such as an averaging method and a majority voting method are not always optimal, because each function can learn fundamentally different optimal trajectories from exploration.
In this paper, we propose a Temporal Difference Weighted (TDW) algorithm, an ensemble method that adjusts weights of each contribution based on accumulated temporal difference errors.
The advantage of this algorithm is that it improves ensemble performance by reducing weights of Q-functions unfamiliar with current trajectories.
We provide experimental results for Gridworld tasks and Atari tasks that show significant performance improvements compared with baseline algorithms.
Using ensemble methods that combine multiple function approximators can often achieve better performance than a single function by reducing the variance of estimation (Dietterich (2000) ; Kuncheva (2014) ).
Ensemble methods are effective in supervised learning, and also reinforcement learning (Wiering & Van Hasselt (2008) ).
There are two situations where multiple function approximators are combined: combining and learning multiple functions during training (Freund & Schapire (1997) ) and combining individually trained functions to jointly decide actions during testing (Breiman (1996) ).
In this paper, we focus on the second setting of reinforcement learning wherein each function is trained individually and then combined them to achieve better test performance.
Though there is a body of research on ensemble algorithms in reinforcement learning, it is not as sizeable as the research devoted to ensemble methods for supervised learning.
Wiering & Van Hasselt (2008) investigated many ensemble approaches combining several agents with different valuebased algorithms in Gridworld settings.
Faußer & Schwenker (2011; 2015a) have shown that combining value functions approximated by neural networks improves performance greater than using a single agent.
Although previous work dealt with each agent equally contributing to the final output, weighting each contribution based on its accuracy is also a known and accepted approach in supervised learning (Dietterich (2000) ).
However, unlike supervised learning, reinforcement learning agents learn from trajectories resulting from exploration, such that each agent learns from slightly different data.
This characteristic is significant in tasks with high-dimensional state-space, where there are several possible optimal trajectories to maximize cumulative rewards.
In such a situation, the final joint policy function resulting from simple averaging or majority voting is not always optimal if each agent learned different optimal trajectories.
Furthermore, it is difficult to decide constant weights of each contribution as it is possible that agents with poor episode rewards have better performance in specific areas.
In this paper, we propose the temporal difference weighted (TDW) algorithm, an ensemble method for reinforcement learning at test time.
The most important point of this algorithm is that confident agents are prioritized to participate in action selection while contributions of agents unfamiliar with the current trajectory are reduced.
To do so in the TDW algorithm, the weights of the contributions at each Q-function are calculated as softmax probabilities based on accumulated TD errors.
Extending an averaging method and a majority voting method, actions are determined by weighted average or voting methods according to the weights.
The advantage of the TDW algorithm is that arbitrary training algorithms can use this algorithm without any modifications, because the TDW algorithm only cares about the joint decision problem, which could be easily adopted in competitions and development works using reinforcement learning.
In our experiment, we demonstrate that the TDW retains performance in tabular representation Gridworld tasks with multiple possible trajectories, where simple ensemble methods are significantly degraded.
Second, to demonstrate the effectiveness of our TDW algorithm in high-dimensional state-space, we also show that our TDW algorithm can achieve better performance than baseline algorithms in Atari tasks (Bellemare et al. (2013) ).
In this paper, we have introduced the TDW algorithm: an ensemble method that accumulates temporal difference errors as an uncertainties in order to adjust weights of each Q-function, improving performance especially in high-dimensional state-space or situations where there are multiple optimal trajectories.
We have shown performance evaluations in Gridworld tasks and Atari tasks, wherein the TDW algorithms have achieved significantly better performance than non-weighted algorithms and globally weighted algorithms.
However, it is difficult to correctly measure uncertainties with frequent reward occurrences because the intrinsic prediction errors are also accumulated.
Thus, these types of games did not realize the same performance improvements.
In future work, we intend to investigate an extension of this work into continuous action-space tasks because only the joint decision problem of Q-functions is considered in this paper.
We believe a similar algorithm can extend a conventional ensemble method (Huang et al. (2017) ) of Deep Deterministic Policy Gradients (Lillicrap et al. (2015) ) by measuring uncertainties of pairs of a policy function and a Q-function.
We will also consider a separate path, developing an algorithm that measures uncertainties without rewards because reward information is not always available especially in the case of real world application.
A Q-FUNCTION TABLES OBTAINED ON GRIDWORLDS   table
1   0   20   40   60   80   100   table
2   0   20   40   60   80   100   table
3   0   20   40   60   80   100   table
4   0   20   40   60   80   100   table 5   0

assistant: text: Ensemble method for reinforcement learning that weights Q-functions based on accumulated TD errors.