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<span class="md-ellipsis">
|
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|
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函数逼近
|
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优化
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概率论
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计数
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</a>
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概率概念
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分布
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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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机器学习
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</span>
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机器学习
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经典机器学习
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梯度机器学习
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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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</span>
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</a>
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<span class="md-ellipsis">
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分布式深度学习
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</span>
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|
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|
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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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嵌入与序列模型
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</a>
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Transformer 与语言模型
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</span>
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|
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</a>
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<span class="md-ellipsis">
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高级文本生成
|
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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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音频与语音
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数字信号处理
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自动语音识别
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</span>
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语音合成
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</span>
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</a>
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<span class="md-ellipsis">
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说话人与音频分析
|
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</span>
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|
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|
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</a>
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</li>
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源分离与降噪
|
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</span>
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|
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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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图像与视频 Token 化
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</span>
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|
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|
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</a>
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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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</span>
|
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|
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|
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|
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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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感知
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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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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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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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<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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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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几何深度学习
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</span>
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硬件基础
|
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|
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</span>
|
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|
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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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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">多元微积分<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>多元微积分将导数和积分扩展到多变量函数，这对于机器学习模型拥有数百万参数的情形至关重要。本章涵盖偏导数、梯度、雅可比矩阵、海森矩阵以及使反向传播成为可能的多变量链式法则。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>到目前为止，我们的函数都只接受单个输入 <span class="arithmatex">\(x\)</span> 并产生单个输出 <span class="arithmatex">\(f(x)\)</span>。但在机器学习中，我们几乎从不只处理一个变量。</p>
|
||
</li>
|
||
<li>
|
||
<p>考虑一个双变量函数，例如 <span class="arithmatex">\(f(x, y) = x^2 + y^2\)</span>。它在三维空间中定义了一个曲面，形状像一个碗。我们想知道：如果我们在保持 <span class="arithmatex">\(y\)</span> 固定的同时稍微调整 <span class="arithmatex">\(x\)</span>，<span class="arithmatex">\(f\)</span> 会如何变化？这就是<strong>偏导数</strong>。</p>
|
||
</li>
|
||
<li>
|
||
<p><span class="arithmatex">\(f\)</span> 对 <span class="arithmatex">\(x\)</span> 的<strong>偏导数</strong>，记作 <span class="arithmatex">\(\frac{\partial f}{\partial x}\)</span>，将其他所有变量视为常数，然后对 <span class="arithmatex">\(x\)</span> 正常求导。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于 <span class="arithmatex">\(f(x, y) = x^2y + 3x - 2y\)</span>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\frac{\partial f}{\partial x} = 2xy + 3 \qquad \frac{\partial f}{\partial y} = x^2 - 2\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>计算 <span class="arithmatex">\(\frac{\partial f}{\partial x}\)</span> 时，我们将 <span class="arithmatex">\(y\)</span> 视为常数，因此 <span class="arithmatex">\(x^2y\)</span> 求导得 <span class="arithmatex">\(2xy\)</span>，<span class="arithmatex">\(3x\)</span> 求导得 <span class="arithmatex">\(3\)</span>，<span class="arithmatex">\(-2y\)</span> 求导得 <span class="arithmatex">\(0\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>计算 <span class="arithmatex">\(\frac{\partial f}{\partial y}\)</span> 时，我们将 <span class="arithmatex">\(x\)</span> 视为常数，因此 <span class="arithmatex">\(x^2y\)</span> 求导得 <span class="arithmatex">\(x^2\)</span>，<span class="arithmatex">\(3x\)</span> 求导得 <span class="arithmatex">\(0\)</span>，<span class="arithmatex">\(-2y\)</span> 求导得 <span class="arithmatex">\(-2\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>从几何上看，对 <span class="arithmatex">\(x\)</span> 求偏导数就像用一个平行于 <span class="arithmatex">\(xz\)</span> 平面的平面（在固定的 <span class="arithmatex">\(y\)</span> 值处）切割三维曲面，然后求所得曲线的斜率。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="偏导数：固定一个变量来切割曲面" src="../../images/partial_derivative.svg" /></p>
|
||
<ul>
|
||
<li><strong>梯度</strong>将所有偏导数收集到一个向量中：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n}\right)\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>对于 <span class="arithmatex">\(f(x, y) = x^2 + y^2\)</span>：<span class="arithmatex">\(\nabla f(x, y) = (2x, 2y)\)</span>。在点 <span class="arithmatex">\((1, 2)\)</span> 处：<span class="arithmatex">\(\nabla f(1, 2) = (2, 4)\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>梯度有两个关键性质：</p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>方向</strong>：它指向上升最陡的方向。想象一位登山者在山上。他们所在位置的梯度指向正上方，沿着最陡的路径。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>大小</strong>：<span class="arithmatex">\(\|\nabla f\|\)</span> 给出了最陡方向上的变化率。梯度大意味着地形陡峭；梯度小意味着地形近乎平坦。</p>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="梯度向量指向山坡上方，垂直于等高线" src="../../images/gradient_contour.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>由于梯度指向上坡，沿相反方向（<span class="arithmatex">\(-\nabla f\)</span>）移动就是下坡，走向更低的值。这个简单的想法是<strong>梯度下降</strong>的基础，我们将在后续章节中详细探讨这种优化技术。现在，关键要点是：梯度告诉你哪个方向是"上坡"，以及攀登的陡峭程度。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>方向导数</strong>推广了偏导数。它不问"<span class="arithmatex">\(f\)</span> 沿 <span class="arithmatex">\(x\)</span> 轴如何变化？"，而是问"<span class="arithmatex">\(f\)</span> 沿任意方向 <span class="arithmatex">\(\mathbf{u}\)</span> 如何变化？"它通过梯度与单位向量的点积来计算：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>对于 <span class="arithmatex">\(f(x, y) = x^2 + y^2\)</span> 在点 <span class="arithmatex">\((1, 2)\)</span> 处，沿方向 <span class="arithmatex">\(\mathbf{v} = (3, 4)\)</span>：首先归一化得到 <span class="arithmatex">\(\mathbf{u} = (3/5, 4/5)\)</span>，然后 <span class="arithmatex">\(D_{\mathbf{u}} f = (2, 4) \cdot (3/5, 4/5) = 6/5 + 16/5 = 22/5\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>偏导数是方向导数的特例，其中方向沿着坐标轴。如果方向导数在某个方向上为零，则函数在该点沿该方向是平坦的。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>等高线</strong>（或水平曲线）连接函数值相等的点。对于 <span class="arithmatex">\(f(x, y) = x^2 + y^2\)</span>，等高线是以原点为中心的圆：对应不同 <span class="arithmatex">\(c\)</span> 值的 <span class="arithmatex">\(x^2 + y^2 = c\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>等高线永不相交（一个点不可能有两个不同的函数值）。</p>
