- 论文题目：Deterministic Policy Gradient Algorithms
所解决的问题？
stochastic policy的方法由于含有部分随机，所以效率不高，方差大，采用deterministic policy方法比stochastic policy的采样效率高，但是没有办法探索环境，因此只能采用off-policy的方法来进行了。
背景
以往的action是一个动作分布$\pi_{\theta}(a|s)$，作者所提出的是输出一个确定性的策略(deterministic policy) $a =\mu_{\theta}(s)$。
In the stochastic case，the policy gradient integrates over both state and action spaces, whereas in the deterministic case it only integrates over the state space.
- Stochastic Policy Gradient
前人采用off-policy的随机策略方法， behaviour policy $\beta(a|s) \neq \pi_{\theta}(a|s)$：
$$ \begin{aligned} J_{\beta}\left(\pi_{\theta}\right) &=\int_{\mathcal{S}} \rho^{\beta}(s) V^{\pi}(s) \mathrm{d} s \\ &=\int_{\mathcal{S}} \int_{\mathcal{A}} \rho^{\beta}(s) \pi_{\theta}(a | s) Q^{\pi}(s, a) \mathrm{d} a \mathrm{d} s \end{aligned} $$
Differentiating the performance objective and applying an approximation gives the off-policy policy-gradient (Degris et al., 2012b)
$$ \begin{aligned} \nabla_{\theta} J_{\beta}\left(\pi_{\theta}\right) & \approx \int_{\mathcal{S}} \int_{\mathcal{A}} \rho^{\beta}(s) \nabla_{\theta} \pi_{\theta}(a | s) Q^{\pi}(s, a) \mathrm{d} a \mathrm{d} s \\ &=\mathbb{E}_{s \sim \rho^{\beta}, a \sim \beta}\left[\frac{\pi_{\theta}(a | s)}{\beta_{\theta}(a | s)} \nabla_{\theta} \log \pi_{\theta}(a | s) Q^{\pi}(s, a)\right] \end{aligned} $$
This approximation drops a term that depends on the action-value gradient $\nabla_{\theta}Q^{\pi}(s,a)$; (Degris et al., 2012b)
$\mu_{\theta}(s)$ 更新公式：
$$ \theta^{k+1}=\theta^{k}+\alpha \mathbb{E}_{s \sim \rho^{\mu^{k}}} \left[\nabla_{\theta} Q^{\mu^{k}}\left(s, \mu_{\theta}(s)\right)\right] $$
引入链导法则：
$$ \theta^{k+1}=\theta^{k}+\alpha \mathbb{E}_{s \sim \rho^{\mu^{k}}} \left[\nabla_{\theta} \mu_{\theta}(s) \nabla_{a}Q^{\mu^{k}}\left(s, a\right) |_{a=\mu_{\theta}(s)} \right] $$
所采用的方法？
- On-Policy Deterministic Actor-Critic
如果环境有大量噪声帮助智能体做exploration的话，这个算法还是可以的，使用sarsa更新critic，使用 $Q^{w}(s,a)$ 近似true action-value $Q^{\mu}$：
$$ \begin{aligned} \delta_{t} &=r_{t}+\gamma Q^{w}\left(s_{t+1}, a_{t+1}\right)-Q^{w}\left(s_{t}, a_{t}\right) \\ w_{t+1} &=w_{t}+\alpha_{w} \delta_{t} \nabla_{w} Q^{w}\left(s_{t}, a_{t}\right) \\ \theta_{t+1} &=\theta_{t}+\left.\alpha_{\theta} \nabla_{\theta} \mu_{\theta}\left(s_{t}\right) \nabla_{a} Q^{w}\left(s_{t}, a_{t}\right)\right|_{a=\mu_{\theta}(s)} \end{aligned} $$
- Off-Policy Deterministic Actor-Critic
we modify the performance objective to be the value function of the target policy, averaged over the state distribution of the behaviour policy
$$ \begin{aligned} J_{\beta}\left(\mu_{\theta}\right) &=\int_{\mathcal{S}} \rho^{\beta}(s) V^{\mu}(s) \mathrm{d} s \\ &=\int_{\mathcal{S}} \rho^{\beta}(s) Q^{\mu}\left(s, \mu_{\theta}(s)\right) \mathrm{d} s \end{aligned} $$
$$ \begin{aligned} \nabla_{\theta} J_{\beta}\left(\mu_{\theta}\right) & \approx \int_{\mathcal{S}} \rho^{\beta}(s) \nabla_{\theta} \mu_{\theta}(a | s) Q^{\mu}(s, a) \mathrm{d} s \\ &=\mathbb{E}_{s \sim \rho^{\beta}} [\nabla_{\theta} \mu_{\theta}(s) \nabla_{a}Q^{\mu}(s,a)|_{a =\mu_{\theta}(s)}] \end{aligned} $$
得到off-policy deterministic actorcritic (OPDAC) 算法：
$$ \begin{aligned} \delta_{t} &=r_{t}+\gamma Q^{w}\left(s_{t+1}, \mu_{\theta}\left(s_{t+1}\right)\right)-Q^{w}\left(s_{t}, a_{t}\right) \\ w_{t+1} &=w_{t}+\alpha_{w} \delta_{t} \nabla_{w} Q^{w}\left(s_{t}, a_{t}\right) \\ \theta_{t+1} &=\theta_{t}+\left.\alpha_{\theta} \nabla_{\theta} \mu_{\theta}\left(s_{t}\right) \nabla_{a} Q^{w}\left(s_{t}, a_{t}\right)\right|_{a=\mu_{\theta}(s)} \end{aligned} $$
与stochastic off policy算法不同的是由于这里是deterministic policy，所以不需要用重要性采样(importance sampling)。
取得的效果？
所出版信息？作者信息？
这篇文章是ICML2014上面的一篇文章。第一作者David Silver是Google DeepMind的research Scientist，本科和研究生就读于剑桥大学，博士于加拿大阿尔伯特大学就读，2013年加入DeepMind公司，AlphaGo创始人之一，项目领导者。
参考链接
- 参考文献：Degris, T., White, M., and Sutton, R. S. (2012b). Linear off-policy actor-critic. In 29th International Conference on Machine Learning.
扩展阅读
假定真实的action-value function为 $Q^{\pi}(s,a)$，用一个function近似它 $Q^{w}(s,a) \approx Q^{\pi}(s,a)$。However, if the function approximator is compatible such that 1. $Q^{w}(s, a)=\nabla_{\theta} \log \pi_{\theta}(a | s)^{\top} w$ (linear in "fearure") 2. the parameters $w$ are chosen to minimise the mean-squared error $\varepsilon^{2}(w) = \mathbb{E}_{s \sim \rho^{\pi},a \sim \pi_{\theta}}[(Q^{w}(s,a)-Q^{\pi}(s,a))^{2}]$ (linear regression problem form these feature )，then there is no bias (Sutton et al., 1999),
$$ \nabla_{\theta} J\left(\pi_{\theta}\right)=\mathbb{E}_{s \sim \rho^{\pi}, a \sim \pi_{\theta}}\left[\nabla_{\theta} \log \pi_{\theta}(a | s) Q^{w}(s, a)\right] $$
最后，论文给出了DPG的采用线性函数逼近定理，以及一些理论证明基础。
- 参考文献：Sutton, R.S., McAllester D. A., Singh, S. P., and Mansour, Y. (1999). Policy gradient methods for reinforcement learning with function approximation. In Neural Information Processing Systems 12, pages 1057–1063.
这篇文章以后有时间再读一遍吧，里面还是有些证明需要仔细推敲一下。
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