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【5分钟 Paper】(TD3) Addressing Function Approximation Error in Actor-Critic Methods

  • 论文题目:Addressing Function Approximation Error in Actor-Critic Methods

标题及作者信息

所解决的问题?

  value-base的强化学习值函数的近似估计会过估计值函数(DQN),作者将Double Q-Learning处理过拟合的思想引入actor critic算法中。(过估计的问题就在于累计误差会使得某些不好的statevalue变地很高(exploration 不充分所导致的))。还花了很大的心血在处理过估计问题修正后带来的方差过高的问题。

  作者将过估计的问题引入到continuous action space中,在continuous action space中处理过估计问题的难点在于policychange非常缓慢,导致currenttargetvalue差距不大, too similar to avoid maximization bias

背景

  以往的算法解决过估计问题的就是Double Q Learning那一套,但是这种方法虽然说会降低bias但是会引入高的variance(在选择下一个时刻s‘action的时候,不确定性变得更大才将以往DQNmax这一步变得不是那么max,与之带来的问题就是方差会变大),仍然会对policy的优化起负面作用。作者是用clipped double q learning来解决这个问题。

所采用的方法?

  作者所采用的很多components用于减少方差:

  1. DQN 中的 target network 用于variance reduction by reducing the accumulation of errors(不使用target network的使用是振荡更新的)。
  2. 为了解决valuepolicy耦合的关系,提出了延迟更新(delaying policy updates)的方式。(to address the coupling of value and policy, we propose delaying policy updates until the value estimate has converged)
  3. 提出了novel regularization的更新方式SARSA-style ( the variance reduction by averaging over valueestimates)。这种方法参考的是18Nachum的将值函数smooth能够减少方差的算法。
  • Nachum, O., Norouzi, M., Tucker, G., and Schuurmans, D. Smoothed action value functions for learning gaussian policies. arXiv preprint arXiv:1803.02348, 2018.

  当然multi-step return也能够去权衡方差与偏差之间的关系,还有一些放在文末扩展阅读里面了。

  作者将上述修正方法用于Deep Deterministic Policy Gradient算法中并将其命名为Twin Delayed Deep Deterministic policy gradient (TD3)算法中。一种考虑了在policyvalue 函数近似过程中所带来的一些误差对AC框架所带来的影响。

前人算法回顾

  首先回顾一下DPG算法的更新公式:

\nabla_{\phi} J(\phi)=\mathbb{E}_{s \sim p_{\pi}}\left[\left.\nabla_{a} Q^{\pi}(s, a)\right|_{a=\pi(s)} \nabla_{\phi} \pi_{\phi}(s)\right]

  其中 Q^{\pi}(s,a) = r+\gamma \mathbb{E}_{s^{\prime},a^{\prime}}[Q^{\pi}(s^{\prime},a^{\prime})]Q^{\pi}(s,a)可以用参数 \theta 近似,在DQN中还使用了frozen target network Q_{\theta^{\prime}}(s,a),更新的目标为:

y=r+\gamma Q_{\theta^{\prime}}\left(s^{\prime}, a^{\prime}\right), \quad a^{\prime} \sim \pi_{\phi^{\prime}}\left(s^{\prime}\right)

  如果受误差\varepsilon 干扰,则有:

\mathbb{E}_{\varepsilon}[\max_{a^{\prime}}(Q(s^{\prime},a^{\prime})+\varepsilon)] \geq \max_{a^{\prime}}Q(s^{\prime},a^{\prime})

  在AC框架下,用\phi_{approx}表示actor能获得近似值函数Q_{\theta}(s,a)的近似策略参数(Q_{\theta}(s,a)所对应的那个策略参数),\phi_{true}表示actor能获得真实准确Q^{\pi}(s,a)的参数(which is not known during learning)。

\begin{aligned}
\phi_{\text {approx }} &=\phi+\frac{\alpha}{Z_{1}} \mathbb{E}_{s \sim p_{\pi}}[\nabla_{\phi} \pi_{\phi}(s)   \nabla_{a}  Q_{\theta}(s, a)|_{a=\pi_{\phi} (s)}]\\
\phi_{\text {true }} &=\phi+\frac{\alpha}{Z_{2}} \mathbb{E}_{s \sim p_{\pi}}[\nabla_{\phi} \pi_{\phi}(s)   \nabla_{a}  Q^{\pi}(s, a)|_{a=\pi_{\phi} (s)}]
\end{aligned}

  其中 Z_{1},Z_{2} 是梯度归一化参数,有 Z^{-1}||\mathbb{E[\cdot]}|| =1。这里做归一化的原因就是更容易保证收敛(Without normalized gradients, overestimation bias is still guaranteed to occur with slightly stricter conditions. )。

  由于梯度方向是局部最大化的方向,存在一个足够小的 \varepsilon_{1},使得\alpha \leq \varepsilon_{1}approximate value of \pi_{approx} 会有一个下界 approximate value of \pi_{true}(approximate 会存在过估计问题,就是下面这个式子所描述的)。

\mathbb{E}[Q_{\theta}(s,\pi_{approx}(s))] \geq \mathbb{E}[Q_{\theta}(s,\pi_{true}(s))]

  相反的,存在一个足够小的 \varepsilon_{2} 使得 \alpha \leq \varepsilon_{2}时,the true value of \pi_{approx} 会有一个上界 the true value of\pi_{true} (approximate policy所得出来的动作在真实的action value function中无法达到最优):