|
||
</li>
|
||
<li>
|
||
<p>梯度始终垂直于等高线，从低值指向高值。</p>
|
||
</li>
|
||
<li>
|
||
<p>等高线密集表示地形陡峭；等高线稀疏表示坡度平缓。</p>
|
||
</li>
|
||
<li>
|
||
<p>到目前为止，我们的函数都只产生单个输出。但许多函数会产生多个输出。函数 <span class="arithmatex">\(\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^m\)</span> 接收 <span class="arithmatex">\(n\)</span> 个输入并产生 <span class="arithmatex">\(m\)</span> 个输出。<strong>雅可比矩阵</strong>组织了这样一个向量值函数的所有偏导数：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[
|
||
J = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n} \end{bmatrix}
|
||
\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>雅可比矩阵的每一行是一个输出分量的梯度。对于一个有 3 个输入和 2 个输出的函数，雅可比矩阵是一个 <span class="arithmatex">\(2 \times 3\)</span> 矩阵。</p>
|
||
</li>
|
||
<li>
|
||
<p>雅可比矩阵将导数推广到向量值函数。</p>
|
||
</li>
|
||
<li>
|
||
<p>就像标量函数的导数告诉你每单位输入变化对应的输出变化量一样，雅可比矩阵告诉你每个输出相对于每个输入的变化情况。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>雅可比行列式</strong>衡量一个变换局部拉伸或压缩空间的程度。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果行列式为 2，小区域的面积加倍。如果行列式为 0，该变换将空间压缩到更低维度（回想我们在矩阵章节中学到的：行列式为零意味着奇异变换，不可逆）。</p>
|
||
</li>
|
||
<li>
|
||
<p>当多个变换组合在一起（一个变换的输出作为下一个变换的输入）时，整体映射的雅可比矩阵是各个雅可比矩阵的乘积。我们将在后续章节中看到这个思想变得至关重要。</p>
|
||
</li>
|
||
<li>
|
||
<p>梯度捕获一阶信息（斜率），而<strong>海森矩阵</strong>捕获二阶信息（曲率）。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于标量函数 <span class="arithmatex">\(f(x_1, \ldots, x_n)\)</span>，海森矩阵是所有二阶偏导数的 <span class="arithmatex">\(n \times n\)</span> 矩阵：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[
|
||
H = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots \\ \vdots & \vdots & \ddots \end{bmatrix}
|
||
\]</div>
|
||
<ul>
|
||
<li>对于 <span class="arithmatex">\(f(x, y) = x^3 + 2xy^2 - y^3\)</span>，梯度为 <span class="arithmatex">\((3x^2 + 2y^2,\; 4xy - 3y^2)\)</span>，海森矩阵为：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[
|
||
H = \begin{bmatrix} 6x & 4y \\ 4y & 4x - 6y \end{bmatrix}
|
||
\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>对角线元素（<span class="arithmatex">\(6x\)</span> 和 <span class="arithmatex">\(4x - 6y\)</span>）告诉你 <span class="arithmatex">\(x\)</span> 方向的斜率随 <span class="arithmatex">\(x\)</span> 移动如何变化，<span class="arithmatex">\(y\)</span> 方向同理。</p>
|
||
</li>
|
||
<li>
|
||
<p>非对角线元素（<span class="arithmatex">\(4y\)</span>）告诉你一个方向的斜率随另一个方向的移动如何变化。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>克莱罗定理</strong>保证：对于具有连续二阶导数的函数，混合偏导数相等：<span class="arithmatex">\(\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>这意味着海森矩阵是对称的，这（正如我们在矩阵章节中看到的）保证了实特征值和正交特征向量。</p>
|
||
</li>
|
||
<li>
|
||
<p>海森矩阵告诉我们临界点（梯度为零的点）附近函数的形状：</p>
|
||
<ul>
|
||
<li>如果 <span class="arithmatex">\(H\)</span> 是正定的（所有特征值为正），则该点是<strong>局部极小值点</strong>，曲面像碗一样向各个方向向上弯曲。</li>
|
||
<li>如果 <span class="arithmatex">\(H\)</span> 是负定的（所有特征值为负），则该点是<strong>局部极大值点</strong>，曲面像倒扣的碗一样向各个方向向下弯曲。</li>
|
||
<li>如果 <span class="arithmatex">\(H\)</span> 同时具有正负特征值，则该点是<strong>鞍点</strong>，曲面在某些方向上向上弯曲，在另一些方向上向下弯曲，就像山坳一样。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p><strong>多变量链式法则</strong>将链式法则扩展到多变量函数。如果 <span class="arithmatex">\(z = f(x, y)\)</span>，其中 <span class="arithmatex">\(x = g(t)\)</span> 且 <span class="arithmatex">\(y = h(t)\)</span>，则：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>从 <span class="arithmatex">\(t\)</span> 到 <span class="arithmatex">\(z\)</span> 的每条路径都贡献一项：沿该路径的偏导数乘以中间变量对 <span class="arithmatex">\(t\)</span> 的导数。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，如果 <span class="arithmatex">\(z = x^2 y + 3x - y^2\)</span>，<span class="arithmatex">\(x = \cos(t)\)</span>，<span class="arithmatex">\(y = \sin(t)\)</span>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\frac{dz}{dt} = (2xy + 3)(-\sin t) + (x^2 - 2y)(\cos t)\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>除了手动计算导数，还有三种方法：</p>
|
||
<ul>
|
||
<li><strong>数值微分</strong>：用 <span class="arithmatex">\(f'(x) \approx \frac{f(x+h) - f(x-h)}{2h}\)</span>（取很小的 <span class="arithmatex">\(h\)</span>）来近似。简单但有噪声且不精确。</li>
|
||
<li><strong>符号微分</strong>：通过代数地应用求导法则产生精确表达式。可能导致表达式呈指数级膨胀。</li>
|
||
