8303 lines
313 KiB
Plaintext
8303 lines
313 KiB
Plaintext
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"# 学习率调度器\n",
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":label:`sec_scheduler`\n",
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"\n",
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"到目前为止，我们主要关注如何更新权重向量的优化算法，而不是它们的更新速率。\n",
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"然而，调整学习率通常与实际算法同样重要，有如下几方面需要考虑：\n",
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"\n",
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"* 首先，学习率的大小很重要。如果它太大，优化就会发散；如果它太小，训练就会需要过长时间，或者我们最终只能得到次优的结果。我们之前看到问题的条件数很重要（有关详细信息，请参见 :numref:`sec_momentum`）。直观地说，这是最不敏感与最敏感方向的变化量的比率。\n",
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"* 其次，衰减速率同样很重要。如果学习率持续过高，我们可能最终会在最小值附近弹跳，从而无法达到最优解。 :numref:`sec_minibatch_sgd`比较详细地讨论了这一点，在 :numref:`sec_sgd`中我们则分析了性能保证。简而言之，我们希望速率衰减，但要比$\\mathcal{O}(t^{-\\frac{1}{2}})$慢，这样能成为解决凸问题的不错选择。\n",
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"* 另一个同样重要的方面是初始化。这既涉及参数最初的设置方式（详情请参阅 :numref:`sec_numerical_stability`），又关系到它们最初的演变方式。这被戏称为*预热*（warmup），即我们最初开始向着解决方案迈进的速度有多快。一开始的大步可能没有好处，特别是因为最初的参数集是随机的。最初的更新方向可能也是毫无意义的。\n",
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"* 最后，还有许多优化变体可以执行周期性学习率调整。这超出了本章的范围，我们建议读者阅读 :cite:`Izmailov.Podoprikhin.Garipov.ea.2018`来了解个中细节。例如，如何通过对整个路径参数求平均值来获得更好的解。\n",
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"\n",
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"鉴于管理学习率需要很多细节，因此大多数深度学习框架都有自动应对这个问题的工具。\n",
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"在本章中，我们将梳理不同的调度策略对准确性的影响，并展示如何通过*学习率调度器*（learning rate scheduler）来有效管理。\n",
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"\n",
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"## 一个简单的问题\n",
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"\n",
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"我们从一个简单的问题开始，这个问题可以轻松计算，但足以说明要义。\n",
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"为此，我们选择了一个稍微现代化的LeNet版本（激活函数使用`relu`而不是`sigmoid`，汇聚层使用最大汇聚层而不是平均汇聚层），并应用于Fashion-MNIST数据集。\n",
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"此外，我们混合网络以提高性能。\n",
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"由于大多数代码都是标准的，我们只介绍基础知识，而不做进一步的详细讨论。如果需要，请参阅 :numref:`chap_cnn`进行复习。\n"
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"pytorch"
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"source": [
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"%matplotlib inline\n",
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"import math\n",
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"import torch\n",
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"from torch import nn\n",
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"from torch.optim import lr_scheduler\n",
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"from d2l import torch as d2l\n",
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"\n",
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"\n",
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"def net_fn():\n",
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" model = nn.Sequential(\n",
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" nn.Conv2d(1, 6, kernel_size=5, padding=2), nn.ReLU(),\n",
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" nn.MaxPool2d(kernel_size=2, stride=2),\n",
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" nn.Conv2d(6, 16, kernel_size=5), nn.ReLU(),\n",
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" nn.MaxPool2d(kernel_size=2, stride=2),\n",
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" nn.Flatten(),\n",
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" nn.Linear(16 * 5 * 5, 120), nn.ReLU(),\n",
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" nn.Linear(120, 84), nn.ReLU(),\n",
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" nn.Linear(84, 10))\n",
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"\n",
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" return model\n",
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"\n",
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"loss = nn.CrossEntropyLoss()\n",
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"device = d2l.try_gpu()\n",
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"\n",
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"batch_size = 256\n",
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"train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size=batch_size)\n",
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"\n",
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"# 代码几乎与d2l.train_ch6定义在卷积神经网络一章LeNet一节中的相同\n",
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"def train(net, train_iter, test_iter, num_epochs, loss, trainer, device,\n",
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" scheduler=None):\n",
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" net.to(device)\n",
