12.11. 学习率调度¶ 在 SageMaker Studio Lab 中打开 Notebook
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*。 然而，调整学习率有时与实际的优化算法同样重要。有以下几方面需要考虑。
首先，学习率的*量级*是重要的。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（详情请参阅 :numref:`sec_momentum` ）。直观地说，这是最不敏感与最敏感方向的变化量之比。
其次，衰减率同样很重要。如果学习率保持很大，我们可能会在最小值附近“弹跳”，从而无法达到最优。 :numref:`sec_minibatch-sgd` 对此进行了详细讨论，在 :numref:`sec_sgd` 中我们分析了性能保证。简而言之，我们希望速率衰减，但可能比 \(\mathcal{O}(t^{-\frac{1}{2}})\) 慢，这对于凸问题是一个不错的选择。
另一个同样重要的方面是*初始化*。这既关系到参数最初的设定方式（回顾 :numref:`sec-numerical-stability` 了解详情），也关系到它们最初的演变方式。这被归结为*预热*（warmup）的概念，即我们最初朝解决方案移动的速度有多快。一开始的大步可能没有好处，特别是当初始参数集是随机的时候。最初的更新方向也可能相当没有意义。
最后，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。我们建议读者回顾 :cite:`Izmailov.Podoprikhin.Garipov.ea.2018` 中的细节，例如，如何通过对整个参数*路径*进行平均来获得更好的解。
鉴于管理学习率需要很多细节，大多数深度学习框架都有自动处理的工具。在本章中，我们将回顾不同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。
12.11.1. 演示问题¶
我们从一个“演示”问题开始，这个问题计算成本低廉，但又足够复杂，足以说明一些关键问题。为此，我们选择了一个稍微现代化版本的LeNet（使用relu代替sigmoid激活函数，使用MaxPooling代替AveragePooling），并应用于Fashion-MNIST。此外，我们为了性能对网络进行了混合化。由于大部分代码都是标准的，我们只介绍基础知识，不作进一步的详细讨论。如果需要，请参阅 :numref:`chap-cnn` 进行复习。
%matplotlib inline
import math
import torch
from torch import nn
from torch.optim import lr_scheduler
from d2l import torch as d2l
def net_fn():
model = nn.Sequential(
nn.Conv2d(1, 6, kernel_size=5, padding=2), nn.ReLU(),
nn.MaxPool2d(kernel_size=2, stride=2),
nn.Conv2d(6, 16, kernel_size=5), nn.ReLU(),
nn.MaxPool2d(kernel_size=2, stride=2),
nn.Flatten(),
nn.Linear(16 * 5 * 5, 120), nn.ReLU(),
nn.Linear(120, 84), nn.ReLU(),
nn.Linear(84, 10))
return model
loss = nn.CrossEntropyLoss()
device = d2l.try_gpu()
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size=batch_size)
# The code is almost identical to `d2l.train_ch6` defined in the
# lenet section of chapter convolutional neural networks
def train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
scheduler=None):
net.to(device)
animator = d2l.Animator(xlabel='epoch', xlim=[0, num_epochs],
legend=['train loss', 'train acc', 'test acc'])
for epoch in range(num_epochs):
metric = d2l.Accumulator(3) # train_loss, train_acc, num_examples
for i, (X, y) in enumerate(train_iter):
net.train()
trainer.zero_grad()
X, y = X.to(device), y.to(device)
y_hat = net(X)
l = loss(y_hat, y)
l.backward()
trainer.step()
with torch.no_grad():
metric.add(l * X.shape[0], d2l.accuracy(y_hat, y), X.shape[0])
train_loss = metric[0] / metric[2]
train_acc = metric[1] / metric[2]
if (i + 1) % 50 == 0:
animator.add(epoch + i / len(train_iter),
(train_loss, train_acc, None))
test_acc = d2l.evaluate_accuracy_gpu(net, test_iter)
animator.add(epoch+1, (None, None, test_acc))
if scheduler:
if scheduler.__module__ == lr_scheduler.__name__:
# Using PyTorch In-Built scheduler
scheduler.step()
else:
# Using custom defined scheduler
for param_group in trainer.param_groups:
param_group['lr'] = scheduler(epoch)
print(f'train loss {train_loss:.3f}, train acc {train_acc:.3f}, '