<li><strong>自动微分（autodiff）</strong>：跟踪运算链并高效地计算精确导数。JAX、PyTorch 和 TensorFlow 都使用这种方法。它能给出精确的数值（而非近似值），且不会产生臃肿的符号表达式。</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colab-notebook">编程练习（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>使用 <code>jax.grad</code> 计算函数 <span class="arithmatex">\(f(x, y) = x^2 y + 3x - 2y\)</span> 在点 <span class="arithmatex">\((1, 2)\)</span> 处的梯度。由于 <span class="arithmatex">\(f\)</span> 接收向量输入，请使用带 <code>argnums</code> 参数的 <code>jax.grad</code>。
|
||
<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</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">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-3" 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="k">def</span><span class="w"> </span><span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">y</span> <span class="o">+</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">df_dx</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">argnums</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">df_dy</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">argnums</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"∂f/∂x = </span><span class="si">{</span><span class="n">df_dx</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> (期望: </span><span class="si">{</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">3</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"∂f/∂y = </span><span class="si">{</span><span class="n">df_dy</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> (期望: </span><span class="si">{</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>使用 <code>jax.jacobian</code> 计算向量值函数的雅可比矩阵，并与手动计算结果进行比较。
|
||
<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>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="k">def</span><span class="w"> </span><span class="nf">F</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a> <span class="k">return</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">**</span><span class="mi">2</span><span class="p">])</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">J</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">jacobian</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">x</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">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"在 (1,2) 处的雅可比矩阵:</span><span class="se">\n</span><span class="si">{</span><span class="n">J</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="c1"># 期望: [[2*x[0], 1], [x[1]**2, 2*x[0]*x[1]]] = [[2, 1], [4, 4]]</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>使用 <code>jax.hessian</code> 计算 <span class="arithmatex">\(f(x, y) = x^3 + 2xy^2 - y^3\)</span> 的海森矩阵，并验证其对称性。
|
||
<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>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="k">def</span><span class="w"> </span><span class="nf">f</span><span class="p">(</span><span class="n">xy</span><span class="p">):</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">xy</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">xy</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a> <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">y</span><span class="o">**</span><span class="mi">3</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">H</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">hessian</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">point</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">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">hess</span> <span class="o">=</span> <span class="n">H</span><span class="p">(</span><span class="n">point</span><span class="p">)</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"海森矩阵:</span><span class="se">\n</span><span class="si">{</span><span class="n">hess</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"是否对称: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">hess</span><span class="p">,</span><span class="w"> </span><span class="n">hess</span><span class="o">.</span><span class="n">T</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="c1"># 期望: [[6x, 4y], [4y, 4x-6y]] = [[6, 8], [8, -8]]</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>从头构建一个极简的自动微分引擎。</p>
|
||
<ul>
|
||
<li>每个 <code>Var</code> 追踪其值以及如何通过链式法则反向传播梯度。</li>
|
||
<li>尝试扩展更多运算（除法、幂运算等）。</li>
|
||
<li>这是 JAX、PyTorch 和 Numpy 的设计基础。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">Var</span><span class="p">:</span>
|
||
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a> <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">children</span><span class="o">=</span><span class="p">(),</span> <span class="n">backward_fn</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
|
||
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a> <span class="bp">self</span><span class="o">.</span><span class="n">val</span> <span class="o">=</span> <span class="n">val</span>
