http://blog.csdn.net/ljp1919/article/details/78241809
DeepLearing学习笔记-改善深层神经网络(第二周作业-优化方法)
0- 背景：
本文将介绍几种常用的优化方法，用以加快神经网络的学习速度
本文需要用到的库如下：
import numpy as npimport matplotlib.pyplot as pltimport scipy.ioimport mathimport sklearnimport sklearn.datasetsfrom opt_utils import load_params_and_grads, initialize_parameters, forward_propagation, backward_propagationfrom opt_utils import compute_cost, predict, predict_dec, plot_decision_boundary, load_datasetfrom testCases import *%matplotlib inlineplt.rcParams['figure.figsize'] = (7.0, 4.0) plt.rcParams['image.interpolation'] = 'nearest'plt.rcParams['image.cmap'] = 'gray'
1- 梯度下降法
梯度下降是每次处理完 m个样本后对参数进行一次更新操作,也叫做Batch Gradient Descent。
对于L层模型，梯度下降法对于各层参数的更新： l=1,...,L:
W[l]=W[l]−α dW[l](1)
b[l]=b[l]−α db[l](2)
L表示层数，α 是学习率。所有的这些参数都存在 parameters 字典中。注意，循环是从L1开始。
def update_parameters_with_gd(parameters, grads, learning_rate): """ Update parameters using one step of gradient descent Arguments: parameters -- python dictionary containing your parameters to be updated: parameters['W' + str(l)] = Wl parameters['b' + str(l)] = bl grads -- python dictionary containing your gradients to update each parameters: grads['dW' + str(l)] = dWl grads['db' + str(l)] = dbl learning_rate -- the learning rate, scalar. Returns: parameters -- python dictionary containing your updated parameters """ L = len(parameters) // 2 for l in range(L): parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * grads['dW' + str(l+1)] parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * grads['db' + str(l+1)] return parameters
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测试代码：
parameters, grads, learning_rate = update_parameters_with_gd_test_case()parameters = update_parameters_with_gd(parameters, grads, learning_rate)print("W1 = " + str(parameters["W1"]))print("b1 = " + str(parameters["b1"]))print("W2 = " + str(parameters["W2"]))print("b2 = " + str(parameters["b2"]))
测试代码运行如下：
W1 = [[ 1.63535156 -0.62320365 -0.53718766] [-1.07799357 0.85639907 -2.29470142]]b1 = [[ 1.74604067] [-0.75184921]]W2 = [[ 0.32171798 -0.25467393 1.46902454] [-2.05617317 -0.31554548 -0.3756023 ] [ 1.1404819 -1.09976462 -0.1612551 ]]b2 = [[-0.88020257] [ 0.02561572] [ 0.57539477]]
梯度下降的一种变体是随机梯度下降法Stochastic Gradient Descent (SGD)。这等同于mini-batch中每个mini-batch只有一个样本的梯度下降法。此时，梯度下降的更新法则就变成，每个样本都要计算一次，而不是此前的对整个样本集计算一次。
两者代码如下：
- (Batch) Gradient Descent:
X = data_inputY = labelsparameters = initialize_parameters(layers_dims)for i in range(0, num_iterations): a, caches = forward_propagation(X, parameters) cost = compute_cost(a, Y) grads = backward_propagation(a, caches, parameters) parameters = update_parameters(parameters, grads)
- Stochastic Gradient Descent:
X = data_inputY = labelsparameters = initialize_parameters(layers_dims)for i in range(0, num_iterations): for j in range(0, m): a, caches = forward_propagation(X[:,j], parameters) cost = compute_cost(a, Y[:,j]) grads = backward_propagation(a, caches, parameters) parameters = update_parameters(parameters, grads)
在随机梯度下降中, 我们对于每个样本都更新梯度。当训练集很大时，这种方法可以明显提高运行速度，但是参数会沿着最小方向震荡，而不是平滑地收敛。
Figure 1 : SGD vs GD
“+” 表示代价最小值。SGD在收敛前出现很多震荡，但是由于每步都只有一个样本，所以每步都比梯度下降GD要来得快。 (vs. the whole batch for GD).注意 SGD 共需要三个循环:
1. 最外层的迭代次数
2. m个训练样本
3. 每层参数的更新 ( (W[1],b[1]) to (W[L],b[L]))
在实际情况中，我们一般是折中，即所谓的 Mini-batch gradient descent。将整体的训练集分成子数据集，然后每个子训练集计算一次梯度下降。
Figure 2 : SGD vs Mini-Batch GD
“+” 表示最小代价值。
谨记:
- gradient descent, mini-batch gradient descent 和 stochastic gradient descent之间的区别在于梯度更新所用到的样本数据量。
- 超参数学习率 α是需要调整获取到
- mini-batch的尺寸也是调整获取到的，所以也是一个超参数。一般情况下这种方式比另外两者更好，特别是当训练集特别大的时候。
2 - Mini-Batch梯度下降
mini-batches用于训练集 (X, Y)，一般有以下两个步骤：
Shuffle(洗牌): 随机洗牌的方式将训练样本的数据顺序随机打散，注意：X和Y的随机要一致，否则Y值不能与X匹配，出现张冠李戴。随机化的洗牌操作能够将样本切分成不同的mini-batches。洗牌方式如下图所示：
Partition(分割): 将已经随机化的数据集(X, Y)分割成 mini_batch_size (本文= 64)大小的子数据集。尾部的数据可能小于一个mini_batch_size，所以对于最后一个mini-batch要注意处理。
我们定义 random_mini_batches函数来实现上述功能。在采用索引切片的时候，操作1st and 2nd mini-batches如下，其他依次。
first_mini_batch_X = shuffled_X[:, 0 : mini_batch_size]second_mini_batch_X = shuffled_X[:, mini_batch_size : 2 * mini_batch_size]...
当样本数无法被mini_batch_size整除的时候，最后一个mini-batch< mini_batch_size=64。 ⌊s⌋ 表示 s向下取整 (Python中实现：math.floor(s))。所以 ⌊mmini_batch_size⌋ 个mini-batches中的样本数量是= 64，最后一个min-batch中样本数量= (m−mini_batch_size×⌊mmini_batch_size⌋)。
代码实现如下：
def random_mini_batches(X, Y, mini_batch_size = 64, seed = 0): """ Creates a list of random minibatches from (X, Y) Arguments: X -- input data, of shape (input size, number of examples) Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (1, number of examples) mini_batch_size -- size of the mini-batches, integer Returns: mini_batches -- list of synchronous (mini_batch_X, mini_batch_Y) """ np.random.seed(seed) m = X.shape[1] mini_batches = [] permutation = list(np.random.permutation(m)) shuffled_X = X[:, permutation] shuffled_Y = Y[:, permutation].reshape((1,m)) num_complete_minibatches = math.floor(m/mini_batch_size) for k in range(0, num_complete_minibatches): mini_batch_X = shuffled_X[:, k * mini_batch_size : (k+1) * mini_batch_size] mini_batch_Y = shuffled_Y[:, k * mini_batch_size : (k+1) * mini_batch_size] mini_batch = (mini_batch_X, mini_batch_Y) mini_batches.append(mini_batch) if m % mini_batch_size != 0: mini_batch_X = shuffled_X[:, num_complete_minibatches * mini_batch_size : ] mini_batch_Y = shuffled_Y[:, num_complete_minibatches * mini_batch_size : ] mini_batch = (mini_batch_X, mini_batch_Y) mini_batches.append(mini_batch) return mini_batches