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" animator = d2l.Animator(xlabel='epoch', xlim=[0, num_epochs],\n",
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" legend=['train loss', 'train acc', 'test acc'])\n",
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"\n",
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" for epoch in range(num_epochs):\n",
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" metric = d2l.Accumulator(3) # train_loss,train_acc,num_examples\n",
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" for i, (X, y) in enumerate(train_iter):\n",
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" net.train()\n",
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" trainer.zero_grad()\n",
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" X, y = X.to(device), y.to(device)\n",
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" y_hat = net(X)\n",
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" l = loss(y_hat, y)\n",
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" l.backward()\n",
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" trainer.step()\n",
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" with torch.no_grad():\n",
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" metric.add(l * X.shape[0], d2l.accuracy(y_hat, y), X.shape[0])\n",
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" train_loss = metric[0] / metric[2]\n",
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" train_acc = metric[1] / metric[2]\n",
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" if (i + 1) % 50 == 0:\n",
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" animator.add(epoch + i / len(train_iter),\n",
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" (train_loss, train_acc, None))\n",
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"\n",
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" test_acc = d2l.evaluate_accuracy_gpu(net, test_iter)\n",
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" animator.add(epoch+1, (None, None, test_acc))\n",
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"\n",
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" if scheduler:\n",
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" if scheduler.__module__ == lr_scheduler.__name__:\n",
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" # UsingPyTorchIn-Builtscheduler\n",
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" scheduler.step()\n",
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" else:\n",
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" # Usingcustomdefinedscheduler\n",
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" for param_group in trainer.param_groups:\n",
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" param_group['lr'] = scheduler(epoch)\n",
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"\n",
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" print(f'train loss {train_loss:.3f}, train acc {train_acc:.3f}, '\n",
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" f'test acc {test_acc:.3f}')"
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]
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"source": [
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"让我们来看看如果使用默认设置，调用此算法会发生什么。\n",
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||
"例如设学习率为$0.3$并训练$30$次迭代。\n",
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||
"留意在超过了某点、测试准确度方面的进展停滞时，训练准确度将如何继续提高。\n",
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"两条曲线之间的间隙表示过拟合。\n"
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"train loss 0.128, train acc 0.951, test acc 0.885\n"
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]
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|
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" def __init__(self, lr=0.1):\n",
|
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" self.lr = lr\n",
|
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"\n",
|
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},
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"source": [
|
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"source": [
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"net = net_fn()\n",
|
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"trainer = torch.optim.SGD(net.parameters(), lr)\n",
|
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"train(net, train_iter, test_iter, num_epochs, loss, trainer, device,\n",
|
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" scheduler)"
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]
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},
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"这比以前好一些：曲线比以前更加平滑，并且过拟合更小了。\n",
|
||
"遗憾的是，关于为什么在理论上某些策略会导致较轻的过拟合，有一些观点认为，较小的步长将导致参数更接近零，因此更简单。\n",
|
||
"但是，这并不能完全解释这种现象，因为我们并没有真正地提前停止，而只是轻柔地降低了学习率。\n",
|
||
"\n",
|
||
"## 策略\n",
|
||
"\n",
|
||
"虽然我们不可能涵盖所有类型的学习率调度器，但我们会尝试在下面简要概述常用的策略：多项式衰减和分段常数表。\n",
|
||
"此外，余弦学习率调度在实践中的一些问题上运行效果很好。\n",
|
||
"在某些问题上，最好在使用较高的学习率之前预热优化器。\n",
|
||
"\n",
|
