Merge pull request #505 from wanghaoshuang/rl

Add a Policy Gradients example implemented by Paddle Fluid.

fluid/policy_gradient/README.md

+# Policy Gradient RL by PaddlePaddle
+本文介绍了如何使用PaddlePaddle通过policy-based的强化学习方法来训练一个player（actor model）, 我们希望这个player可以完成简单的走阶梯任务。
+
+ 内容分为:
+
+ - 任务描述
+ -  模型
+ -  策略（目标函数）
+ -  算法（Gradient ascent）
+ -  PaddlePaddle实现
+
+
+## 1. 任务描述
+假设有一个阶梯，连接A、B点，player从A点出发，每一步只能向前走一步或向后走一步，到达B点即为完成任务。我们希望训练一个聪明的player，它知道怎么最快的从A点到达B点。
+我们在命令行以下边的形式模拟任务：
+```
+A - O - - - - - B
+```
+一个‘-'代表一个阶梯，A点在行头，B点在行末，O代表player当前在的位置。
+
+## 2. Policy Gradient
+### 2.1 模型
+#### inputyer
+模型的输入是player观察到的当前阶梯的状态$S$, 要包含阶梯的长度和player当前的位置信息。
+在命令行模拟的情况下，player的位置和阶梯长度连个变量足以表示当前的状态，但是我们为了便于将这个demo推广到更复杂的任务场景，我们这里用一个向量来表示游戏状态$S$.
+向量$S$的长度为阶梯的长度，每一维代表一个阶梯，player所在的位置为1，其它位置为0.
+下边是一个例子：
+```
+S = [0, 1, 0, 0]  // 阶梯长度为4，player在第二个阶梯上。
+```
+#### hidden layer
+隐藏层采用两个全连接layer `FC_1`和`FC_2`, 其中`FC_1` 的size为10， `FC_2`的size为2.
+
+#### output layer
+我们使用softmax将`FC_2`的output映射为所有可能的动作（前进或后退）的概率分布（Probability of taking the action）,即为一个二维向量`act_probs`, 其中，`act_probs[0]` 为后退的概率，`act_probs[1]`为前进的概率。
+
+#### 模型表示
+我将我们的player模型(actor)形式化表示如下：
+$$a = \pi_\theta(s)$$
+其中$\theta$表示模型的参数，$s$是输入状态。
+
+
+### 2.2 策略（目标函数）
+我们怎么评估一个player(模型)的好坏呢？首先我们定义几个术语：
+我们让$\pi_\theta(s)$来玩一局游戏，$s_t$表示第$t$时刻的状态，$a_t$表示在状态$s_t$做出的动作，$r_t$表示做过动作$a_t$后得到的奖赏。
+一局游戏的过程可以表示如下：
+$$\tau = [s_1, a_1, r_1, s_2, a_2, r_2 ... s_T, a_T, r_T] \tag{1}$$
+
+一局游戏的奖励表示如下：
+$$R(\tau) = \sum_{t=1}^Tr_t$$
+
+player玩一局游戏，可能会出现多种操作序列$\tau$ ,某个$\tau$出现的概率是依赖于player model的$\theta$, 记做：
+$$P(\tau | \theta)$$
+那么，给定一个$\theta$(player model), 玩一局游戏，期望得到的奖励是：
+$$\overline {R}_\theta = \sum_\tau R(\tau)\sum_\tau R(\tau) P(\tau|\theta)$$
+大多数情况，我们无法穷举出所有的$\tau$，所以我们就抽取N个$\tau$来计算近似的期望：
+$$\overline {R}_\theta = \sum_\tau R(\tau) P(\tau|\theta) \approx \frac{1}{N} \sum_{n=1}^N R(\tau^n)$$
+
+$\overline {R}_\theta$就是我们需要的目标函数，它表示了一个参数为$\theta$的player玩一局游戏得分的期望，这个期望越大，代表这个player能力越强。
+### 2.3 算法（Gradient ascent）
+我们的目标函数是$\overline {R}_\theta$, 我们训练的任务就是, 我们训练的任务就是:
+$$\theta^* = \arg\max_\theta  \overline {R}_\theta$$
