3201 lines
120 KiB
Plaintext
3201 lines
120 KiB
Plaintext
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"# RMSProp算法\n",
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":label:`sec_rmsprop`\n",
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"\n",
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" :numref:`sec_adagrad`中的关键问题之一，是学习率按预定时间表$\\mathcal{O}(t^{-\\frac{1}{2}})$显著降低。\n",
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"虽然这通常适用于凸问题，但对于深度学习中遇到的非凸问题，可能并不理想。\n",
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"但是，作为一个预处理器，Adagrad算法按坐标顺序的适应性是非常可取的。\n",
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"\n",
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" :cite:`Tieleman.Hinton.2012`建议以RMSProp算法作为将速率调度与坐标自适应学习率分离的简单修复方法。\n",
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"问题在于，Adagrad算法将梯度$\\mathbf{g}_t$的平方累加成状态矢量$\\mathbf{s}_t = \\mathbf{s}_{t-1} + \\mathbf{g}_t^2$。\n",
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"因此，由于缺乏规范化，没有约束力，$\\mathbf{s}_t$持续增长，几乎上是在算法收敛时呈线性递增。\n",
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"\n",
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"解决此问题的一种方法是使用$\\mathbf{s}_t / t$。\n",
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"对$\\mathbf{g}_t$的合理分布来说，它将收敛。\n",
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"遗憾的是，限制行为生效可能需要很长时间，因为该流程记住了值的完整轨迹。\n",
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"另一种方法是按动量法中的方式使用泄漏平均值，即$\\mathbf{s}_t \\leftarrow \\gamma \\mathbf{s}_{t-1} + (1-\\gamma) \\mathbf{g}_t^2$，其中参数$\\gamma > 0$。\n",
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"保持所有其它部分不变就产生了RMSProp算法。\n",
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"\n",
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"## 算法\n",
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"\n",
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"让我们详细写出这些方程式。\n",
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"\n",
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"$$\\begin{aligned}\n",
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" \\mathbf{s}_t & \\leftarrow \\gamma \\mathbf{s}_{t-1} + (1 - \\gamma) \\mathbf{g}_t^2, \\\\\n",
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" \\mathbf{x}_t & \\leftarrow \\mathbf{x}_{t-1} - \\frac{\\eta}{\\sqrt{\\mathbf{s}_t + \\epsilon}} \\odot \\mathbf{g}_t.\n",
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"\\end{aligned}$$\n",
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"\n",
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"常数$\\epsilon > 0$通常设置为$10^{-6}$，以确保我们不会因除以零或步长过大而受到影响。\n",
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"鉴于这种扩展，我们现在可以自由控制学习率$\\eta$，而不考虑基于每个坐标应用的缩放。\n",
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"就泄漏平均值而言，我们可以采用与之前在动量法中适用的相同推理。\n",
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"扩展$\\mathbf{s}_t$定义可获得\n",
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"\n",
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"$$\n",
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"\\begin{aligned}\n",
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"\\mathbf{s}_t & = (1 - \\gamma) \\mathbf{g}_t^2 + \\gamma \\mathbf{s}_{t-1} \\\\\n",
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"& = (1 - \\gamma) \\left(\\mathbf{g}_t^2 + \\gamma \\mathbf{g}_{t-1}^2 + \\gamma^2 \\mathbf{g}_{t-2} + \\ldots, \\right).\n",
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"\\end{aligned}\n",
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"$$\n",
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"\n",
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"同之前在 :numref:`sec_momentum`小节一样，我们使用$1 + \\gamma + \\gamma^2 + \\ldots, = \\frac{1}{1-\\gamma}$。\n",
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"因此，权重总和标准化为$1$且观测值的半衰期为$\\gamma^{-1}$。\n",
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"让我们图像化各种数值的$\\gamma$在过去40个时间步长的权重。\n"
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"import math\n",
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"import torch\n",
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"from d2l import torch as d2l"
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]
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"## 从零开始实现\n",
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"\n",
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"和之前一样，我们使用二次函数$f(\\mathbf{x})=0.1x_1^2+2x_2^2$来观察RMSProp算法的轨迹。\n",
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"RMSProp算法中不会发生这种情况，因为$\\eta$是单独控制的。\n"
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"\n",
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"## 小结\n",
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"\n",
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"* RMSProp算法与Adagrad算法非常相似，因为两者都使用梯度的平方来缩放系数。\n",
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"\n",
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"1. 如果我们设置$\\gamma = 1$，实验会发生什么？为什么？\n",
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"1. 随着优化的进展，需要调整$\\gamma$吗？RMSProp算法对此有多敏感？\n"
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|
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]
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