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机器学习
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计算语言学
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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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|
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|
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</a>
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高级文本生成
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</span>
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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">优化<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>优化是模型训练的数学核心——寻找使损失函数最小的参数。本文涵盖驻点、凸性、梯度下降、牛顿法、带拉格朗日乘数的约束优化，以及驱动现代深度学习的主流优化器（SGD、Adam）。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>训练神经网络、拟合回归线、调优超参数：几乎所有机器学习算法的核心都是一个<strong>优化</strong>问题。</p>
|
||
</li>
|
||
<li>
|
||
<p>我们有一个函数（损失函数、代价函数、目标函数），希望找到使其尽可能小（或大）的输入。</p>
|
||
</li>
|
||
<li>
|
||
<p>在优化之前，我们需要理解函数的<strong>零点</strong>（或根）。<span class="arithmatex">\(f(x)\)</span> 的零点是指满足 <span class="arithmatex">\(f(x) = 0\)</span> 的 <span class="arithmatex">\(x\)</span> 值。从图形上看，这些点就是与 x 轴的交点。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，<span class="arithmatex">\(f(x) = x^2 - 3x + 2 = (x-1)(x-2)\)</span> 的零点在 <span class="arithmatex">\(x = 1\)</span> 和 <span class="arithmatex">\(x = 2\)</span> 处。在两个零点之间，函数为负（<span class="arithmatex">\(f(1.5) = -0.25\)</span>）；在零点之外，函数为正。零点将数轴分割成若干个区域，在每个区域中函数保持相同符号。</p>
|
||
</li>
|
||
<li>
|
||
<p>零点的<strong>重数</strong>是指对应因式出现的次数。</p>
|
||
</li>
|
||
<li>
|
||
<p>在单零点（重数为 1）处，图像穿过 x 轴。在二重零点（重数为 2）处，图像接触 x 轴但反弹回去而不穿过，在该点处看起来是"平坦"的。</p>
|
||
</li>
|
||
<li>
|
||
<p>寻找零点之所以重要，是因为导数 <span class="arithmatex">\(f'(x)\)</span> 的零点正是 <span class="arithmatex">\(f(x)\)</span> 的<strong>驻点</strong>——即极大值或极小值的候选点。</p>
|
||
</li>
|
||
<li>
|
||
<p>在极大值或极小值处，切线是水平的（斜率为 0），因此 <span class="arithmatex">\(f'(x) = 0\)</span>。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="驻点：当导数为零时，函数出现波峰、波谷或鞍点" src="../../images/critical_points.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>但并非每个驻点都是极大值或极小值。<span class="arithmatex">\(f'(x) = 0\)</span> 的点也可能是<strong>拐点</strong>（如 <span class="arithmatex">\(f(x) = x^3\)</span> 在 <span class="arithmatex">\(x = 0\)</span> 处），函数在该点暂时变平但并未改变方向。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>二阶导数检验</strong>可以解决这个问题。在驻点 <span class="arithmatex">\(x = c\)</span>（即 <span class="arithmatex">\(f'(c) = 0\)</span>）处：</p>
|
||
<ul>
|
||
<li>若 <span class="arithmatex">\(f''(c) > 0\)</span>：曲线向下凸（碗状），因此 <span class="arithmatex">\(c\)</span> 是<strong>局部极小值</strong>。</li>
|
||
<li>若 <span class="arithmatex">\(f''(c) < 0\)</span>：曲线向上凸（山丘状），因此 <span class="arithmatex">\(c\)</span> 是<strong>局部极大值</strong>。</li>
|
||
<li>若 <span class="arithmatex">\(f''(c) = 0\)</span>：检验无效，需要使用更高阶导数或其他方法。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>例如，<span class="arithmatex">\(f(x) = x^3 - 3x\)</span>。导数为 <span class="arithmatex">\(f'(x) = 3x^2 - 3 = 3(x-1)(x+1)\)</span>，因此驻点在 <span class="arithmatex">\(x = -1\)</span> 和 <span class="arithmatex">\(x = 1\)</span> 处。二阶导数为 <span class="arithmatex">\(f''(x) = 6x\)</span>。在 <span class="arithmatex">\(x = -1\)</span> 处：<span class="arithmatex">\(f''(-1) = -6 < 0\)</span>（局部极大值）。在 <span class="arithmatex">\(x = 1\)</span> 处：<span class="arithmatex">\(f''(1) = 6 > 0\)</span>（局部极小值）。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果连接函数图像上任意两点的线段位于图像之上（或与之重合），则该函数是<strong>凸的</strong>。可以想象成一个碗形，处处向上弯曲。数学上，若对所有 <span class="arithmatex">\(x\)</span> 有 <span class="arithmatex">\(f''(x) \geq 0\)</span>，则 <span class="arithmatex">\(f\)</span> 是凸函数。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="凸函数具有唯一的全局最小值；非凸函数可能有多个局部最小值" src="../../images/convex_nonconvex.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>凸性的强大之处在于凸函数有一个卓越的性质：每个局部极小值同时也是<strong>全局最小值</strong>。不存在会让人陷入的欺骗性局部低谷。如果你把一个球滚入凸碗中，它总是会到达底部。</p>
