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统计学
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推断
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概率论
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计数
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概率概念
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</a>
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分布
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</span>
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</a>
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贝叶斯
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</span>
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</a>
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信息论
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</span>
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机器学习
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</span>
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经典机器学习
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梯度机器学习
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</span>
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</a>
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深度学习
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</span>
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</a>
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强化学习
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强化学习
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编程任务（使用CoLab或笔记本）
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分布式深度学习
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计算语言学
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计算语言学
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语言学基础
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</a>
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文本处理与经典 NLP
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</span>
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|
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</a>
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嵌入与序列模型
|
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</span>
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|
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|
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</a>
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<span class="md-ellipsis">
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Transformer 与语言模型
|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2007%3A%20computational%20linguistics/05.%20advanced%20text%20generation/" class="md-nav__link">
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高级文本生成
|
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|
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|
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|
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</span>
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|
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|
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|
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</a>
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计算机视觉
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图像基础
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目标检测与分割
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ViT 与生成模型
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视频与 3D 视觉
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音频与语音
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数字信号处理
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</a>
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<span class="md-ellipsis">
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自动语音识别
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</span>
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</a>
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</li>
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<a href="../../chapter%2009%3A%20audio%20and%20speech/03.%20text%20to%20speech%20and%20voice/" class="md-nav__link">
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<span class="md-ellipsis">
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语音合成
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</span>
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</a>
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</li>
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<span class="md-ellipsis">
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说话人与音频分析
|
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|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<span class="md-ellipsis">
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源分离与降噪
|
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|
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</span>
|
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|
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|
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|
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</a>
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</ul>
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</nav>
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<span class="md-ellipsis">
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多模态学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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多模态学习
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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多模态表征
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</span>
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</a>
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视觉语言模型
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</span>
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</a>
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<span class="md-ellipsis">
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图像与视频 Token 化
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</span>
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|
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|
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|
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</a>
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</li>
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跨模态生成
|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2010%3A%20multimodal%20learning/05.%20unified%20multimodal%20architectures/" class="md-nav__link">
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<span class="md-ellipsis">
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统一多模态架构
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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自主系统