f'test acc {test_acc:.3f}')
%matplotlib inline
from mxnet import autograd, gluon, init, lr_scheduler, np, npx
from mxnet.gluon import nn
from d2l import mxnet as d2l
npx.set_np()
net = nn.HybridSequential()
net.add(nn.Conv2D(channels=6, kernel_size=5, padding=2, activation='relu'),
nn.MaxPool2D(pool_size=2, strides=2),
nn.Conv2D(channels=16, kernel_size=5, activation='relu'),
nn.MaxPool2D(pool_size=2, strides=2),
nn.Dense(120, activation='relu'),
nn.Dense(84, activation='relu'),
nn.Dense(10))
net.hybridize()
loss = gluon.loss.SoftmaxCrossEntropyLoss()
device = d2l.try_gpu()
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size=batch_size)
# The code is almost identical to `d2l.train_ch6` defined in the
# lenet section of chapter convolutional neural networks
def train(net, train_iter, test_iter, num_epochs, loss, trainer, device):
net.initialize(force_reinit=True, ctx=device, init=init.Xavier())
animator = d2l.Animator(xlabel='epoch', xlim=[0, num_epochs],
legend=['train loss', 'train acc', 'test acc'])
for epoch in range(num_epochs):
metric = d2l.Accumulator(3) # train_loss, train_acc, num_examples
for i, (X, y) in enumerate(train_iter):
X, y = X.as_in_ctx(device), y.as_in_ctx(device)
with autograd.record():
y_hat = net(X)
l = loss(y_hat, y)
l.backward()
trainer.step(X.shape[0])
metric.add(l.sum(), d2l.accuracy(y_hat, y), X.shape[0])
train_loss = metric[0] / metric[2]
train_acc = metric[1] / metric[2]
if (i + 1) % 50 == 0:
animator.add(epoch + i / len(train_iter),
(train_loss, train_acc, None))
test_acc = d2l.evaluate_accuracy_gpu(net, test_iter)
animator.add(epoch + 1, (None, None, test_acc))
print(f'train loss {train_loss:.3f}, train acc {train_acc:.3f}, '
f'test acc {test_acc:.3f}')
[22:48:45] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU
%matplotlib inline
import math
import tensorflow as tf
from tensorflow.keras.callbacks import LearningRateScheduler
from d2l import tensorflow as d2l
def net():
return tf.keras.models.Sequential([
tf.keras.layers.Conv2D(filters=6, kernel_size=5, activation='relu',
padding='same'),
tf.keras.layers.AvgPool2D(pool_size=2, strides=2),
tf.keras.layers.Conv2D(filters=16, kernel_size=5,
activation='relu'),
tf.keras.layers.AvgPool2D(pool_size=2, strides=2),
tf.keras.layers.Flatten(),
tf.keras.layers.Dense(120, activation='relu'),
tf.keras.layers.Dense(84, activation='sigmoid'),
tf.keras.layers.Dense(10)])
batch_size = 256
train_iter, test_iter = d2l.load_data_fashion_mnist(batch_size=batch_size)
# The code is almost identical to `d2l.train_ch6` defined in the
# lenet section of chapter convolutional neural networks
def train(net_fn, train_iter, test_iter, num_epochs, lr,
device=d2l.try_gpu(), custom_callback = False):
device_name = device._device_name
strategy = tf.distribute.OneDeviceStrategy(device_name)
with strategy.scope():
optimizer = tf.keras.optimizers.SGD(learning_rate=lr)
