We consider the problem of learning in a non-stationary reinforcement learning (RL) environment, where the setting can be fully described by a piecewise stationary discrete-time Markov decision process (MDP). We introduce a variant of the Restarted Bayesian Online Change-Point Detection algorithm (R-BOCPD) that operates on input streams originating from the more general multinomial distribution and provides near-optimal theoretical guarantees in terms of false-alarm rate and detection delay. Based on this, we propose an improved version of the UCRL2 algorithm for MDPs with state transition kernel sampled from a multinomial distribution, which we call R-BOCPD-UCRL2. We perform a finite-time performance analysis and show that R-BOCPD-UCRL2 enjoys a favorable regret bound of $O\left(D O \sqrt{A T K_T \log\left (\frac{T}{\delta} \right) + \frac{K_T \log \frac{K_T}{\delta}}{\min\limits_\ell \: \mathbf{KL}\left( {\mathbf{\theta}^{(\ell+1)}}\mid\mid{\mathbf{\theta}^{(\ell)}}\right)}}\right)$, where $D$ is the largest MDP diameter from the set of MDPs defining the piecewise stationary MDP setting, $O$ is the finite number of states (constant over all changes), $A$ is the finite number of actions (constant over all changes), $K_T$ is the number of change points up to horizon $T$, and $\mathbf{\theta}^{(\ell)}$ is the transition kernel during the interval $[c_\ell, c_{\ell+1})$, which we assume to be multinomially distributed over the set of states $\mathbb{O}$. Interestingly, the performance bound does not directly scale with the variation in MDP state transition distributions and rewards, ie. can also model abrupt changes. In practice, R-BOCPD-UCRL2 outperforms the state-of-the-art in a variety of scenarios in synthetic environments. We provide a detailed experimental setup along with a code repository (upon publication) that can be used to easily reproduce our experiments.
翻译:本文考虑在非平稳强化学习环境下的学习问题,其中该环境可以完全由一个分段平稳离散时间马尔可夫决策过程描述。我们引入了一种重启贝叶斯在线变点检测算法(Restarted Bayesian Online Change-Point Detection,简称R-BOCPD),该算法适用于来自更一般的多项式分布的输入流,并提供了在误警率和检测延迟方面的近乎最优理论保证。基于此,我们提出了一种针对具有从多项式分布中采样的状态转移核的马尔可夫决策过程的改进版本的UCRL2算法,称为R-BOCPD-UCRL2。我们进行了有限时间性能分析,并展示了R-BOCPD-UCRL2具有有利的遗憾界$O\left(D O \sqrt{A T K_T \log\left (\frac{T}{\delta} \right) + \frac{K_T \log \frac{K_T}{\delta}}{\min\limits_\ell \: \mathbf{KL}\left( {\mathbf{\theta}^{(\ell+1)}}\mid\mid{\mathbf{\theta}^{(\ell)}}\right)}}\right)$,其中$D$是定义分段平稳MDP设置的MDP集合中的最大MDP直径,$O$是有限状态数(在所有变化中恒定),$A$是有限的动作数(在所有变化中恒定),$K_T$是到horizon $T$的变化点数,$\mathbf{\theta}^{(\ell)}$ 表示在时间区间$[c_\ell,c_{\ell+1})$内的转移核,假设其在状态集$\mathbb{O}$上服从多项式分布。有趣的是,该性能上界不直接与MDP状态转移分布和奖励的变化程度相关,也就是可以模型化突然的变化。在实践中,R-BOCPD-UCRL2在各种情景中胜过了现有技术。我们提供了一个详细的实验设置以及一个可用于轻松重现我们实验的代码库(在刊物出版后)。