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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自主系统
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感知
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</span>
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</a>
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机器人学习
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</span>
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</a>
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<span class="md-ellipsis">
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视觉-语言-动作模型
|
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</span>
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|
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|
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|
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</a>
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<a href="../../chapter%2011%3A%20autonomous%20systems/04.%20self-driving/" class="md-nav__link">
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自动驾驶
|
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</span>
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2011%3A%20autonomous%20systems/05.%20space%20and%20extreme%20robotics/" class="md-nav__link">
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<span class="md-ellipsis">
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太空与极端机器人
|
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|
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|
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|
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</span>
|
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|
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|
||
|
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</a>
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图神经网络
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</span>
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<span class="md-nav__icon md-icon"></span>
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图神经网络
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</label>
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几何深度学习
|
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<span class="md-ellipsis">
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|
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硬件基础
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</a>
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ARM 与 NEON
|
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x86 与 AVX
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Triton、TPU 与 Pallas
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AI 推理
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量化
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边缘推理
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扩缩与部署
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ML 系统设计
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应用 AI
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前沿 AI
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<h1 id="_1">强化学习<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>强化学习通过试错法最大化累积奖励来训练智能体做出序列决策。本文件涵盖MDP、价值函数、贝尔曼方程、Q学习、策略梯度、演员-评论家方法、PPO和RLHF——这些是游戏智能体和语言模型对齐背后的框架。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>监督学习需要标注数据。无监督学习在无标注数据中发现模式。<strong>强化学习（RL）</strong> 与两者都不同：智能体通过与环境的交互、采取行动和接收奖励来学习。没有正确的标签；智能体必须通过试错来发现好的行为。</p>
|
||
</li>
|
||
<li>
|
||
<p>想象教狗一个新把戏。你不会给它展示一个正确行为的数据集。相反，它尝试各种动作，你对好的行为给予奖励，随着时间的推移它明白了你想要什么。RL将这个形式化。</p>
|
||
</li>
|
||
<li>
|
||
<p>RL设置包含五个核心组件。<strong>智能体（agent）</strong> 是学习者和决策者。<strong>环境（environment）</strong> 是智能体之外与之交互的一切。在每个时间步，智能体观察一个<strong>状态（state）</strong> <span class="arithmatex">\(s_t\)</span>，选择一个<strong>动作（action）</strong> <span class="arithmatex">\(a_t\)</span>，接收一个<strong>奖励（reward）</strong> <span class="arithmatex">\(r_t\)</span>，并转移到新状态 <span class="arithmatex">\(s_{t+1}\)</span>。智能体的目标是最大化其随时间收集的总奖励。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="智能体-环境循环：智能体观察状态，采取动作，接收奖励，环境转移到新状态" src="../../images/mdp_agent_loop.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>策略（policy）</strong> <span class="arithmatex">\(\pi\)</span> 是智能体的策略：从状态到动作的映射。确定性策略对每个状态给出一个动作：<span class="arithmatex">\(a = \pi(s)\)</span>。随机策略给出动作上的概率分布：<span class="arithmatex">\(\pi(a \mid s)\)</span>。RL的目标是找到最优策略，即最大化期望累积奖励的策略。</p>
|
||
</li>
|
||
<li>
|
||
<p>RL的数学框架是<strong>马尔可夫决策过程（MDP）</strong>，由元组 <span class="arithmatex">\((S, A, P, R, \gamma)\)</span> 定义：一组状态 <span class="arithmatex">\(S\)</span>，一组动作 <span class="arithmatex">\(A\)</span>，转移概率 <span class="arithmatex">\(P(s' \mid s, a)\)</span>，奖励函数 <span class="arithmatex">\(R(s, a)\)</span>，以及折扣因子 <span class="arithmatex">\(\gamma\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>马尔可夫性质</strong>（来自第05章）指出未来仅取决于当前状态，而不是如何到达那里的历史：<span class="arithmatex">\(P(s_{t+1} \mid s_t, a_t, s_{t-1}, \ldots) = P(s_{t+1} \mid s_t, a_t)\)</span>。这意味着状态包含了做出决策所需的全部信息。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>折扣因子</strong> <span class="arithmatex">\(\gamma \in [0, 1)\)</span> 决定了智能体对未来奖励相对于即时奖励的重视程度。从时间 <span class="arithmatex">\(t\)</span> 开始的折扣回报为：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[G_t = r_t + \gamma r_{t+1} + \gamma^2 r_{t+2} + \cdots = \sum_{k=0}^{\infty} \gamma^k r_{t+k}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>当 <span class="arithmatex">\(\gamma = 0\)</span> 时，智能体完全短视，只关心下一个奖励。当 <span class="arithmatex">\(\gamma\)</span> 接近1时，智能体具有长远眼光。折扣因子还确保了求和收敛（如果奖励有界），这对数学上的良定义性很重要。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>价值函数</strong>估计处于某个状态（或在某个状态下采取某个动作）有多好。<strong>状态价值函数</strong> <span class="arithmatex">\(V^\pi(s)\)</span> 是从状态 <span class="arithmatex">\(s\)</span> 开始并按照策略 <span class="arithmatex">\(\pi\)</span> 行动所获得的期望回报：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[V^\pi(s) = \mathbb{E}_\pi \left[ G_t \mid s_t = s \right]\]</div>
|
||
<ul>
|
||
<li><strong>动作价值函数</strong> <span class="arithmatex">\(Q^\pi(s, a)\)</span> 是从状态 <span class="arithmatex">\(s\)</span> 开始，采取动作 <span class="arithmatex">\(a\)</span>，然后按照 <span class="arithmatex">\(\pi\)</span> 行动所获得的期望回报：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[Q^\pi(s, a) = \mathbb{E}_\pi \left[ G_t \mid s_t = s, a_t = a \right]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>两者关系：<span class="arithmatex">\(V^\pi(s) = \sum_a \pi(a \mid s) \, Q^\pi(s, a)\)</span>。状态价值是动作价值按策略加权的平均值。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>贝尔曼方程</strong>表达了递归关系：一个状态的价值等于即时奖励加上下一个状态的折扣价值。对于状态价值函数：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[V^\pi(s) = \sum_a \pi(a \mid s) \sum_{s'} P(s' \mid s, a) \left[ R(s, a) + \gamma \, V^\pi(s') \right]\]</div>
|
||
<ul>
|
||