loss = tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True)
net = net_fn()
net.compile(optimizer=optimizer, loss=loss, metrics=['accuracy'])
callback = d2l.TrainCallback(net, train_iter, test_iter, num_epochs,
device_name)
if custom_callback is False:
net.fit(train_iter, epochs=num_epochs, verbose=0,
callbacks=[callback])
else:
net.fit(train_iter, epochs=num_epochs, verbose=0,
callbacks=[callback, custom_callback])
return net
让我们看看，如果我们使用默认设置（例如学习率为 \(0.3\)）并训练 \(30\) 个迭代周期，会发生什么。注意，训练准确率持续提高，而测试准确率在超过某个点后停滞不前。两条曲线之间的差距表明存在过拟合。
lr, num_epochs = 0.3, 30
net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr=lr)
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)
train loss 0.145, train acc 0.944, test acc 0.877
lr, num_epochs = 0.3, 30
net.initialize(force_reinit=True, ctx=device, init=init.Xavier())
trainer = gluon.Trainer(net.collect_params(), 'sgd', {'learning_rate': lr})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)
train loss 0.160, train acc 0.939, test acc 0.884
lr, num_epochs = 0.3, 30
train(net, train_iter, test_iter, num_epochs, lr)
loss 0.228, train acc 0.916, test acc 0.890
51109.0 examples/sec on /GPU:0
<keras.engine.sequential.Sequential at 0x7f631399ea40>
12.11.2. 调度器¶
调整学习率的一种方法是在每一步都明确地设置它。这可以通过set_learning_rate方法方便地实现。我们可以在每个迭代周期之后（甚至在每个小批量之后）将其向下调整，例如，以一种动态的方式来响应优化的进展情况。
lr = 0.1
trainer.param_groups[0]["lr"] = lr
print(f'learning rate is now {trainer.param_groups[0]["lr"]:.2f}')
learning rate is now 0.10
trainer.set_learning_rate(0.1)
print(f'learning rate is now {trainer.learning_rate:.2f}')
learning rate is now 0.10
lr = 0.1
dummy_model = tf.keras.models.Sequential([tf.keras.layers.Dense(10)])
dummy_model.compile(tf.keras.optimizers.SGD(learning_rate=lr), loss='mse')
print(f'learning rate is now ,', dummy_model.optimizer.lr.numpy())
learning rate is now , 0.1
更一般地，我们希望定义一个调度器。当调度器被更新次数调用时，它会返回适当的学习率值。让我们定义一个简单的调度器，将学习率设置为 \(\eta = \eta_0 (t + 1)^{-\frac{1}{2}}\)。
class SquareRootScheduler:
def __init__(self, lr=0.1):
self.lr = lr
def __call__(self, num_update):
return self.lr * pow(num_update + 1.0, -0.5)
class SquareRootScheduler:
def __init__(self, lr=0.1):
self.lr = lr
def __call__(self, num_update):
return self.lr * pow(num_update + 1.0, -0.5)
class SquareRootScheduler:
def __init__(self, lr=0.1):
self.lr = lr
def __call__(self, num_update):
return self.lr * pow(num_update + 1.0, -0.5)
让我们在一系列值上绘制它的行为。
scheduler = SquareRootScheduler(lr=0.1)
d2l.plot(torch.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])
scheduler = SquareRootScheduler(lr=0.1)
d2l.plot(np.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])
scheduler = SquareRootScheduler(lr=0.1)
d2l.plot(tf.range(num_epochs), [scheduler(t) for t in range(num_epochs)])
现在，让我们看看这在Fashion-MNIST上的训练效果如何。我们只需将调度器作为额外的参数传递给训练算法。
net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr)
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