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代码测试如下：
X_assess, Y_assess, mini_batch_size = random_mini_batches_test_case()mini_batches = random_mini_batches(X_assess, Y_assess, mini_batch_size)print ("shape of the 1st mini_batch_X: " + str(mini_batches[0][0].shape))print ("shape of the 2nd mini_batch_X: " + str(mini_batches[1][0].shape))print ("shape of the 3rd mini_batch_X: " + str(mini_batches[2][0].shape))print ("shape of the 1st mini_batch_Y: " + str(mini_batches[0][1].shape))print ("shape of the 2nd mini_batch_Y: " + str(mini_batches[1][1].shape)) print ("shape of the 3rd mini_batch_Y: " + str(mini_batches[2][1].shape))print ("mini batch sanity check: " + str(mini_batches[0][0][0][0:3]))
测试代码运行输出结果如下：
shape of the 1st mini_batch_X: (12288, 64)shape of the 2nd mini_batch_X: (12288, 64)shape of the 3rd mini_batch_X: (12288, 20)shape of the 1st mini_batch_Y: (1, 64)shape of the 2nd mini_batch_Y: (1, 64)shape of the 3rd mini_batch_Y: (1, 20)mini batch sanity check: [ 0.90085595 -0.7612069 0.2344157 ]
PS：一般mini-batch size的取值是2n，如 16, 32, 64, 128等
Momentum(动量梯度下降法)
由于min-batch梯度下降法是在看过训练集的一部分子数据集之后，就开始了参数的更新，那么就会在参数更新过程中出现偏差震荡。采用动量梯度下降法可以减缓震荡的出现。
momentum方式是在参数更新时候，参考历史的参数值，以平滑参数的更新。我们以变量 v存储梯度变化的历史方向。一般情况下，这个 v值是历史梯度值的指数加权平均结果。我们可以将 v视为球下坡滚动的”velocity”。
红色箭头表示在momentum作用下每个mini-batch梯度下降的方向，而蓝色则是没有momentum作用的mini-batch梯度下降方向。
velocity值初始化：
velocity, v,在Python中是一个字典，初始为0矩阵，其尺寸与 grads 一致:
for l=1,...,L:
v["dW" + str(l+1)] = ... v["db" + str(l+1)] = ...
initialize_velocity代码实现如下：
def initialize_velocity(parameters): """ Initializes the velocity as a python dictionary with: - keys: "dW1", "db1", ..., "dWL", "dbL" - values: numpy arrays of zeros of the same shape as the corresponding gradients/parameters. Arguments: parameters -- python dictionary containing your parameters. parameters['W' + str(l)] = Wl parameters['b' + str(l)] = bl Returns: v -- python dictionary containing the current velocity. v['dW' + str(l)] = velocity of dWl v['db' + str(l)] = velocity of dbl """ L = len(parameters) // 2 v = {} for l in range(L): v["dW" + str(l+1)] = np.zeros((parameters['W' + str(l+1)].shape[0], parameters['W' + str(l+1)].shape[1])) v["db" + str(l+1)] = np.zeros((parameters['b' + str(l+1)].shape[0], parameters['b' + str(l+1)].shape[1])) return v
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初始化函数测试：
parameters = initialize_velocity_test_case()v = initialize_velocity(parameters)print("v[\"dW1\"] = " + str(v["dW1"]))print("v[\"db1\"] = " + str(v["db1"]))print("v[\"dW2\"] = " + str(v["dW2"]))print("v[\"db2\"] = " + str(v["db2"]))
测试结果如下：
v["dW1"] = [[ 0. 0. 0.] [ 0. 0. 0.]]v["db1"] = [[ 0.] [ 0.]]v["dW2"] = [[ 0. 0. 0.] [ 0. 0. 0.] [ 0. 0. 0.]]v["db2"] = [[ 0.] [ 0.] [ 0.]]