|
||
</li>
|
||
<li>
|
||
<p>若 <span class="arithmatex">\(-f\)</span> 是凸的，则函数是<strong>凹的</strong>（向下弯曲）。函数从凹性过渡到凸性的点称为<strong>拐点</strong>，出现在 <span class="arithmatex">\(f''(x) = 0\)</span> 处。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>牛顿法</strong>利用切线寻找函数的零点（进而也可用于寻找其导数的驻点）。从初始猜测 <span class="arithmatex">\(x_0\)</span> 出发，迭代更新：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]</div>
|
||
<p><img alt="牛顿法：沿切线方向逼近根的更好近似值" src="../../images/newtons_method.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>其思想是：在 <span class="arithmatex">\(x_n\)</span> 处画出切线，找到它与 x 轴的交点，该交点即为 <span class="arithmatex">\(x_{n+1}\)</span>。对于性质良好且初始点选取恰当的函数，牛顿法收敛非常快（二次收敛，即每步正确位数大致翻倍）。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，求 <span class="arithmatex">\(\sqrt{5}\)</span>（即 <span class="arithmatex">\(f(x) = x^2 - 5\)</span> 的零点）：<span class="arithmatex">\(f'(x) = 2x\)</span>，因此 <span class="arithmatex">\(x_{n+1} = x_n - \frac{x_n^2 - 5}{2x_n}\)</span>。从 <span class="arithmatex">\(x_0 = 2\)</span> 开始：<span class="arithmatex">\(x_1 = 2.25\)</span>，<span class="arithmatex">\(x_2 = 2.2361\ldots\)</span>，已精确到小数点后四位。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果初始猜测离根太远、根附近 <span class="arithmatex">\(f'(x) = 0\)</span>，或函数在附近有拐点，牛顿法可能会失败。此外，它还需要计算导数，这可能代价高昂。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于优化（寻找极小值而非零点），我们将牛顿法应用于 <span class="arithmatex">\(f'(x) = 0\)</span>，得到更新公式：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[x_{n+1} = x_n - \frac{f'(x_n)}{f''(x_n)}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>在多维情形下，这变为 <span class="arithmatex">\(\mathbf{x}_{n+1} = \mathbf{x}_n - H^{-1} \nabla f(\mathbf{x}_n)\)</span>，其中 <span class="arithmatex">\(H\)</span> 是 Hessian 矩阵。这正是上一节中二阶泰勒近似的实际应用：将函数近似为二次型，跳到该二次型的极小值点，然后重复。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>拉格朗日乘数</strong>用于求解<strong>约束优化</strong>：在约束条件 <span class="arithmatex">\(g(x, y) = c\)</span> 下求 <span class="arithmatex">\(f(x, y)\)</span> 的最优值。我们不是在 <span class="arithmatex">\(\mathbb{R}^n\)</span> 中全域搜索，而是限制在约束条件成立的集合（一条曲线或曲面）上。</p>
|
||
</li>
|
||
<li>
|
||
<p>关键见解是几何层面的：在约束最优解处，<span class="arithmatex">\(f\)</span> 的梯度必须与 <span class="arithmatex">\(g\)</span> 的梯度平行。如果它们不平行，我们可以沿着约束条件朝某个方向移动，从而继续改进 <span class="arithmatex">\(f\)</span> 的值，这意味着还没有达到最优。</p>
|
||
</li>
|
||
<li>
|
||
<p>我们引入一个新变量 <span class="arithmatex">\(\lambda\)</span>（拉格朗日乘数），定义<strong>拉格朗日函数</strong>：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\mathcal{L}(x, y, \lambda) = f(x, y) - \lambda(g(x, y) - c)\]</div>
|
||
<ul>
|
||
<li>令所有偏导数为零，得到一个方程组，其解即为约束最优解：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[\frac{\partial \mathcal{L}}{\partial x} = 0, \quad \frac{\partial \mathcal{L}}{\partial y} = 0, \quad \frac{\partial \mathcal{L}}{\partial \lambda} = 0\]</div>
|
||
<p><img alt="拉格朗日乘数：在最优解处，f 和 g 的梯度平行" src="../../images/lagrange_multiplier.svg" /></p>
|
||
<ul>
|
||
<li>例如，在 <span class="arithmatex">\(x^2 + y^2 = 1\)</span> 的约束下最大化 <span class="arithmatex">\(f(x,y) = x^2 y\)</span>。拉格朗日函数为 <span class="arithmatex">\(\mathcal{L} = x^2 y - \lambda(x^2 + y^2 - 1)\)</span>。求偏导：</li>
|
||
</ul>
|
||