scheduler)
train loss 0.273, train acc 0.900, test acc 0.886
trainer = gluon.Trainer(net.collect_params(), 'sgd',
{'lr_scheduler': scheduler})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)
train loss 0.523, train acc 0.810, test acc 0.815
train(net, train_iter, test_iter, num_epochs, lr,
custom_callback=LearningRateScheduler(scheduler))
loss 0.388, train acc 0.858, test acc 0.847
51521.6 examples/sec on /GPU:0
<keras.engine.sequential.Sequential at 0x7f6134289330>
这比之前效果好多了。有两个方面很突出：曲线比之前平滑得多。其次，过拟合更少了。不幸的是，为什么某些策略在*理论上*会导致更少的过拟合，这并不是一个得到很好解决的问题。有一些观点认为，较小的步长将导致参数更接近于零，因此更简单。然而，这并不能完全解释这种现象，因为我们并不是真的提前停止，而只是温和地降低学习率。
12.11.3. 策略¶
虽然我们不可能涵盖所有种类的学习率调度器，但我们试图在下面简要概述流行的策略。常见的选择是多项式衰减和分段常数调度。除此之外，余弦学习率调度在某些问题上根据经验被发现效果很好。最后，在某些问题上，在使用大学习率之前对优化器进行预热是有益的。
12.11.3.1. 因子调度器¶
多项式衰减的另一种替代方案是乘法衰减，即 \(\eta_{t+1} \leftarrow \eta_t \cdot \alpha\) 对于 \(\alpha \in (0, 1)\)。为防止学习率衰减超过一个合理的下限，更新方程通常被修改为 \(\eta_{t+1} \leftarrow \mathop{\mathrm{max}}(\eta_{\mathrm{min}}, \eta_t \cdot \alpha)\)。
class FactorScheduler:
def __init__(self, factor=1, stop_factor_lr=1e-7, base_lr=0.1):
self.factor = factor
self.stop_factor_lr = stop_factor_lr
self.base_lr = base_lr
def __call__(self, num_update):
self.base_lr = max(self.stop_factor_lr, self.base_lr * self.factor)
return self.base_lr
scheduler = FactorScheduler(factor=0.9, stop_factor_lr=1e-2, base_lr=2.0)
d2l.plot(torch.arange(50), [scheduler(t) for t in range(50)])
class FactorScheduler:
def __init__(self, factor=1, stop_factor_lr=1e-7, base_lr=0.1):
self.factor = factor
self.stop_factor_lr = stop_factor_lr
self.base_lr = base_lr
def __call__(self, num_update):
self.base_lr = max(self.stop_factor_lr, self.base_lr * self.factor)
return self.base_lr
scheduler = FactorScheduler(factor=0.9, stop_factor_lr=1e-2, base_lr=2.0)
d2l.plot(np.arange(50), [scheduler(t) for t in range(50)])
class FactorScheduler:
def __init__(self, factor=1, stop_factor_lr=1e-7, base_lr=0.1):
self.factor = factor
self.stop_factor_lr = stop_factor_lr
self.base_lr = base_lr
def __call__(self, num_update):
self.base_lr = max(self.stop_factor_lr, self.base_lr * self.factor)
return self.base_lr
scheduler = FactorScheduler(factor=0.9, stop_factor_lr=1e-2, base_lr=2.0)
d2l.plot(tf.range(50), [scheduler(t) for t in range(50)])
这也可以通过MXNet中的内置调度器lr_scheduler.FactorScheduler对象来实现。它需要一些额外的参数，例如预热期、预热模式（线性或常数）、期望的最大更新次数等。接下来，我们将酌情使用内置的调度器，并只在这里解释它们的功能。如所演示的，如果需要，构建自己的调度器是相当直接的。
12.11.3.2. 多因子调度器¶
训练深度网络的一个常见策略是保持学习率分段常数，并每隔一段时间将其降低一个给定的量。也就是说，给定一组降低速率的时间点，例如 \(s = \{5, 10, 20\}\)，每当 \(t \in s\) 时，就降低 \(\eta_{t+1} \leftarrow \eta_t \cdot \alpha\)。假设值在每一步都减半，我们可以实现如下。
net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr=0.5)
scheduler = lr_scheduler.MultiStepLR(trainer, milestones=[15, 30], gamma=0.5)
def get_lr(trainer, scheduler):
lr = scheduler.get_last_lr()[0]
trainer.step()
scheduler.step()
return lr
d2l.plot(torch.arange(num_epochs), [get_lr(trainer, scheduler)