带momentum的参数更新:
更新规则如下：
for l=1,...,L:
{vdW[l]=βvdW[l]+(1−β)dW[l]W[l]=W[l]−αvdW[l](3)
{vdb[l]=βvdb[l]+(1−β)db[l]b[l]=b[l]−αvdb[l](4)
其中 L 表示层数, β 是momentum值，α 是学习率。 这些参数都存于 parameters 字典中。注意W[1] and b[1]是从第1层开始的。
update_parameters_with_momentum函数代码实现如下：
def update_parameters_with_momentum(parameters, grads, v, beta, learning_rate): """ Update parameters using Momentum Arguments: parameters -- python dictionary containing your parameters: parameters['W' + str(l)] = Wl parameters['b' + str(l)] = bl grads -- python dictionary containing your gradients for each parameters: grads['dW' + str(l)] = dWl grads['db' + str(l)] = dbl v -- python dictionary containing the current velocity: v['dW' + str(l)] = ... v['db' + str(l)] = ... beta -- the momentum hyperparameter, scalar learning_rate -- the learning rate, scalar Returns: parameters -- python dictionary containing your updated parameters v -- python dictionary containing your updated velocities """ L = len(parameters) // 2 for l in range(L): v["dW" + str(l+1)] = beta * v["dW" + str(l+1)] + (1-beta) * grads['dW' + str(l+1)] v["db" + str(l+1)] = beta * v["db" + str(l+1)] + (1-beta) * grads['db' + str(l+1)] parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * v["dW" + str(l+1)] parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * v["db" + str(l+1)] return parameters, v
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函数测试代码：
parameters, grads, v = update_parameters_with_momentum_test_case()parameters, v = update_parameters_with_momentum(parameters, grads, v, beta = 0.9, learning_rate = 0.01)print("W1 = " + str(parameters["W1"]))print("b1 = " + str(parameters["b1"]))print("W2 = " + str(parameters["W2"]))print("b2 = " + str(parameters["b2"]))print("v[\"dW1\"] = " + str(v["dW1"]))print("v[\"db1\"] = " + str(v["db1"]))print("v[\"dW2\"] = " + str(v["dW2"]))print("v[\"db2\"] = " + str(v["db2"]))
测试结果如下：
W1 = [[ 1.62544598 -0.61290114 -0.52907334] [-1.07347112 0.86450677 -2.30085497]]b1 = [[ 1.74493465] [-0.76027113]]W2 = [[ 0.31930698 -0.24990073 1.4627996 ] [-2.05974396 -0.32173003 -0.38320915] [ 1.13444069 -1.0998786 -0.1713109 ]]b2 = [[-0.87809283] [ 0.04055394] [ 0.58207317]]v["dW1"] = [[-0.11006192 0.11447237 0.09015907] [ 0.05024943 0.09008559 -0.06837279]]v["db1"] = [[-0.01228902] [-0.09357694]]v["dW2"] = [[-0.02678881 0.05303555 -0.06916608] [-0.03967535 -0.06871727 -0.08452056] [-0.06712461 -0.00126646 -0.11173103]]v["db2"] = [[ 0.02344157] [ 0.16598022] [ 0.07420442]]
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注意 :
- velocity初始化为zeros，所以算法需要迭代一定次数以建立起速度，实现每次迭代的bigger steps。
- 当β=0，则退化成标准的梯度下降法。
β值得选取：
- β 越大，历史梯度值引入到当前值的权重越大，更新就会越平滑。但是如果 β太大，则会导致更新平滑过度。
- β一般取值在0.8 到 0.999之间，常取β=0.9。
- 可以通过尝试几个β值，然后看哪个值在降低cost function J效果最好，来获取最优值。
4 - Adam算法
Adam算法应该是目前在神经网络领域最有效的优化算法了，该算法联合了RMSProp算法和Momentum算法。
Adam算法流程：
1.先计算历史梯度的指数加权平均值，存于变量 v ，vcorrected 表示校正后的值。
2. 计算历史梯度平方值的指数加权平均值，存于变量 s ，scorrected表示校正后的值。
3. 联合”1” and “2”更新参数
更新规则如下, for l=1,...,L:
⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪vdW[l]=β1vdW[l]+(1−β1)∂J∂W[l]vcorrecteddW[l]=vdW[l]1−(β1)tsdW[l]=β2sdW[l]+(1−β2)(∂J∂W[l])2scorrecteddW[l]=sdW[l]1−(β1)tW[l]=W[l]−αvcorrecteddW[l]scorrecteddW[l]√+ε
其中：- t 表示迭代的次数
- L 表示层数
- β1 and β2都是超参数，控制指数加权的权重
- α 是学习率
- ε 是一个很小的值，为了避免除0操作
变量 v,s 的初始化如下:
for l=1,...,L:
v["dW" + str(l+1)] = ... v["db" + str(l+1)] = ... s["dW" + str(l+1)] = ... s["db" + str(l+1)] = ...