+
+为了找到理想的$\theta$，我们使用Gradient ascent方法不断在$\overline {R}_\theta$的梯度方向更新$\theta$，可表示如下：
+$$\theta' = \theta + \eta * \bigtriangledown \overline {R}_\theta$$
+
+$$ \bigtriangledown \overline {R}_\theta =  \sum_\tau R(\tau)  \bigtriangledown P(\tau|\theta)\\
+= \sum_\tau R(\tau) P(\tau|\theta) \frac{\bigtriangledown P(\tau|\theta)}{P(\tau|\theta)} \\
+=\sum_\tau R(\tau) P(\tau|\theta) {\bigtriangledown \log P(\tau|\theta)} $$
+
+
+$$P(\tau|\theta) = P(s_1)P(a_1|s_1,\theta)P(s_2, r_1|s_1,a_1)P(a_2|s_2,\theta)P(s_3,r_2|s_2,a_2)...P(a_t|s_t,\theta)P(s_{t+1}, r_t|s_t,a_t)\\
+=P(s_1) \sum_{t=1}^T P(a_t|s_t,\theta)P(s_{t+1}, r_t|s_t,a_t)$$
+
+$$\log P(\tau|\theta) =  \log P(s_1)  + \sum_{t=1}^T [\log P(a_t|s_t,\theta) + \log P(s_{t+1}, r_t|s_t,a_t)]$$
+
+$$ \bigtriangledown \log P(\tau|\theta) =  \sum_{t=1}^T \bigtriangledown \log P(a_t|s_t,\theta)$$
+
+$$ \bigtriangledown \overline {R}_\theta =  \sum_\tau R(\tau) P(\tau|\theta) {\bigtriangledown \log P(\tau|\theta)} \\
+\approx  \frac{1}{N} \sum_{n=1}^N R(\tau^n) {\bigtriangledown \log P(\tau|\theta)} \\
+= \frac{1}{N} \sum_{n=1}^N R(\tau^n) {\sum_{t=1}^T \bigtriangledown \log P(a_t|s_t,\theta)} \\
+= \frac{1}{N} \sum_{n=1}^N \sum_{t=1}^T R(\tau^n) { \bigtriangledown \log P(a_t|s_t,\theta)} \tag{11}$$
+
+#### 2.3.2 导数解释
+
+在使用深度学习框架进行训练求解时，一般用梯度下降方法，所以我们把Gradient ascent转为Gradient
+descent, 重写等式$(5)(6)$为：
+
+$$\theta^* = \arg\min_\theta  (-\overline {R}_\theta \tag{13}$$
+$$\theta' = \theta - \eta * \bigtriangledown (-\overline {R}_\theta)) \tag{14}$$
+
+根据上一节的推导，$ (-\bigtriangledown \overline {R}_\theta) $结果如下：
+
+$$ -\bigtriangledown \overline {R}_\theta
+= \frac{1}{N} \sum_{n=1}^N \sum_{t=1}^T R(\tau^n) { \bigtriangledown -\log P(a_t|s_t,\theta)} \tag{15}$$
+
+根据等式(14), 我们的player的模型可以设计为：
+
+<p align="center">
+<img src="images/PG_1.svg" width="620" hspace='10'/> <br/>
+图 1
+</p>
+
+用户的在一局游戏中的一次操作可以用元组$(s_t, a_t)$, 就是在状态$s_t$状态下做了动作$a_t$, 我们通过图(1)中的前向网络计算出来cross entropy cost为$−\log P(a_t|s_t,\theta)$， 恰好是等式(15)中我们需要微分的一项。