for t in range(num_epochs)])
scheduler = lr_scheduler.MultiFactorScheduler(step=[15, 30], factor=0.5,
base_lr=0.5)
d2l.plot(np.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])
class MultiFactorScheduler:
def __init__(self, step, factor, base_lr):
self.step = step
self.factor = factor
self.base_lr = base_lr
def __call__(self, epoch):
if epoch in self.step:
self.base_lr = self.base_lr * self.factor
return self.base_lr
else:
return self.base_lr
scheduler = MultiFactorScheduler(step=[15, 30], factor=0.5, base_lr=0.5)
d2l.plot(tf.range(num_epochs), [scheduler(t) for t in range(num_epochs)])
这种分段常数学习率调度背后的直觉是，让优化进行，直到在权重向量分布方面达到一个稳定点。然后（也只有在那时），我们才降低速率，以获得一个更高质量的局部最小值代理。下面的例子展示了这如何能产生越来越好的解。
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
scheduler)
train loss 0.194, train acc 0.927, test acc 0.869
trainer = gluon.Trainer(net.collect_params(), 'sgd',
{'lr_scheduler': scheduler})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)
train loss 0.194, train acc 0.927, test acc 0.887
train(net, train_iter, test_iter, num_epochs, lr,
custom_callback=LearningRateScheduler(scheduler))
loss 0.234, train acc 0.912, test acc 0.891
51585.5 examples/sec on /GPU:0
<keras.engine.sequential.Sequential at 0x7f61980fbe20>
12.11.3.3. 余弦调度器¶
Loshchilov和Hutter :cite:`Loshchilov.Hutter.2016` 提出了一个相当令人困惑的启发式方法。它依赖于一个观察，即我们可能不希望在开始时过快地降低学习率，而且，我们可能希望在最后用一个非常小的学习率来“完善”解。这导致了一个类似余弦的调度，在 \(t \in [0, T]\) 范围内具有以下函数形式的学习率。
(12.11.1)¶\[\eta_t = \eta_T + \frac{\eta_0 - \eta_T}{2} \left(1 + \cos(\pi t/T)\right)\]
这里 \(\eta_0\) 是初始学习率， \(\eta_T\) 是在时间 \(T\) 的目标率。此外，对于 \(t > T\)，我们简单地将值固定在 \(\eta_T\)，而不再增加它。在下面的例子中，我们设置最大更新步数 \(T = 20\)。
class CosineScheduler:
def __init__(self, max_update, base_lr=0.01, final_lr=0,
warmup_steps=0, warmup_begin_lr=0):
self.base_lr_orig = base_lr
self.max_update = max_update
self.final_lr = final_lr
self.warmup_steps = warmup_steps
self.warmup_begin_lr = warmup_begin_lr
self.max_steps = self.max_update - self.warmup_steps
def get_warmup_lr(self, epoch):
increase = (self.base_lr_orig - self.warmup_begin_lr) \
* float(epoch) / float(self.warmup_steps)
return self.warmup_begin_lr + increase
def __call__(self, epoch):
if epoch < self.warmup_steps:
return self.get_warmup_lr(epoch)
if epoch <= self.max_update:
self.base_lr = self.final_lr + (
self.base_lr_orig - self.final_lr) * (1 + math.cos(
math.pi * (epoch - self.warmup_steps) / self.max_steps)) / 2
return self.base_lr
scheduler = CosineScheduler(max_update=20, base_lr=0.3, final_lr=0.01)
d2l.plot(torch.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])
scheduler = lr_scheduler.CosineScheduler(max_update=20, base_lr=0.3,
final_lr=0.01)
d2l.plot(np.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])
class CosineScheduler:
def __init__(self, max_update, base_lr=0.01, final_lr=0,
warmup_steps=0, warmup_begin_lr=0):
self.base_lr_orig = base_lr
self.max_update = max_update