<li>对于最优价值函数 <span class="arithmatex">\(V^{*}(s)\)</span>，智能体总是选择最佳动作：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[V^{*}(s) = \max_a \sum_{s'} P(s' \mid s, a) \left[ R(s, a) + \gamma \, V^{*}(s') \right]\]</div>
|
||
<ul>
|
||
<li>类似地，<span class="arithmatex">\(Q^{*}\)</span> 的<strong>贝尔曼最优方程</strong>为：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[Q^{*}(s, a) = \sum_{s'} P(s' \mid s, a) \left[ R(s, a) + \gamma \max_{a'} Q^{*}(s', a') \right]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>一旦你有了 <span class="arithmatex">\(Q^{*}\)</span>，最优策略就很简单了：总是选择Q值最高的动作：<span class="arithmatex">\(\pi^{*}(s) = \arg\max_a Q^{*}(s, a)\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>动态规划</strong>方法在已知转移概率和奖励（完整模型）时求解MDP。<strong>策略评估</strong>通过迭代应用贝尔曼方程直到收敛来计算给定策略的 <span class="arithmatex">\(V^\pi\)</span>。<strong>策略改进</strong>利用价值函数并通过对最优动作贪心来构建更好的策略：<span class="arithmatex">\(\pi'(s) = \arg\max_a \sum_{s'} P(s' \mid s, a)[R(s,a) + \gamma V^\pi(s')]\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>策略迭代</strong>在评估和改进之间交替，直到策略停止变化。它保证收敛到最优策略。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>价值迭代</strong>将两个步骤合并为一个：重复应用贝尔曼最优方程直到 <span class="arithmatex">\(V^{*}\)</span> 收敛，然后提取策略。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[V(s) \leftarrow \max_a \sum_{s'} P(s' \mid s, a) \left[ R(s, a) + \gamma \, V(s') \right]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>动态规划需要知道 <span class="arithmatex">\(P(s' \mid s, a)\)</span>，这通常不可行。在大多数真实问题中，智能体不知道环境的动态；它只能与环境交互。这就是<strong>无模型</strong>方法发挥作用的地方。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>时序差分（TD）学习</strong>在不了解模型的情况下从经验中学习。关键思想是<strong>引导（bootstrapping）</strong>：不等情节结束才计算实际回报 <span class="arithmatex">\(G_t\)</span>，而是使用当前的价值函数对其进行估计：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[V(s_t) \leftarrow V(s_t) + \alpha \left[ r_t + \gamma \, V(s_{t+1}) - V(s_t) \right]\]</div>
|
||
<ul>
|
||
<li>括号中的项是<strong>TD误差</strong>：<strong>TD目标</strong>（<span class="arithmatex">\(r_t + \gamma V(s_{t+1})\)</span>）与当前估计 <span class="arithmatex">\(V(s_t)\)</span> 之间的差异。如果TD误差为正，说明该状态比预期好，我们增加其价值。如果为负，则减少其价值。</li>
|
||
</ul>
|
||
<p><img alt="状态转移展示TD目标：当前价值、奖励以及引导的下一状态价值，附更新公式" src="../../images/td_update.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>TD学习在每一步之后（而不是完成整个情节后）进行更新，这使其比蒙特卡洛方法高效得多。它也适用于持续（非情节式）环境。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>SARSA</strong>（状态-动作-奖励-状态-动作）是将TD学习应用于Q值。智能体在状态 <span class="arithmatex">\(s\)</span> 下采取动作 <span class="arithmatex">\(a\)</span>，观察奖励 <span class="arithmatex">\(r\)</span> 和下一状态 <span class="arithmatex">\(s'\)</span>，然后根据其策略选择下一个动作 <span class="arithmatex">\(a'\)</span>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[Q(s, a) \leftarrow Q(s, a) + \alpha \left[ r + \gamma \, Q(s', a') - Q(s, a) \right]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>SARSA是<strong>在策略（on-policy）</strong>：它使用智能体实际采取的动作进行更新，这包括了探索。这使得SARSA更为保守；它学习一个考虑自身探索噪声的策略。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>Q学习</strong>是最著名的RL算法。它类似于SARSA，但不同的是它使用最佳可能动作而非智能体实际采取的动作：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[Q(s, a) \leftarrow Q(s, a) + \alpha \left[ r + \gamma \max_{a'} Q(s', a') - Q(s, a) \right]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>Q学习是<strong>离策略（off-policy）</strong>：它学习最优Q值，与正在执行的策略无关。智能体可以随机探索，同时仍然学习最优动作价值。这使得Q学习更具攻击性，通常收敛更快，但可能高估值。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>探索 vs 利用</strong>是基本困境：智能体应该利用已知信息（选择估计价值最高的动作）还是探索未知动作（可能发现更好的）？</p>
|
||
</li>
|
||
<li>
|
||
<p>最简单的策略是<strong>ε-贪心</strong>：以概率 <span class="arithmatex">\(\epsilon\)</span> 采取随机动作（探索）；以概率 <span class="arithmatex">\(1 - \epsilon\)</span> 采取贪心动作（利用）。一种常见的时间表是从高 <span class="arithmatex">\(\epsilon\)</span>（大量探索）开始，随时间衰减。</p>
|
||
</li>
|
||
<li>
|
||
<p>表格方法（在表中存储每个状态-动作对的价值）适用于小的离散状态空间。对于大或连续的状态空间，需要函数近似。<strong>深度Q网络（DQN）</strong> 使用神经网络来近似 <span class="arithmatex">\(Q(s, a; \theta)\)</span>，其中 <span class="arithmatex">\(\theta\)</span> 是网络权重。</p>
|
||
</li>
|
||
<li>
|
||
<p>DQN引入了两个关键的稳定技术。<strong>经验回放</strong>：不是从连续的转移中学习（高度相关），而是将转移存储在回放缓冲区中，并采样随机小批次进行训练。这打破了相关性并高效地重用数据。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>目标网络</strong>：使用一个单独的、缓慢更新的网络副本来计算TD目标。没有这个，每次更新网络时目标都会移动，造成"追自己尾巴"的不稳定性。目标网络定期更新（每 <span class="arithmatex">\(N\)</span> 步硬更新）或连续更新（软更新：<span class="arithmatex">\(\theta^{-} \leftarrow \tau\theta + (1-\tau)\theta^{-}\)</span>）。</p>
|
||
</li>
|
||
<li>
|
||
<p>DQN损失只是预测Q值与TD目标之间的均方误差：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathcal{L}(\theta) = \mathbb{E} \left[ \left( r + \gamma \max_{a'} Q(s', a'; \theta^{-}) - Q(s, a; \theta) \right)^2 \right]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>到目前为止的所有方法都学习价值函数并从中推导策略。<strong>策略梯度</strong>方法采用不同方法：它们直接参数化策略 <span class="arithmatex">\(\pi(a \mid s; \theta)\)</span> 并通过梯度上升优化期望回报。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>策略梯度定理</strong>给出了期望回报相对于策略参数的梯度：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\nabla_\theta J(\theta) = \mathbb{E}_\pi \left[ \nabla_\theta \log \pi(a \mid s; \theta) \cdot G_t \right]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>这说明：增加导致高回报的动作的概率，减少导致低回报的动作的概率。对数概率梯度给出了改变策略的方向，<span class="arithmatex">\(G_t\)</span> 则缩放改变的程度。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>REINFORCE</strong>是最简单的策略梯度算法。运行一个情节，为每一步计算回报 <span class="arithmatex">\(G_t\)</span>，然后更新：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\theta \leftarrow \theta + \alpha \, \nabla_\theta \log \pi(a_t \mid s_t; \theta) \cdot G_t\]</div>
|
||
<ul>
|
||
<li>REINFORCE方差很高，因为 <span class="arithmatex">\(G_t\)</span> 是期望回报的噪声单样本估计。一个常见修复是减去一个<strong>基线（baseline）</strong>（通常是平均回报或学习到的价值函数）来降低方差而不引入偏差：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\theta \leftarrow \theta + \alpha \, \nabla_\theta \log \pi(a_t \mid s_t; \theta) \cdot (G_t - b)\]</div>
|
||
<ul>
|
||
<li><strong>演员-评论家（Actor-Critic）</strong> 方法使用两个网络。<strong>演员（actor）</strong> 是策略 <span class="arithmatex">\(\pi(a \mid s; \theta)\)</span>。<strong>评论家（critic）</strong> 是价值函数 <span class="arithmatex">\(V(s; \phi)\)</span>，作为基线。优势 <span class="arithmatex">\(A_t = r_t + \gamma V(s_{t+1}) - V(s_t)\)</span> 替代了 <span class="arithmatex">\(G_t - b\)</span>：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\theta \leftarrow \theta + \alpha \, \nabla_\theta \log \pi(a_t \mid s_t; \theta) \cdot A_t\]</div>
|
||
<ul>
|
||
<li>评论家通过最小化TD误差来更新，与基于价值的方法相同。演员使用策略梯度更新，评论家的优势估计降低了方差。这是两全其美。</li>
|
||
</ul>
|
||
<p><img alt="双头架构：演员输出动作概率，评论家输出价值估计，优势信号指导演员更新" src="../../images/actor_critic.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>PPO</strong>（近端策略优化）是实践中使用最广泛的策略梯度算法。它解决了一个关键问题：如果策略更新过大，性能可能灾难性地崩溃。</p>
|
||
</li>
|
||
<li>
|
||