+图1是我们需要的player模型，我用这个网络的前向计算可以预测任何状态下该做什么动作。但是怎么去训练学习这个网络呢？在等式(15)中还有一项$R(\tau^n)$, 我做反向梯度传播的时候要加上这一项，所以我们需要在图1基础上再加上$R(\tau^n)$， 如 图2 所示：
+
+<p align="center">
+<img src="images/PG_2.svg" width="620" hspace='10'/> <br/>
+图 2
+</p>
+
+图2就是我们最终的网络结构。
+
+#### 2.3.3 直观理解
+对于等式(15),我只看游戏中的一步操作，也就是这一项: $R(\tau^n) { \bigtriangledown -\log P(a_t|s_t,\theta)}$,  我们可以简单的认为我们训练的目的是让 $R(\tau^n) {[ -\log P(a_t|s_t,\theta)]}$尽可能的小，也就是$R(\tau^n) \log P(a_t|s_t,\theta)$尽可能的大。
+
+-  如果我们当前游戏局的奖励$R(\tau^n)$为正，那么我们希望当前操作的出现的概率$P(a_t|s_t,\theta)$尽可能大。
+-  如果我们当前游戏局的奖励$R(\tau^n)$为负，那么我们希望当前操作的出现的概率$P(a_t|s_t,\theta)$尽可能小。
+
+#### 2.3.4 一个问题
+
+一人犯错，诛连九族。一人得道，鸡犬升天。如果一局游戏得到奖励，我们希望帮助获得奖励的每一次操作都被重视；否则，导致惩罚的操作都要被冷落一次。
+是不是很有道理的样子？但是，如果有些游戏场景只有奖励，没有惩罚，怎么办？也就是所有的$R(\tau^n)$都为正。
+针对不同的游戏场景，我们有不同的解决方案：
+
+1. 每局游戏得分不一样：将每局的得分减去一个bias，结果就有正有负了。
+2. 每局游戏得分一样：把完成一局的时间作为计分因素，并减去一个bias.
+
+我们在第一章描述的游戏场景，需要用第二种 ，player每次到达终点都会收到1分的奖励，我们可以按完成任务所用的步数来定义奖励R.
+更进一步，我们认为一局游戏中每步动作对结局的贡献是不同的，有聪明的动作，也有愚蠢的操作。直观的理解，一般是靠前的动作是愚蠢的，靠后的动作是聪明的。既然有了这个价值观，那么我们拿到1分的奖励，就不能平均分给每个动作了。
+如图3所示，让所有动作按先后排队，从后往前衰减地给每个动作奖励，然后再每个动作的奖励再减去所有动作奖励的平均值：
+
+<p align="center">
+<img src="images/PG_3.svg" width="620" hspace='10'/> <br/>
+图 3
+</p>
+
+## 3. 训练效果
+
+demo运行训练效果如下，经过1000轮尝试，我们的player就学会了如何有效的完成任务了：
+
+```
+---------O    epoch: 0; steps: 42
+---------O    epoch: 1; steps: 77
+---------O    epoch: 2; steps: 82
+---------O    epoch: 3; steps: 64
+---------O    epoch: 4; steps: 79
+---------O    epoch: 501; steps: 19
+---------O    epoch: 1001; steps: 9
+---------O    epoch: 1501; steps: 9
+---------O    epoch: 2001; steps: 11
+---------O    epoch: 2501; steps: 9
+---------O    epoch: 3001; steps: 9
+---------O    epoch: 3002; steps: 9
+---------O    epoch: 3003; steps: 9
+---------O    epoch: 3004; steps: 9
+---------O    epoch: 3005; steps: 9
+---------O    epoch: 3006; steps: 9
+---------O    epoch: 3007; steps: 9
+---------O    epoch: 3008; steps: 9
+---------O    epoch: 3009; steps: 9
+---------O    epoch: 3010; steps: 11
+---------O    epoch: 3011; steps: 9
+---------O    epoch: 3012; steps: 9
+---------O    epoch: 3013; steps: 9
+---------O    epoch: 3014; steps: 9
+```