|
||
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a> <span class="bp">self</span><span class="o">.</span><span class="n">grad</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a> <span class="bp">self</span><span class="o">.</span><span class="n">children</span> <span class="o">=</span> <span class="n">children</span>
|
||
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a> <span class="bp">self</span><span class="o">.</span><span class="n">backward_fn</span> <span class="o">=</span> <span class="n">backward_fn</span>
|
||
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a>
|
||
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a> <span class="k">def</span><span class="w"> </span><span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
|
||
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a> <span class="n">out</span> <span class="o">=</span> <span class="n">Var</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">val</span> <span class="o">+</span> <span class="n">other</span><span class="o">.</span><span class="n">val</span><span class="p">,</span> <span class="n">children</span><span class="o">=</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">))</span>
|
||
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a> <span class="k">def</span><span class="w"> </span><span class="nf">_backward</span><span class="p">():</span>
|
||
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a> <span class="bp">self</span><span class="o">.</span><span class="n">grad</span> <span class="o">+=</span> <span class="n">out</span><span class="o">.</span><span class="n">grad</span> <span class="c1"># d(a+b)/da = 1</span>
|
||
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a> <span class="n">other</span><span class="o">.</span><span class="n">grad</span> <span class="o">+=</span> <span class="n">out</span><span class="o">.</span><span class="n">grad</span> <span class="c1"># d(a+b)/db = 1</span>
|
||
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a> <span class="n">out</span><span class="o">.</span><span class="n">backward_fn</span> <span class="o">=</span> <span class="n">_backward</span>
|
||
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a> <span class="k">return</span> <span class="n">out</span>
|
||
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a>
|
||
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a> <span class="k">def</span><span class="w"> </span><span class="fm">__mul__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
|
||
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a> <span class="n">out</span> <span class="o">=</span> <span class="n">Var</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">val</span> <span class="o">*</span> <span class="n">other</span><span class="o">.</span><span class="n">val</span><span class="p">,</span> <span class="n">children</span><span class="o">=</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">))</span>
|
||
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a> <span class="k">def</span><span class="w"> </span><span class="nf">_backward</span><span class="p">():</span>
|
||
<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a> <span class="bp">self</span><span class="o">.</span><span class="n">grad</span> <span class="o">+=</span> <span class="n">other</span><span class="o">.</span><span class="n">val</span> <span class="o">*</span> <span class="n">out</span><span class="o">.</span><span class="n">grad</span> <span class="c1"># d(a*b)/da = b</span>
|
||
<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a> <span class="n">other</span><span class="o">.</span><span class="n">grad</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">val</span> <span class="o">*</span> <span class="n">out</span><span class="o">.</span><span class="n">grad</span> <span class="c1"># d(a*b)/db = a</span>
|
||
<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a> <span class="n">out</span><span class="o">.</span><span class="n">backward_fn</span> <span class="o">=</span> <span class="n">_backward</span>
|
||
<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a> <span class="k">return</span> <span class="n">out</span>
|
||
<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a>
|
||
<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a> <span class="k">def</span><span class="w"> </span><span class="nf">backward</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
|
||
<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a> <span class="c1"># 拓扑排序，然后传播梯度</span>
|
||