<p>PPO使用一个<strong>裁剪的替代目标</strong>。令 <span class="arithmatex">\(r_t(\theta) = \frac{\pi(a_t | s_t; \theta)}{\pi(a_t | s_t; \theta_{\text{old}})}\)</span> 为新旧策略之间的概率比。损失为：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathcal{L}^{\text{CLIP}}(\theta) = \mathbb{E} \left[ \min\!\left( r_t(\theta) A_t, \; \text{clip}(r_t(\theta), 1-\epsilon, 1+\epsilon) A_t \right) \right]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>裁剪（通常 <span class="arithmatex">\(\epsilon = 0.2\)</span>）防止比率远离1，使更新保持小而稳定。如果优势为正（动作好），比率上限为 <span class="arithmatex">\(1 + \epsilon\)</span>。如果为负（动作差），比率下限为 <span class="arithmatex">\(1 - \epsilon\)</span>。这比早期的信任区域方法（TRPO）更简单、更稳定。</p>
|
||
</li>
|
||
<li>
|
||
<p>PPO被用于通过<strong>RLHF</strong>（基于人类反馈的强化学习）训练ChatGPT风格的模型。在RLHF中，一个奖励模型在人类偏好数据（人类更喜欢两个输出中的哪一个？）上训练，然后PPO优化语言模型策略以最大化这个学习到的奖励。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>DPO</strong>（直接偏好优化）通过完全消除奖励模型来简化RLHF。DPO不训练奖励模型然后运行RL，而是推导出一个闭式损失，直接从偏好数据优化策略：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathcal{L}_{\text{DPO}}(\theta) = -\mathbb{E} \left[ \log \sigma\!\left( \beta \log \frac{\pi_\theta(y_w \mid x)}{\pi_{\text{ref}}(y_w \mid x)} - \beta \log \frac{\pi_\theta(y_l \mid x)}{\pi_{\text{ref}}(y_l \mid x)} \right) \right]\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>这里 <span class="arithmatex">\(y_w\)</span> 是偏好的（胜出）回答，<span class="arithmatex">\(y_l\)</span> 是不被偏好的（失败）回答。DPO增加偏好输出的相对概率，并且比基于PPO的RLHF实现起来简单得多。</p>
|
||
</li>
|
||
<li>
|
||
<p>RL算法中有两个重要区分。<strong>在策略 vs 离策略</strong>：在策略方法（SARSA, PPO）从当前策略生成的数据中学习；离策略方法（Q学习, DQN）可以从任何策略生成的数据中学习。离策略方法样本效率更高（它们重用旧数据），但可能不那么稳定。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>基于模型 vs 无模型</strong>：无模型方法（到目前为止讨论的所有方法）直接从经验中学习价值或策略。基于模型的方法学习环境的模型（<span class="arithmatex">\(P(s' \mid s, a)\)</span> 和 <span class="arithmatex">\(R(s, a)\)</span>）并用其进行规划（想象未来的轨迹而不实际采取动作）。基于模型的方法样本效率更高，但增加了学习精确模型的复杂性。</p>
|
||
</li>
|
||
<li>
|
||
<p>总结RL领域：</p>
|
||
</li>
|
||
</ul>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>方法</th>
|
||
<th>类型</th>
|
||
<th>核心思想</th>
|
||
<th>优势</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td>价值迭代</td>
|
||
<td>DP, 基于模型</td>
|
||
<td>贝尔曼最优性</td>
|
||
<td>精确解（小MDP）</td>
|
||
</tr>
|
||
<tr>
|
||
<td>SARSA</td>
|
||
<td>TD, 在策略</td>
|
||
<td>在策略学习Q</td>
|
||
<td>保守、安全</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Q学习</td>
|
||
<td>TD, 离策略</td>
|
||
<td>学习Q*, 贪心目标</td>
|
||
<td>简单、有效</td>
|
||
</tr>
|
||
<tr>
|
||
<td>DQN</td>
|
||
<td>深度, 离策略</td>
|
||
<td>神经Q + 回放 + 目标网络</td>
|
||
<td>扩展到高维状态</td>
|
||
</tr>
|
||
<tr>
|
||
<td>REINFORCE</td>
|
||
<td>策略梯度</td>
|
||
<td>log-概率 * 回报的梯度</td>
|
||
<td>简单的策略优化</td>
|
||
</tr>
|
||
<tr>
|
||
<td>演员-评论家</td>
|
||
<td>PG + 价值</td>
|
||
<td>演员 + 评论家降低方差</td>
|
||
<td>实用且灵活</td>
|
||
</tr>
|
||
<tr>
|
||
<td>PPO</td>
|
||
<td>PG, 裁剪</td>
|
||
<td>信任区域般的稳定性</td>
|
||
<td>行业标准</td>
|
||
</tr>
|
||
<tr>
|
||
<td>DPO</td>
|
||
<td>直接偏好</td>
|
||
<td>跳过奖励模型</td>
|
||
<td>更简单的RLHF</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h2 id="colab">编程任务（使用CoLab或笔记本）<a class="headerlink" href="#colab" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>为简单的网格世界实现价值迭代。计算最优价值函数并提取最优策略。将两者可视化为热力图和箭头图。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># 4x4网格世界：目标在(3,3)，每步奖励-1，目标处为0</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">grid_size</span> <span class="o">=</span> <span class="mi">4</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">gamma</span> <span class="o">=</span> <span class="mf">0.99</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">goal</span> <span class="o">=</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="c1"># 动作：上、下、左、右</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">actions</span> <span class="o">=</span> <span class="p">[(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)]</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="n">action_names</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'up'</span><span class="p">,</span> <span class="s1">'down'</span><span class="p">,</span> <span class="s1">'left'</span><span class="p">,</span> <span class="s1">'right'</span><span class="p">]</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">action_arrows</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'</span><span class="se">\u2191</span><span class="s1">'</span><span class="p">,</span> <span class="s1">'</span><span class="se">\u2193</span><span class="s1">'</span><span class="p">,</span> <span class="s1">'</span><span class="se">\u2190</span><span class="s1">'</span><span class="p">,</span> <span class="s1">'</span><span class="se">\u2192</span><span class="s1">'</span><span class="p">]</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="k">def</span><span class="w"> </span><span class="nf">step</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w"> </span><span class="sd">"""确定性转移。"""</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a> <span class="n">ns</span> <span class="o">=</span> <span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">grid_size</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">])),</span>
|
||
<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a> <span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">grid_size</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">])))</span>
|
||
<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a> <span class="k">return</span> <span class="n">ns</span>
|
||
<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>
|
||
<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="c1"># 价值迭代</span>
|
||
<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="n">V</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">grid_size</span><span class="p">,</span> <span class="n">grid_size</span><span class="p">))</span>
|
||
<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="k">for</span> <span class="n">iteration</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">):</span>
|
||
<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a> <span class="n">V_new</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">V</span><span class="p">)</span>
|
||
<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a> <span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">==</span> <span class="n">goal</span><span class="p">:</span>
|
||
<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a> <span class="k">continue</span>
|
||
<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a> <span class="n">values</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">actions</span><span class="p">:</span>
|
||
<a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a> <span class="n">ns</span> <span class="o">=</span> <span class="n">step</span><span class="p">((</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">),</span> <span class="n">a</span><span class="p">)</span>
|
||
<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a> <span class="n">values</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span> <span class="o">+</span> <span class="n">gamma</span> <span class="o">*</span> <span class="nb">float</span><span class="p">(</span><span class="n">V</span><span class="p">[</span><span class="n">ns</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ns</span><span class="p">[</span><span class="mi">1</span><span class="p">]]))</span>