Adam的初始化代码：
# GRADED FUNCTION: initialize_adamdef initialize_adam(parameters) : """ Initializes v and s as two python dictionaries with: - keys: "dW1", "db1", ..., "dWL", "dbL" - values: numpy arrays of zeros of the same shape as the corresponding gradients/parameters. Arguments: parameters -- python dictionary containing your parameters. parameters["W" + str(l)] = Wl parameters["b" + str(l)] = bl Returns: v -- python dictionary that will contain the exponentially weighted average of the gradient. v["dW" + str(l)] = ... v["db" + str(l)] = ... s -- python dictionary that will contain the exponentially weighted average of the squared gradient. s["dW" + str(l)] = ... s["db" + str(l)] = ... """ L = len(parameters) v = {} s = {} # Initialize v, s. Input: "parameters". Outputs: "v, s". for l in range(L): ### START CODE HERE ### (approx. 4 lines) v["dW" + str(l+1)] = np.zeros((parameters['W' + str(l+1)].shape[0], parameters['W' + str(l+1)].shape[1])) v["db" + str(l+1)] = np.zeros((parameters['b' + str(l+1)].shape[0], parameters['b' + str(l+1)].shape[1])) s["dW" + str(l+1)] = np.zeros((parameters['W' + str(l+1)].shape[0], parameters['W' + str(l+1)].shape[1])) s["db" + str(l+1)] = np.zeros((parameters['b' + str(l+1)].shape[0], parameters['b' + str(l+1)].shape[1])) ### END CODE HERE ### return v, s
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代码测试：
parameters = initialize_adam_test_case()v, s = initialize_adam(parameters)print("v[\"dW1\"] = " + str(v["dW1"]))print("v[\"db1\"] = " + str(v["db1"]))print("v[\"dW2\"] = " + str(v["dW2"]))print("v[\"db2\"] = " + str(v["db2"]))print("s[\"dW1\"] = " + str(s["dW1"]))print("s[\"db1\"] = " + str(s["db1"]))print("s[\"dW2\"] = " + str(s["dW2"]))print("s[\"db2\"] = " + str(s["db2"]))
测试代码输出：
v["dW1"] = [[ 0. 0. 0.] [ 0. 0. 0.]]v["db1"] = [[ 0.] [ 0.]]v["dW2"] = [[ 0. 0. 0.] [ 0. 0. 0.] [ 0. 0. 0.]]v["db2"] = [[ 0.] [ 0.] [ 0.]]s["dW1"] = [[ 0. 0. 0.] [ 0. 0. 0.]]s["db1"] = [[ 0.] [ 0.]]s["dW2"] = [[ 0. 0. 0.] [ 0. 0. 0.] [ 0. 0. 0.]]s["db2"] = [[ 0.] [ 0.] [ 0.]]