<div class="arithmatex">\[2xy - 2\lambda x = 0, \quad x^2 - 2\lambda y = 0, \quad x^2 + y^2 = 1\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>由第一个方程（假设 <span class="arithmatex">\(x \neq 0\)</span>）：<span class="arithmatex">\(\lambda = y\)</span>。代入第二个方程：<span class="arithmatex">\(x^2 = 2y^2\)</span>。结合约束条件：<span class="arithmatex">\(2y^2 + y^2 = 1\)</span>，得 <span class="arithmatex">\(y = \frac{1}{\sqrt{3}}\)</span>。最大值为 <span class="arithmatex">\(f = \frac{2}{3\sqrt{3}}\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于不等式约束（<span class="arithmatex">\(g(x,y) \leq c\)</span> 而非 <span class="arithmatex">\(= c\)</span>），<strong>Karush-Kuhn-Tucker（KKT）条件</strong>推广了拉格朗日乘数法。约束要么是激活的（有效约束，按等式处理），要么是非激活的（解在内部，约束无关紧要）。</p>
|
||
</li>
|
||
<li>
|
||
<p>在实践中，我们很少手工进行优化。以下是主要的算法家族：</p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>一阶方法</strong>（仅使用梯度）：梯度下降、随机梯度下降（SGD）、Adam。这些方法每步计算成本低，但收敛可能较慢，尤其是在病态问题上。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>二阶方法</strong>（使用梯度和 Hessian 矩阵）：牛顿法收敛快，但计算和求逆 Hessian 矩阵代价高昂（对于 <span class="arithmatex">\(n\)</span> 个参数为 <span class="arithmatex">\(O(n^3)\)</span>）。<strong>拟牛顿法</strong>（如 BFGS 和 L-BFGS）仅利用梯度信息近似 Hessian 矩阵，比一阶方法收敛更快，又无需承担完全的二阶方法计算成本。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>共轭梯度法</strong>：适用于大型稀疏系统，仅需矩阵-向量乘积，无需存储完整的 Hessian 矩阵。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>高斯-牛顿法</strong>和<strong>莱文贝格-马夸尔特法</strong>：专门用于最小二乘问题（在回归中常见），通过 Jacobian 矩阵近似 Hessian 矩阵。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>自然梯度下降</strong>：利用 Fisher 信息矩阵考虑参数空间的几何结构，对概率模型可能更有效。</p>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>优化器的选择取决于具体问题。对于深度学习，一阶方法（尤其是 Adam）占主导地位，因为参数量巨大（数百万到数十亿），计算 Hessian 矩阵不切实际。对于目标函数光滑的小规模问题，二阶方法可能快得多。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colab-notebook">编程练习（在 CoLab 或 notebook 中完成）<a class="headerlink" href="#colab-notebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>实现牛顿法求 <span class="arithmatex">\(\sqrt{7}\)</span>（即 <span class="arithmatex">\(f(x) = x^2 - 7\)</span> 的零点）。观察其快速收敛。
|
||
<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>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">7</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="n">df</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">x</span> <span class="o">=</span> <span class="mf">3.0</span> <span class="c1"># 初始猜测</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></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="mi">6</span><span class="p">):</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">/</span> <span class="n">df</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"step </span><span class="si">{</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="si">}</span><span class="s2">: x = </span><span class="si">{</span><span class="n">x</span><span class="si">:</span><span class="s2">.10f</span><span class="si">}</span><span class="s2"> (error: </span><span class="si">{</span><span class="nb">abs</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">jnp</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mf">7.0</span><span class="p">))</span><span class="si">:</span><span class="s2">.2e</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>使用梯度下降最小化 <span class="arithmatex">\(f(x, y) = (x - 3)^2 + (y + 1)^2\)</span>。最小值在 <span class="arithmatex">\((3, -1)\)</span> 处。尝试不同的学习率。
|
||