|
||
<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a> <span class="n">V_new</span> <span class="o">=</span> <span class="n">V_new</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="n">values</span><span class="p">))</span>
|
||
<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a> <span class="k">if</span> <span class="n">jnp</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">V_new</span> <span class="o">-</span> <span class="n">V</span><span class="p">))</span> <span class="o"><</span> <span class="mf">1e-6</span><span class="p">:</span>
|
||
<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"在</span><span class="si">{</span><span class="n">iteration</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s2">次迭代后收敛"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a> <span class="k">break</span>
|
||
<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a> <span class="n">V</span> <span class="o">=</span> <span class="n">V_new</span>
|
||
<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a>
|
||
<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a><span class="c1"># 提取策略</span>
|
||
<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a><span class="n">policy</span> <span class="o">=</span> <span class="p">[[</span><span class="s1">''</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">)]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">)]</span>
|
||
<a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-0-41" name="__codelineno-0-41" href="#__codelineno-0-41"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-0-42" name="__codelineno-0-42" href="#__codelineno-0-42"></a> <span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span> <span class="o">==</span> <span class="n">goal</span><span class="p">:</span>
|
||
<a id="__codelineno-0-43" name="__codelineno-0-43" href="#__codelineno-0-43"></a> <span class="n">policy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="s1">'G'</span>
|
||
<a id="__codelineno-0-44" name="__codelineno-0-44" href="#__codelineno-0-44"></a> <span class="k">continue</span>
|
||
<a id="__codelineno-0-45" name="__codelineno-0-45" href="#__codelineno-0-45"></a> <span class="n">best_a</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">a</span><span class="p">:</span> <span class="o">-</span><span class="mi">1</span> <span class="o">+</span> <span class="n">gamma</span> <span class="o">*</span> <span class="nb">float</span><span class="p">(</span><span class="n">V</span><span class="p">[</span><span class="n">step</span><span class="p">((</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">),</span> <span class="n">actions</span><span class="p">[</span><span class="n">a</span><span class="p">])[</span><span class="mi">0</span><span class="p">],</span> <span class="n">step</span><span class="p">((</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">),</span> <span class="n">actions</span><span class="p">[</span><span class="n">a</span><span class="p">])[</span><span class="mi">1</span><span class="p">]]))</span>
|
||
<a id="__codelineno-0-46" name="__codelineno-0-46" href="#__codelineno-0-46"></a> <span class="n">policy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">action_arrows</span><span class="p">[</span><span class="n">best_a</span><span class="p">]</span>
|
||
<a id="__codelineno-0-47" name="__codelineno-0-47" href="#__codelineno-0-47"></a>
|
||
<a id="__codelineno-0-48" name="__codelineno-0-48" href="#__codelineno-0-48"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<a id="__codelineno-0-49" name="__codelineno-0-49" href="#__codelineno-0-49"></a><span class="n">im</span> <span class="o">=</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'YlOrRd_r'</span><span class="p">)</span>
|
||
<a id="__codelineno-0-50" name="__codelineno-0-50" href="#__codelineno-0-50"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"最优价值函数"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-51" name="__codelineno-0-51" href="#__codelineno-0-51"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-0-52" name="__codelineno-0-52" href="#__codelineno-0-52"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-0-53" name="__codelineno-0-53" href="#__codelineno-0-53"></a> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">V</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">]</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2">"</span><span class="p">,</span> <span class="n">ha</span><span class="o">=</span><span class="s1">'center'</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">'center'</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
|
||
<a id="__codelineno-0-54" name="__codelineno-0-54" href="#__codelineno-0-54"></a><span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">im</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||
<a id="__codelineno-0-55" name="__codelineno-0-55" href="#__codelineno-0-55"></a>
|
||
<a id="__codelineno-0-56" name="__codelineno-0-56" href="#__codelineno-0-56"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">grid_size</span><span class="p">,</span> <span class="n">grid_size</span><span class="p">)),</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">'Greys'</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<a id="__codelineno-0-57" name="__codelineno-0-57" href="#__codelineno-0-57"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"最优策略"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-58" name="__codelineno-0-58" href="#__codelineno-0-58"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-0-59" name="__codelineno-0-59" href="#__codelineno-0-59"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-0-60" name="__codelineno-0-60" href="#__codelineno-0-60"></a> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">j</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">policy</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">ha</span><span class="o">=</span><span class="s1">'center'</span><span class="p">,</span> <span class="n">va</span><span class="o">=</span><span class="s1">'center'</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<a id="__codelineno-0-61" name="__codelineno-0-61" href="#__codelineno-0-61"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span> <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>在简单的网格世界上实现表格Q学习。训练智能体，绘制学习曲线，显示学习到的Q值。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">grid_size</span> <span class="o">=</span> <span class="mi">5</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">goal</span> <span class="o">=</span> <span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">actions</span> <span class="o">=</span> <span class="p">[(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">)]</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="c1"># Q表</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">Q</span> <span class="o">=</span> <span class="p">{}</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a> <span class="n">Q</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">)]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">4</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.1</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="n">gamma</span> <span class="o">=</span> <span class="mf">0.95</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="n">epsilon</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="n">epsilon_decay</span> <span class="o">=</span> <span class="mf">0.995</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="n">min_epsilon</span> <span class="o">=</span> <span class="mf">0.01</span>