self.final_lr = final_lr
self.warmup_steps = warmup_steps
self.warmup_begin_lr = warmup_begin_lr
self.max_steps = self.max_update - self.warmup_steps
def get_warmup_lr(self, epoch):
increase = (self.base_lr_orig - self.warmup_begin_lr) \
* float(epoch) / float(self.warmup_steps)
return self.warmup_begin_lr + increase
def __call__(self, epoch):
if epoch < self.warmup_steps:
return self.get_warmup_lr(epoch)
if epoch <= self.max_update:
self.base_lr = self.final_lr + (
self.base_lr_orig - self.final_lr) * (1 + math.cos(
math.pi * (epoch - self.warmup_steps) / self.max_steps)) / 2
return self.base_lr
scheduler = CosineScheduler(max_update=20, base_lr=0.3, final_lr=0.01)
d2l.plot(tf.range(num_epochs), [scheduler(t) for t in range(num_epochs)])
在计算机视觉的背景下，这种调度*可以*带来改进的结果。但请注意，这种改进并不能保证（如下所示）。
net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr=0.3)
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
scheduler)
train loss 0.159, train acc 0.942, test acc 0.904
trainer = gluon.Trainer(net.collect_params(), 'sgd',
{'lr_scheduler': scheduler})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)
train loss 0.343, train acc 0.878, test acc 0.859
train(net, train_iter, test_iter, num_epochs, lr,
custom_callback=LearningRateScheduler(scheduler))
loss 0.264, train acc 0.904, test acc 0.880
51258.5 examples/sec on /GPU:0
<keras.engine.sequential.Sequential at 0x7f61a18f2650>
12.11.3.4. 预热¶
在某些情况下，初始化参数不足以保证一个好的解。这对于一些可能导致不稳定优化问题的高级网络设计来说尤其是一个问题。我们可以通过选择一个足够小的学习率来解决这个问题，以防止一开始就发散。不幸的是，这意味着进展缓慢。相反，初始学习率过大会导致发散。
解决这个困境的一个相当简单的方法是使用一个预热期，在此期间学习率*增加*到其初始最大值，然后在优化过程结束前冷却速率。为简单起见，通常使用线性增加。这导致了如下所示的调度形式。
scheduler = CosineScheduler(20, warmup_steps=5, base_lr=0.3, final_lr=0.01)
d2l.plot(torch.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])
scheduler = lr_scheduler.CosineScheduler(20, warmup_steps=5, base_lr=0.3,
final_lr=0.01)
d2l.plot(np.arange(num_epochs), [scheduler(t) for t in range(num_epochs)])
scheduler = CosineScheduler(20, warmup_steps=5, base_lr=0.3, final_lr=0.01)
d2l.plot(tf.range(num_epochs), [scheduler(t) for t in range(num_epochs)])
请注意，网络在开始时收敛得更好（特别是观察前5个迭代周期的性能）。
net = net_fn()
trainer = torch.optim.SGD(net.parameters(), lr=0.3)
train(net, train_iter, test_iter, num_epochs, loss, trainer, device,
scheduler)
train loss 0.181, train acc 0.934, test acc 0.901
trainer = gluon.Trainer(net.collect_params(), 'sgd',
{'lr_scheduler': scheduler})
train(net, train_iter, test_iter, num_epochs, loss, trainer, device)
train loss 0.348, train acc 0.874, test acc 0.871
train(net, train_iter, test_iter, num_epochs, lr,
custom_callback=LearningRateScheduler(scheduler))
loss 0.274, train acc 0.899, test acc 0.880
50584.3 examples/sec on /GPU:0
<keras.engine.sequential.Sequential at 0x7f61a2c32f20>
预热可以应用于任何调度器（不仅仅是余弦）。关于学习率调度的更详细讨论和更多实验，请参阅 :cite:`(Gotmare.Keskar.Xiong.ea.2018)`。特别是，他们发现预热阶段限制了非常深的网络中参数的发散程度。这在直觉上是有道理的，因为我们预期在网络中那些需要最长时间才能在开始时取得进展的部分，由于随机初始化会产生显著的发散。