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Adam算法实现：
def update_parameters_with_adam(parameters, grads, v, s, t, learning_rate = 0.01, beta1 = 0.9, beta2 = 0.999, epsilon = 1e-8): """ Update parameters using Adam Arguments: parameters -- python dictionary containing your parameters: parameters['W' + str(l)] = Wl parameters['b' + str(l)] = bl grads -- python dictionary containing your gradients for each parameters: grads['dW' + str(l)] = dWl grads['db' + str(l)] = dbl v -- Adam variable, moving average of the first gradient, python dictionary s -- Adam variable, moving average of the squared gradient, python dictionary learning_rate -- the learning rate, scalar. beta1 -- Exponential decay hyperparameter for the first moment estimates beta2 -- Exponential decay hyperparameter for the second moment estimates epsilon -- hyperparameter preventing division by zero in Adam updates Returns: parameters -- python dictionary containing your updated parameters v -- Adam variable, moving average of the first gradient, python dictionary s -- Adam variable, moving average of the squared gradient, python dictionary """ L = len(parameters) // 2 v_corrected = {} s_corrected = {} for l in range(L): v["dW" + str(l+1)] = beta1 * v["dW" + str(l+1)] + (1-beta1) * grads['dW' + str(l+1)] v["db" + str(l+1)] = beta1 * v["db" + str(l+1)] + (1-beta1) * grads['db' + str(l+1)] v_corrected["dW" + str(l+1)] = v["dW" + str(l+1)]/(1-np.power(beta1,t)) v_corrected["db" + str(l+1)] = v["db" + str(l+1)]/(1-np.power(beta1,t)) s["dW" + str(l+1)] = beta2 * s["dW" + str(l+1)] + (1-beta2) * np.power(grads['dW' + str(l+1)],2) s["db" + str(l+1)] = beta2 * s["db" + str(l+1)] + (1-beta2) * np.power(grads['db' + str(l+1)],2) s_corrected["dW" + str(l+1)] = s["dW" + str(l+1)]/(1-np.power(beta2,t)) s_corrected["db" + str(l+1)] = s["db" + str(l+1)]/(1-np.power(beta2,t)) parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * v_corrected["dW" + str(l+1)]/(np.sqrt(s_corrected["dW" + str(l+1)])+epsilon) parameters["b" + str(l+1)] = parameters["b" + str(l+1)] - learning_rate * v_corrected["db" + str(l+1)]/(np.sqrt(s_corrected["db" + str(l+1)])+epsilon) return parameters, v, s
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Adam算法测试：
parameters, grads, v, s = update_parameters_with_adam_test_case()parameters, v, s = update_parameters_with_adam(parameters, grads, v, s, t = 2)print("W1 = " + str(parameters["W1"]))print("b1 = " + str(parameters["b1"]))print("W2 = " + str(parameters["W2"]))print("b2 = " + str(parameters["b2"]))print("v[\"dW1\"] = " + str(v["dW1"]))print("v[\"db1\"] = " + str(v["db1"]))print("v[\"dW2\"] = " + str(v["dW2"]))print("v[\"db2\"] = " + str(v["db2"]))print("s[\"dW1\"] = " + str(s["dW1"]))print("s[\"db1\"] = " + str(s["db1"]))print("s[\"dW2\"] = " + str(s["dW2"]))print("s[\"db2\"] = " + str(s["db2"]))
测试代码运行如下：
W1 = [[ 1.63178673 -0.61919778 -0.53561312] [-1.08040999 0.85796626 -2.29409733]]b1 = [[ 1.75225313] [-0.75376553]]W2 = [[ 0.32648046 -0.25681174 1.46954931] [-2.05269934 -0.31497584 -0.37661299] [ 1.14121081 -1.09244991 -0.16498684]]b2 = [[-0.88529979] [ 0.03477238] [ 0.57537385]]v["dW1"] = [[-0.11006192 0.11447237 0.09015907] [ 0.05024943 0.09008559 -0.06837279]]v["db1"] = [[-0.01228902] [-0.09357694]]v["dW2"] = [[-0.02678881 0.05303555 -0.06916608] [-0.03967535 -0.06871727 -0.08452056] [-0.06712461 -0.00126646 -0.11173103]]v["db2"] = [[ 0.02344157] [ 0.16598022] [ 0.07420442]]s["dW1"] = [[ 0.00121136 0.00131039 0.00081287] [ 0.0002525 0.00081154 0.00046748]]s["db1"] = [[ 1.51020075e-05] [ 8.75664434e-04]]s["dW2"] = [[ 7.17640232e-05 2.81276921e-04 4.78394595e-04] [ 1.57413361e-04 4.72206320e-04 7.14372576e-04] [ 4.50571368e-04 1.60392066e-07 1.24838242e-03]]s["db2"] = [[ 5.49507194e-05] [ 2.75494327e-03] [ 5.50629536e-04]]