|
||
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a>
|
||
<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="k">def</span><span class="w"> </span><span class="nf">step</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a_idx</span><span class="p">):</span>
|
||
<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a> <span class="n">a</span> <span class="o">=</span> <span class="n">actions</span><span class="p">[</span><span class="n">a_idx</span><span class="p">]</span>
|
||
<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a> <span class="n">ns</span> <span class="o">=</span> <span class="p">(</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">grid_size</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">])),</span>
|
||
<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a> <span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">grid_size</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">])))</span>
|
||
<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a> <span class="n">r</span> <span class="o">=</span> <span class="mf">0.0</span> <span class="k">if</span> <span class="n">ns</span> <span class="o">==</span> <span class="n">goal</span> <span class="k">else</span> <span class="o">-</span><span class="mf">1.0</span>
|
||
<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a> <span class="n">done</span> <span class="o">=</span> <span class="n">ns</span> <span class="o">==</span> <span class="n">goal</span>
|
||
<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a> <span class="k">return</span> <span class="n">ns</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">done</span>
|
||
<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a>
|
||
<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="n">rewards_per_episode</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a>
|
||
<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="k">for</span> <span class="n">ep</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">500</span><span class="p">):</span>
|
||
<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a> <span class="n">s</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
|
||
<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a> <span class="n">total_reward</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">):</span>
|
||
<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a> <span class="n">key</span><span class="p">,</span> <span class="n">subkey</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
|
||
<a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a> <span class="k">if</span> <span class="nb">float</span><span class="p">(</span><span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">subkey</span><span class="p">))</span> <span class="o"><</span> <span class="n">epsilon</span><span class="p">:</span>
|
||
<a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a> <span class="n">key</span><span class="p">,</span> <span class="n">subkey</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
|
||
<a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a> <span class="n">a</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="n">subkey</span><span class="p">,</span> <span class="p">(),</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a> <span class="k">else</span><span class="p">:</span>
|
||
<a id="__codelineno-1-41" name="__codelineno-1-41" href="#__codelineno-1-41"></a> <span class="n">a</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">i</span><span class="p">:</span> <span class="n">Q</span><span class="p">[</span><span class="n">s</span><span class="p">][</span><span class="n">i</span><span class="p">])</span>
|
||
<a id="__codelineno-1-42" name="__codelineno-1-42" href="#__codelineno-1-42"></a>
|
||
<a id="__codelineno-1-43" name="__codelineno-1-43" href="#__codelineno-1-43"></a> <span class="n">ns</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">done</span> <span class="o">=</span> <span class="n">step</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
|
||
<a id="__codelineno-1-44" name="__codelineno-1-44" href="#__codelineno-1-44"></a> <span class="n">total_reward</span> <span class="o">+=</span> <span class="n">r</span>
|
||
<a id="__codelineno-1-45" name="__codelineno-1-45" href="#__codelineno-1-45"></a> <span class="c1"># Q学习更新</span>
|
||
<a id="__codelineno-1-46" name="__codelineno-1-46" href="#__codelineno-1-46"></a> <span class="n">Q</span><span class="p">[</span><span class="n">s</span><span class="p">][</span><span class="n">a</span><span class="p">]</span> <span class="o">+=</span> <span class="n">alpha</span> <span class="o">*</span> <span class="p">(</span><span class="n">r</span> <span class="o">+</span> <span class="n">gamma</span> <span class="o">*</span> <span class="nb">max</span><span class="p">(</span><span class="n">Q</span><span class="p">[</span><span class="n">ns</span><span class="p">])</span> <span class="o">-</span> <span class="n">Q</span><span class="p">[</span><span class="n">s</span><span class="p">][</span><span class="n">a</span><span class="p">])</span>
|
||
<a id="__codelineno-1-47" name="__codelineno-1-47" href="#__codelineno-1-47"></a> <span class="n">s</span> <span class="o">=</span> <span class="n">ns</span>
|
||
<a id="__codelineno-1-48" name="__codelineno-1-48" href="#__codelineno-1-48"></a> <span class="k">if</span> <span class="n">done</span><span class="p">:</span>
|
||
<a id="__codelineno-1-49" name="__codelineno-1-49" href="#__codelineno-1-49"></a> <span class="k">break</span>
|
||
<a id="__codelineno-1-50" name="__codelineno-1-50" href="#__codelineno-1-50"></a> <span class="n">rewards_per_episode</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">total_reward</span><span class="p">)</span>
|
||