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"### 单因子调度器\n",
|
||
"\n",
|
||
"多项式衰减的一种替代方案是乘法衰减，即$\\eta_{t+1} \\leftarrow \\eta_t \\cdot \\alpha$其中$\\alpha \\in (0, 1)$。\n",
|
||
"为了防止学习率衰减到一个合理的下界之下，\n",
|
||
"更新方程经常修改为$\\eta_{t+1} \\leftarrow \\mathop{\\mathrm{max}}(\\eta_{\\mathrm{min}}, \\eta_t \\cdot \\alpha)$。\n"
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"net = net_fn()\n",
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"trainer = torch.optim.SGD(net.parameters(), lr=0.5)\n",
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"scheduler = lr_scheduler.MultiStepLR(trainer, milestones=[15, 30], gamma=0.5)\n",
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"\n",
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"def get_lr(trainer, scheduler):\n",
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" lr = scheduler.get_last_lr()[0]\n",
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" trainer.step()\n",
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" scheduler.step()\n",
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" return lr\n",
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"\n",
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"d2l.plot(torch.arange(num_epochs), [get_lr(trainer, scheduler)\n",
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" for t in range(num_epochs)])"
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]
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},
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"source": [
|
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"这种分段恒定学习率调度背后的直觉是，让优化持续进行，直到权重向量的分布达到一个驻点。\n",
|
||
"此时，我们才将学习率降低，以获得更高质量的代理来达到一个良好的局部最小值。\n",
|
||
"下面的例子展示了如何使用这种方法产生更好的解决方案。\n"
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]
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},
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{
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"tab": [
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"pytorch"
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]
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"text": [
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"train loss 0.191, train acc 0.928, test acc 0.889\n"
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"### 余弦调度器\n",
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||
"\n",
|
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"余弦调度器是 :cite:`Loshchilov.Hutter.2016`提出的一种启发式算法。\n",
|
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"它所依据的观点是：我们可能不想在一开始就太大地降低学习率，而且可能希望最终能用非常小的学习率来“改进”解决方案。\n",
|
||
"这产生了一个类似于余弦的调度，函数形式如下所示，学习率的值在$t \\in [0, T]$之间。\n",
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"\n",
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"$$\\eta_t = \\eta_T + \\frac{\\eta_0 - \\eta_T}{2} \\left(1 + \\cos(\\pi t/T)\\right)$$\n",
|
||
"\n",
|
||
"这里$\\eta_0$是初始学习率，$\\eta_T$是当$T$时的目标学习率。\n",
|
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"此外，对于$t > T$，我们只需将值固定到$\\eta_T$而不再增加它。\n",
|
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"在下面的示例中，我们设置了最大更新步数$T = 20$。\n"
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"source": [
|
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"\n",
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|
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"预热可以应用于任何调度器，而不仅仅是余弦。\n",
|
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"有关学习率调度的更多实验和更详细讨论，请参阅 :cite:`Gotmare.Keskar.Xiong.ea.2018`。\n",
|
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"其中，这篇论文的点睛之笔的发现：预热阶段限制了非常深的网络中参数的发散程度 。\n",
|
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"这在直觉上是有道理的：在网络中那些一开始花费最多时间取得进展的部分，随机初始化会产生巨大的发散。\n",
|
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"\n",
|
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"## 小结\n",
|
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"\n",
|
||
"* 在训练期间逐步降低学习率可以提高准确性，并且减少模型的过拟合。\n",
|
||
"* 在实验中，每当进展趋于稳定时就降低学习率，这是很有效的。从本质上说，这可以确保我们有效地收敛到一个适当的解，也只有这样才能通过降低学习率来减小参数的固有方差。\n",
|
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"* 余弦调度器在某些计算机视觉问题中很受欢迎。\n",
|
||
"* 优化之前的预热期可以防止发散。\n",
|
||
"* 优化在深度学习中有多种用途。对于同样的训练误差而言，选择不同的优化算法和学习率调度，除了最大限度地减少训练时间，可以导致测试集上不同的泛化和过拟合量。\n",
|
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"\n",
|
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"## 练习\n",
|
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"\n",
|
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"1. 试验给定固定学习率的优化行为。这种情况下可以获得的最佳模型是什么？\n",
|
||
"1. 如果改变学习率下降的指数，收敛性会如何改变？在实验中方便起见，使用`PolyScheduler`。\n",
|
||
"1. 将余弦调度器应用于大型计算机视觉问题，例如训练ImageNet数据集。与其他调度器相比，它如何影响性能？\n",
|
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"1. 预热应该持续多长时间？\n",
|
||
"1. 可以试着把优化和采样联系起来吗？首先，在随机梯度朗之万动力学上使用 :cite:`Welling.Teh.2011`的结果。\n"
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"source": [
|
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]
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}
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