Abstract
In multi-objective reinforcement learning (MORL), much attention is paid to generating optimal solution sets for unknown utility functions of users, based on the stochastic reward vectors only. In online MORL on the other hand, the agent will often be able to elicit preferences from the user, enabling it to learn about the utility function of its user directly. In this paper, we study online MORL with user interaction employing the multi-objective multi-armed bandit (MOMAB) setting — perhaps the most fundamental MORL setting. We use Bayesian learning algorithms to learn about the environment and the user simultaneously. Specifically, we propose two algorithms: Utility-MAP UCB (umap-UCB) and Interactive Thompson Sampling (ITS), and show empirically that the performance of these algorithms in terms of regret closely approximates the regret of UCB and regular Thompson sampling provided with the ground truth utility function of the user from the start, and that ITS outperforms umap-UCB.
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Notes
- 1.
We note that logistic regression based on maximum likelihood can lead to problems in earlier iterations of umap-UCB when there is little data available. We observed this empirically. Specifically, in earlier iterations umap-UCB with ML logistic regression instead of Bayesian logistic regression makes an estimate, \(\bar{\mathbf{w}}\), with a sheer-infinite weight on one objective, such that no comparison will be asked from the user again. This can be prevented with a reasonable choice of prior in Bayesian logistic regression.
- 2.
\(\mathbb {E}[\mathbf{x} \cdot \mathbf{y}]= \mathbb {E}[\mathbf{x}] \cdot \mathbb {E}[\mathbf{y}]\), iff \(\mathbf{x}\) and \(\mathbf{y}\) are independent.
- 3.
Please note that for obtaining 0 regret, it is not necessary that the MAP estimate \(\bar{\mathbf{w}}\) is identical to the ground truth \(\mathbf{w}^*\), as long as it leads to selecting the same arm.
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Acknowledgements
The first author is a postdoctoral fellow of the Research Foundation – Flanders (FWO). This research was in part supported by Innoviris – Brussels Institute for Research and Innovation.
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Roijers, D.M., Zintgraf, L.M., Nowé, A. (2017). Interactive Thompson Sampling for Multi-objective Multi-armed Bandits. In: Rothe, J. (eds) Algorithmic Decision Theory. ADT 2017. Lecture Notes in Computer Science(), vol 10576. Springer, Cham. https://doi.org/10.1007/978-3-319-67504-6_2
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