<a id="__codelineno-1-51" name="__codelineno-1-51" href="#__codelineno-1-51"></a> <span class="n">epsilon</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">min_epsilon</span><span class="p">,</span> <span class="n">epsilon</span> <span class="o">*</span> <span class="n">epsilon_decay</span><span class="p">)</span>
|
||
<a id="__codelineno-1-52" name="__codelineno-1-52" href="#__codelineno-1-52"></a>
|
||
<a id="__codelineno-1-53" name="__codelineno-1-53" href="#__codelineno-1-53"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<a id="__codelineno-1-54" name="__codelineno-1-54" href="#__codelineno-1-54"></a><span class="c1"># 平滑曲线</span>
|
||
<a id="__codelineno-1-55" name="__codelineno-1-55" href="#__codelineno-1-55"></a><span class="n">window</span> <span class="o">=</span> <span class="mi">20</span>
|
||
<a id="__codelineno-1-56" name="__codelineno-1-56" href="#__codelineno-1-56"></a><span class="n">smoothed</span> <span class="o">=</span> <span class="p">[</span><span class="nb">sum</span><span class="p">(</span><span class="n">rewards_per_episode</span><span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">i</span><span class="o">-</span><span class="n">window</span><span class="p">):</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span><span class="o">/</span><span class="nb">min</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">window</span><span class="p">)</span>
|
||
<a id="__codelineno-1-57" name="__codelineno-1-57" href="#__codelineno-1-57"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">rewards_per_episode</span><span class="p">))]</span>
|
||
<a id="__codelineno-1-58" name="__codelineno-1-58" href="#__codelineno-1-58"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">smoothed</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'#3498db'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.5</span><span class="p">)</span>
|
||
<a id="__codelineno-1-59" name="__codelineno-1-59" href="#__codelineno-1-59"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"Episode"</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"Total Reward (smoothed)"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-60" name="__codelineno-1-60" href="#__codelineno-1-60"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Q-Learning on Gridworld"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-61" name="__codelineno-1-61" href="#__codelineno-1-61"></a><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<a id="__codelineno-1-62" name="__codelineno-1-62" href="#__codelineno-1-62"></a>
|
||
<a id="__codelineno-1-63" name="__codelineno-1-63" href="#__codelineno-1-63"></a><span class="c1"># 显示学到的策略</span>
|
||
<a id="__codelineno-1-64" name="__codelineno-1-64" href="#__codelineno-1-64"></a><span class="n">arrow</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'</span><span class="se">\u2191</span><span class="s1">'</span><span class="p">,</span> <span class="s1">'</span><span class="se">\u2193</span><span class="s1">'</span><span class="p">,</span> <span class="s1">'</span><span class="se">\u2190</span><span class="s1">'</span><span class="p">,</span> <span class="s1">'</span><span class="se">\u2192</span><span class="s1">'</span><span class="p">]</span>
|
||
<a id="__codelineno-1-65" name="__codelineno-1-65" href="#__codelineno-1-65"></a><span class="nb">print</span><span class="p">(</span><span class="s2">"学到的策略:"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-66" name="__codelineno-1-66" href="#__codelineno-1-66"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-1-67" name="__codelineno-1-67" href="#__codelineno-1-67"></a> <span class="n">row</span> <span class="o">=</span> <span class="s2">""</span>
|
||
<a id="__codelineno-1-68" name="__codelineno-1-68" href="#__codelineno-1-68"></a> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">):</span>
|
||
<a id="__codelineno-1-69" name="__codelineno-1-69" href="#__codelineno-1-69"></a> <span class="k">if</span> <span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">)</span> <span class="o">==</span> <span class="n">goal</span><span class="p">:</span>
|
||
<a id="__codelineno-1-70" name="__codelineno-1-70" href="#__codelineno-1-70"></a> <span class="n">row</span> <span class="o">+=</span> <span class="s2">" G "</span>
|
||
<a id="__codelineno-1-71" name="__codelineno-1-71" href="#__codelineno-1-71"></a> <span class="k">else</span><span class="p">:</span>
|
||
<a id="__codelineno-1-72" name="__codelineno-1-72" href="#__codelineno-1-72"></a> <span class="n">row</span> <span class="o">+=</span> <span class="sa">f</span><span class="s2">" </span><span class="si">{</span><span class="n">arrow</span><span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span><span class="w"> </span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="n">Q</span><span class="p">[(</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">)][</span><span class="n">a</span><span class="p">])]</span><span class="si">}</span><span class="s2"> "</span>
|
||
<a id="__codelineno-1-73" name="__codelineno-1-73" href="#__codelineno-1-73"></a> <span class="nb">print</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>在多臂老虎机问题上实现REINFORCE。展示策略如何随训练演变以偏向最佳臂。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="c1"># 5臂老虎机，不同期望奖励</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">true_rewards</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">])</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">n_arms</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">true_rewards</span><span class="p">)</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="c1"># 策略：在logits上的softmax</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">logits</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_arms</span><span class="p">)</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="n">lr</span> <span class="o">=</span> <span class="mf">0.1</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a>
|
||
<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="n">policy_history</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="n">reward_history</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a>
|
||
<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="k">for</span> <span class="n">step</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2000</span><span class="p">):</span>
|
||
<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a> <span class="n">probs</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">softmax</span><span class="p">(</span><span class="n">logits</span><span class="p">)</span>
|
||
<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a> <span class="n">policy_history</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">probs</span><span class="p">)</span>
|
||
<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a>
|
||
<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a> <span class="c1"># 采样动作</span>
|
||