<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">params</span><span class="p">):</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">params</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a> <span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">3</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">grad_f</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>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">params</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">0.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">lr</span> <span class="o">=</span> <span class="mf">0.1</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></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="mi">20</span><span class="p">):</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a> <span class="n">g</span> <span class="o">=</span> <span class="n">grad_f</span><span class="p">(</span><span class="n">params</span><span class="p">)</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a> <span class="n">params</span> <span class="o">=</span> <span class="n">params</span> <span class="o">-</span> <span class="n">lr</span> <span class="o">*</span> <span class="n">g</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a> <span class="k">if</span> <span class="n">i</span> <span class="o">%</span> <span class="mi">5</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">19</span><span class="p">:</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"step </span><span class="si">{</span><span class="n">i</span><span class="si">:</span><span class="s2">2d</span><span class="si">}</span><span class="s2">: (</span><span class="si">{</span><span class="n">params</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">, </span><span class="si">{</span><span class="n">params</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">) loss=</span><span class="si">{</span><span class="n">f</span><span class="p">(</span><span class="n">params</span><span class="p">)</span><span class="si">:</span><span class="s2">.6f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>数值求解约束优化问题。在 <span class="arithmatex">\(x + y = 10\)</span> 的约束下最大化 <span class="arithmatex">\(f(x,y) = xy\)</span>，通过参数化 <span class="arithmatex">\(y = 10 - x\)</span> 并求单变量函数的最优值。
|
||
<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="c1"># 代入约束条件：y = 10 - x，所以 f = x(10 - x) = 10x - x²</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span> <span class="o">*</span> <span class="p">(</span><span class="mi">10</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">df</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>
|
||
<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="c1"># 梯度上升（我们要求最大值，所以加上梯度）</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">x</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">lr</span> <span class="o">=</span> <span class="mf">0.1</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></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="mi">20</span><span class="p">):</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">+</span> <span class="n">lr</span> <span class="o">*</span> <span class="n">df</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"x=</span><span class="si">{</span><span class="n">x</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">, y=</span><span class="si">{</span><span class="mi">10</span><span class="o">-</span><span class="n">x</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">, f=</span><span class="si">{</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span> <span class="c1"># 应为 x=5, y=5, f=25</span>
|
||
</code></pre></div></p>
|
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<script src="https://unpkg.com/mathjax@3/es5/tex-mml-chtml.js"></script>
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</body>
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</html> |