fluid/policy_gradient/brain.py

+import numpy as np
+import paddle.v2 as paddle
+import paddle.v2.fluid as fluid
+# reproducible
+np.random.seed(1)
+
+
+class PolicyGradient:
+    def __init__(
+            self,
+            n_actions,
+            n_features,
+            learning_rate=0.01,
+            reward_decay=0.95,
+            output_graph=False, ):
+        self.n_actions = n_actions
+        self.n_features = n_features
+        self.lr = learning_rate
+        self.gamma = reward_decay
+
+        self.ep_obs, self.ep_as, self.ep_rs = [], [], []
+
+        self.place = fluid.CPUPlace()
+        self.exe = fluid.Executor(self.place)
+
+    def build_net(self):
+
+        obs = fluid.layers.data(
+            name='obs', shape=[self.n_features], dtype='float32')
+        acts = fluid.layers.data(name='acts', shape=[1], dtype='int64')
+        vt = fluid.layers.data(name='vt', shape=[1], dtype='float32')
+        # fc1
+        fc1 = fluid.layers.fc(
+            input=obs,
+            size=10,
+            act="tanh"  # tanh activation
+        )
+        # fc2
+        self.all_act_prob = fluid.layers.fc(input=fc1,
+                                            size=self.n_actions,
+                                            act="softmax")
+        # to maximize total reward (log_p * R) is to minimize -(log_p * R)
+        neg_log_prob = fluid.layers.cross_entropy(
+            input=self.all_act_prob,
+            label=acts)  # this is negative log of chosen action
+        neg_log_prob_weight = fluid.layers.elementwise_mul(x=neg_log_prob, y=vt)
+        loss = fluid.layers.reduce_mean(
+            x=neg_log_prob_weight)  # reward guided loss
+
+        sgd_optimizer = fluid.optimizer.SGD(self.lr)
+        sgd_optimizer.minimize(loss)
+        self.exe.run(fluid.default_startup_program())
+
+    def choose_action(self, observation):
+        prob_weights = self.exe.run(
+            fluid.default_main_program().prune(self.all_act_prob),
+            feed={"obs": observation[np.newaxis, :]},
+            fetch_list=[self.all_act_prob])
+        prob_weights = np.array(prob_weights[0])
+        action = np.random.choice(
+            range(prob_weights.shape[1]),
+            p=prob_weights.ravel())  # select action w.r.t the actions prob
+        return action
+
+    def store_transition(self, s, a, r):
+        self.ep_obs.append(s)
+        self.ep_as.append(a)
+        self.ep_rs.append(r)
+
+    def learn(self):
+        # discount and normalize episode reward
+        discounted_ep_rs_norm = self._discount_and_norm_rewards()
+        tensor_obs = np.vstack(self.ep_obs).astype("float32")
+        tensor_as = np.array(self.ep_as).astype("int64")
+        tensor_as = tensor_as.reshape([tensor_as.shape[0], 1])
+        tensor_vt = discounted_ep_rs_norm.astype("float32")[:, np.newaxis]
+        # train on episode
+        self.exe.run(
+            fluid.default_main_program(),
+            feed={
+                "obs": tensor_obs,  # shape=[None, n_obs]
+                "acts": tensor_as,  # shape=[None, ]
+                "vt": tensor_vt  # shape=[None, ]
+            })
+        self.ep_obs, self.ep_as, self.ep_rs = [], [], []  # empty episode data
+        return discounted_ep_rs_norm
+
+    def _discount_and_norm_rewards(self):
+        # discount episode rewards
+        discounted_ep_rs = np.zeros_like(self.ep_rs)
+        running_add = 0
+        for t in reversed(range(0, len(self.ep_rs))):
+            running_add = running_add * self.gamma + self.ep_rs[t]
+            discounted_ep_rs[t] = running_add
+
+        # normalize episode rewards
+        discounted_ep_rs -= np.mean(discounted_ep_rs)
+        discounted_ep_rs /= np.std(discounted_ep_rs)
+        return discounted_ep_rs

fluid/policy_gradient/env.py

+import time
+import sys
+import numpy as np
+
+
+class Env():
+    def __init__(self, stage_len, interval):
+        self.stage_len = stage_len
+        self.end = self.stage_len - 1
+        self.position = 0
+        self.interval = interval
+        self.step = 0
+        self.epoch = -1
+        self.render = False
+
+    def reset(self):
+        self.end = self.stage_len - 1
+        self.position = 0
+        self.epoch += 1
+        self.step = 0
+        if self.render:
+            self.draw(True)
+
+    def status(self):
+        s = np.zeros([self.stage_len]).astype("float32")
+        s[self.position] = 1
+        return s
+
+    def move(self, action):
+        self.step += 1
+        reward = 0.0
+        done = False
+        if action == 0:
+            self.position = max(0, self.position - 1)
+        else:
+            self.position = min(self.end, self.position + 1)
+        if self.render:
+            self.draw()
+        if self.position == self.end:
+            reward = 1.0
+            done = True
+        return reward, done, self.status()
+
+    def draw(self, new_line=False):
+        if new_line:
+            print ""
+        else:
+            print "\r",
+        for i in range(self.stage_len):
+            if i == self.position:
+                sys.stdout.write("O")
+            else:
+                sys.stdout.write("-")
+        sys.stdout.write("    epoch: %d; steps: %d" % (self.epoch, self.step))
+        sys.stdout.flush()
+        time.sleep(self.interval)

fluid/policy_gradient/images/PG_1.svg

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fluid/policy_gradient/images/PG_3.svg

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fluid/policy_gradient/run.py

+from brain import PolicyGradient
+from env import Env
+import numpy as np
+
+n_actions = 2
+interval = 0.01
+stage_len = 10
+epoches = 10000
+
+if __name__ == "__main__":
+
+    brain = PolicyGradient(n_actions, stage_len)
+    e = Env(stage_len, interval)
+    brain.build_net()
+    done = False
+
+    for epoch in range(epoches):
+        if (epoch % 500 == 1) or epoch < 5 or epoch > 3000:
+            e.render = True
+        else:
+            e.render = False
+        e.reset()
+        while not done:
+            s = e.status()
+            action = brain.choose_action(s)
+            r, done, _ = e.move(action)
+            brain.store_transition(s, action, r)
+        done = False
+        brain.learn()