\mathbb{E}[Q^{\pi}(s,\pi_{true}(s))] \geq \mathbb{E}[Q^{\pi}(s,\pi_{approx}(s))]

  the value estimate 会大于等于true value \mathbb{E}[Q_{\theta}(s,\pi_{true}(s))] \geq \mathbb{E}[Q^{\pi}(s,\pi_{true}(s))],三式联立有:

\mathbb{E}[Q_{\theta}(s,\pi_{approx}(s))] \geq \mathbb{E}[Q^{\pi}(s,\pi_{approx}(s))]

Clipped Double Q-Learning

  Double DQN中的target

y = r + \gamma Q_{\theta^{\prime}}(s^{\prime},\pi_{\phi}(s^{\prime}))

  Double Q-learning

\begin{array}{l}
y_{1}=r+\gamma Q_{\theta_{2}^{\prime}}\left(s^{\prime}, \pi_{\phi_{1}}\left(s^{\prime}\right)\right) \\
y_{2}=r+\gamma Q_{\theta_{1}^{\prime}}\left(s^{\prime}, \pi_{\phi_{2}}\left(s^{\prime}\right)\right)
\end{array}

  Clipped Double Q-learning

y_{1} = r + \gamma \min_{i=1,2}Q_{\theta_{i}^{\prime}}(s^{\prime},\pi_{\phi_{1}}(s^{\prime}))

  这里的\phi_{1}指的是target actor(可参见伪代码,只用了一个actor)。这种方法会underestimation bias,由于underestimation bias 这种方法就需要加大探索度,不然算法的效率就会很低。

  如果 Q_{\theta_{2}} > Q_{\theta_{1}},那么就相当于辅助的Q_{\theta_{2}}没用到,那么就no additional bias;如果 Q_{\theta_{1}} > Q_{\theta_{2}}那么就会取到Q_{\theta_{2}},作者原文附录里面有证明收敛性

Addressing Variance

  设置target network用于减小policy更新所带的的方差,不然state value approx会很容易发散,不收敛。

  作者使用policy相比于value做延迟更新(Delayed Policy Updates),这样保证策略更新的时候,先将TD误差最小化,这样不会使得policy更新的时候受误差影响,导致其方差高。

Target Policy Smoothing Regularization

  作者认为similar actions should have similar value,所以对某个action周围加上少许噪声能够使得模型泛化能力更强。

\begin{aligned}
y &=r+\gamma Q_{\theta^{\prime}}\left(s^{\prime}, \pi_{\phi^{\prime}}\left(s^{\prime}\right)+\epsilon\right) \\
\epsilon & \sim \operatorname{clip}(\mathcal{N}(0, \sigma),-c, c)
\end{aligned}

  相似的想法在Nachum et al.(2018)上也有设计,不过是smoothing Q_{\theta},不是Q_{\theta^{\prime}}

  • Nachum, O., Norouzi, M., Tucker, G., and Schuurmans, D. Smoothed action value functions for learning gaussian policies. arXiv preprint arXiv:1803.02348, 2018.

算法伪代码:

TD3算法伪代码

取得的效果?

  作者与当前的sota算法对比,结果如下:

过估计实验

image

  作者还验证了target neteork对收敛性的影响:

在这里插入图片描述

  最终的实验:

实验效果

所出版信息?作者信息?

  ICML2018上的一篇文章,Scott Fujimoto is a PhD student at McGill University and Mila. He is the author of TD3 as well as some of the recent developments in batch deep reinforcement learning.

  他还有俩篇论文比较有意思:Off-Policy Deep Reinforcement Learning without ExplorationBenchmarking Batch Deep Reinforcement Learning Algorithms

扩展阅读

  作者为了验证论文的复现性,参考了2017Henderson, P的文章实验了很多随机种子。

  • 参考文献:Henderson, P., Islam, R., Bachman, P., Pineau, J., Precup, D., and Meger, D. Deep Reinforcement Learning that Matters. arXiv preprint arXiv:1709.06560, 2017

  还有一些平衡biasvariance的方法,比如:

  1. importance sampling
  • Precup, D., Sutton, R. S., and Dasgupta, S. Off-policy temporal-difference learning with function approximation. In International Conference on Machine Learning, pp. 417–424, 2001.
  • Munos, R., Stepleton, T., Harutyunyan, A., and Bellemare, M. Safe and efficient off-policy reinforcement learning. In Advances in Neural Information Processing Systems, pp. 1054–1062, 2016.
  1. distributed methods
  • Mnih, V., Badia, A. P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., and Kavukcuoglu, K. Asynchronous methods for deep reinforcement learning. In Internationa lConference on Machine Learning, pp.1928– 1937, 2016.
  • Espeholt, L., Soyer, H., Munos, R., Simonyan, K., Mnih, V., Ward, T., Doron, Y., Firoiu, V., Harley, T., Dunning, I., et al. Impala: Scalable distributed deep-rl with importance weighted actor-learner architectures. arXiv preprint arXiv:1802.01561, 2018.
  1. approximate bounds
  • He, F. S., Liu, Y., Schwing, A. G., and Peng, J. Learning to play in a day: Faster deep reinforcement learning by optimality tightening. arXiv preprint arXiv:1611.01606, 2016.
  1. reduce discount factor to reduce the contribution of each error
  • Petrik, M. and Scherrer, B. Biasing approximate dynamic programming with a lower discount factor. In Advancesin Neural Information Processing Systems, pp. 1265–1272, 2009.

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