<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a> <span class="c1"># 我们将在数据结构与算法章节中详细介绍</span>
|
||
<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a> <span class="n">order</span><span class="p">,</span> <span class="n">visited</span> <span class="o">=</span> <span class="p">[],</span> <span class="nb">set</span><span class="p">()</span>
|
||
<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a> <span class="k">def</span><span class="w"> </span><span class="nf">topo</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
|
||
<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a> <span class="k">if</span> <span class="n">v</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">visited</span><span class="p">:</span>
|
||
<a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a> <span class="n">visited</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
|
||
<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">v</span><span class="o">.</span><span class="n">children</span><span class="p">:</span>
|
||
<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a> <span class="n">topo</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
|
||
<a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a> <span class="n">order</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
|
||
<a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a> <span class="n">topo</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span>
|
||
<a id="__codelineno-3-35" name="__codelineno-3-35" href="#__codelineno-3-35"></a> <span class="bp">self</span><span class="o">.</span><span class="n">grad</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||
<a id="__codelineno-3-36" name="__codelineno-3-36" href="#__codelineno-3-36"></a> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">reversed</span><span class="p">(</span><span class="n">order</span><span class="p">):</span>
|
||
<a id="__codelineno-3-37" name="__codelineno-3-37" href="#__codelineno-3-37"></a> <span class="k">if</span> <span class="n">v</span><span class="o">.</span><span class="n">backward_fn</span><span class="p">:</span>
|
||
<a id="__codelineno-3-38" name="__codelineno-3-38" href="#__codelineno-3-38"></a> <span class="n">v</span><span class="o">.</span><span class="n">backward_fn</span><span class="p">()</span>
|
||
<a id="__codelineno-3-39" name="__codelineno-3-39" href="#__codelineno-3-39"></a>
|
||
<a id="__codelineno-3-40" name="__codelineno-3-40" href="#__codelineno-3-40"></a><span class="c1"># f(x, y) = x*x*y + x 在 (3, 2) 处</span>
|
||
<a id="__codelineno-3-41" name="__codelineno-3-41" href="#__codelineno-3-41"></a><span class="n">x</span> <span class="o">=</span> <span class="n">Var</span><span class="p">(</span><span class="mf">3.0</span><span class="p">)</span>
|
||
<a id="__codelineno-3-42" name="__codelineno-3-42" href="#__codelineno-3-42"></a><span class="n">y</span> <span class="o">=</span> <span class="n">Var</span><span class="p">(</span><span class="mf">2.0</span><span class="p">)</span>
|
||
<a id="__codelineno-3-43" name="__codelineno-3-43" href="#__codelineno-3-43"></a><span class="n">f</span> <span class="o">=</span> <span class="n">x</span> <span class="o">*</span> <span class="n">x</span> <span class="o">*</span> <span class="n">y</span> <span class="o">+</span> <span class="n">x</span> <span class="c1"># = 3*3*2 + 3 = 21</span>
|
||
<a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a>
|
||
<a id="__codelineno-3-45" name="__codelineno-3-45" href="#__codelineno-3-45"></a><span class="n">f</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
|
||
<a id="__codelineno-3-46" name="__codelineno-3-46" href="#__codelineno-3-46"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"f = </span><span class="si">{</span><span class="n">f</span><span class="o">.</span><span class="n">val</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span> <span class="c1"># 21.0</span>
|
||
<a id="__codelineno-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"df/dx = </span><span class="si">{</span><span class="n">x</span><span class="o">.</span><span class="n">grad</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span> <span class="c1"># 2*x*y + 1 = 13.0</span>
|
||
<a id="__codelineno-3-48" name="__codelineno-3-48" href="#__codelineno-3-48"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"df/dy = </span><span class="si">{</span><span class="n">y</span><span class="o">.</span><span class="n">grad</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span> <span class="c1"># x*x = 9.0</span>
|
||
</code></pre></div></li>
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