<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a> <span class="n">key</span><span class="p">,</span> <span class="n">subkey</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
|
||
<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a> <span class="n">action</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">subkey</span><span class="p">,</span> <span class="n">n_arms</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="n">probs</span><span class="p">)</span>
|
||
<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a>
|
||
<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a> <span class="c1"># 获取奖励（伯努利分布）</span>
|
||
<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a> <span class="n">key</span><span class="p">,</span> <span class="n">subkey</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
|
||
<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a> <span class="n">reward</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">subkey</span><span class="p">)</span> <span class="o"><</span> <span class="n">true_rewards</span><span class="p">[</span><span class="n">action</span><span class="p">])</span>
|
||
<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a> <span class="n">reward_history</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">reward</span><span class="p">)</span>
|
||
<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a>
|
||
<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a> <span class="c1"># REINFORCE更新</span>
|
||
<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a> <span class="c1"># grad log pi(a) = e_a - probs（对于softmax参数化）</span>
|
||
<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a> <span class="n">grad_log_pi</span> <span class="o">=</span> <span class="o">-</span><span class="n">probs</span><span class="o">.</span><span class="n">at</span><span class="p">[</span><span class="n">action</span><span class="p">]</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="mf">1.0</span><span class="p">)</span> <span class="c1"># one-hot(a) - probs</span>
|
||
<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a> <span class="n">logits</span> <span class="o">=</span> <span class="n">logits</span> <span class="o">+</span> <span class="n">lr</span> <span class="o">*</span> <span class="n">reward</span> <span class="o">*</span> <span class="n">grad_log_pi</span>
|
||
<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a>
|
||
<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="n">policy_history</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="n">policy_history</span><span class="p">)</span>
|
||
<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a>
|
||
<a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a><span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'#3498db'</span><span class="p">,</span> <span class="s1">'#e74c3c'</span><span class="p">,</span> <span class="s1">'#27ae60'</span><span class="p">,</span> <span class="s1">'#9b59b6'</span><span class="p">,</span> <span class="s1">'#f39c12'</span><span class="p">]</span>
|
||
<a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a><span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_arms</span><span class="p">):</span>
|
||
<a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">policy_history</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="n">i</span><span class="p">],</span>
|
||
<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s1">'臂</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s1"> (真实=</span><span class="si">{</span><span class="n">true_rewards</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">)'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.5</span><span class="p">)</span>
|
||
<a id="__codelineno-2-42" name="__codelineno-2-42" href="#__codelineno-2-42"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"步骤"</span><span class="p">);</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"P(臂)"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"策略演变 (REINFORCE)"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-44" name="__codelineno-2-44" href="#__codelineno-2-44"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">8</span><span class="p">);</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
|
||
<a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a>
|
||
<a id="__codelineno-2-46" name="__codelineno-2-46" href="#__codelineno-2-46"></a><span class="c1"># 平滑奖励</span>
|
||
<a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a><span class="n">window</span> <span class="o">=</span> <span class="mi">50</span>
|
||
<a id="__codelineno-2-48" name="__codelineno-2-48" href="#__codelineno-2-48"></a><span class="n">smoothed</span> <span class="o">=</span> <span class="p">[</span><span class="nb">sum</span><span class="p">(</span><span class="n">reward_history</span><span class="p">[</span><span class="nb">max</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">i</span><span class="o">-</span><span class="n">window</span><span class="p">):</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span><span class="o">/</span><span class="nb">min</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="n">window</span><span class="p">)</span>
|
||
<a id="__codelineno-2-49" name="__codelineno-2-49" href="#__codelineno-2-49"></a> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">reward_history</span><span class="p">))]</span>
|
||
<a id="__codelineno-2-50" name="__codelineno-2-50" href="#__codelineno-2-50"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">smoothed</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'#27ae60'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">1.5</span><span class="p">)</span>
|
||
<a id="__codelineno-2-51" name="__codelineno-2-51" href="#__codelineno-2-51"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'#e74c3c'</span><span class="p">,</span> <span class="n">linestyle</span><span class="o">=</span><span class="s1">'--'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'最佳臂'</span><span class="p">)</span>
|
||
<a id="__codelineno-2-52" name="__codelineno-2-52" href="#__codelineno-2-52"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"步骤"</span><span class="p">);</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"平均奖励"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-53" name="__codelineno-2-53" href="#__codelineno-2-53"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"奖励随时间变化"</span><span class="p">);</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<a id="__codelineno-2-54" name="__codelineno-2-54" href="#__codelineno-2-54"></a><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
|
||
<a id="__codelineno-2-55" name="__codelineno-2-55" href="#__codelineno-2-55"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">();</span> <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
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