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5 - Model with different optimization algorithms
对于上述几种优化算法的测试，在这里我们采用 “moons” 数据集。
数据加载：
train_X, train_Y = load_dataset()
对于一个3层的神经网络，我们将采用下述三种优化算法来训练：
- Mini-batch Gradient Descent: 通过调用:
update_parameters_with_gd()来实现
- Mini-batch Momentum: 通过调用:
initialize_velocity() 和update_parameters_with_momentum()来实现。
- Mini-batch Adam: 通过调用:
initialize_adam() 和 update_parameters_with_adam()来实现
模型代码：
def model(X, Y, layers_dims, optimizer, learning_rate = 0.0007, mini_batch_size = 64, beta = 0.9, beta1 = 0.9, beta2 = 0.999, epsilon = 1e-8, num_epochs = 10000, print_cost = True): """ 3-layer neural network model which can be run in different optimizer modes. Arguments: X -- input data, of shape (2, number of examples) Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (1, number of examples) layers_dims -- python list, containing the size of each layer learning_rate -- the learning rate, scalar. mini_batch_size -- the size of a mini batch beta -- Momentum hyperparameter beta1 -- Exponential decay hyperparameter for the past gradients estimates beta2 -- Exponential decay hyperparameter for the past squared gradients estimates epsilon -- hyperparameter preventing division by zero in Adam updates num_epochs -- number of epochs print_cost -- True to print the cost every 1000 epochs Returns: parameters -- python dictionary containing your updated parameters """ L = len(layers_dims) costs = [] t = 0 seed = 10 parameters = initialize_parameters(layers_dims) if optimizer == "gd": pass elif optimizer == "momentum": v = initialize_velocity(parameters) elif optimizer == "adam": v, s = initialize_adam(parameters) for i in range(num_epochs): seed = seed + 1 minibatches = random_mini_batches(X, Y, mini_batch_size, seed) for minibatch in minibatches: (minibatch_X, minibatch_Y) = minibatch a3, caches = forward_propagation(minibatch_X, parameters) cost = compute_cost(a3, minibatch_Y) grads = backward_propagation(minibatch_X, minibatch_Y, caches) if optimizer == "gd": parameters = update_parameters_with_gd(parameters, grads, learning_rate) elif optimizer == "momentum": parameters, v = update_parameters_with_momentum(parameters, grads, v, beta, learning_rate) elif optimizer == "adam": t = t + 1 parameters, v, s = update_parameters_with_adam(parameters, grads, v, s, t, learning_rate, beta1, beta2, epsilon) if print_cost and i % 1000 == 0: print ("Cost after epoch %i: %f" %(i, cost)) if print_cost and i % 100 == 0: costs.append(cost) plt.plot(costs) plt.ylabel('cost') plt.xlabel('epochs (per 100)') plt.title("Learning rate = " + str(learning_rate)) plt.show() return parameters
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5-1 Mini-batch Gradient descent
代码如下：
# train 3-layer modellayers_dims = [train_X.shape[0], 5, 2, 1]parameters = model(train_X, train_Y, layers_dims, optimizer = "gd")# Predictpredictions = predict(train_X, train_Y, parameters)# Plot decision boundaryplt.title("Model with Gradient Descent optimization")axes = plt.gca()axes.set_xlim([-1.5,2.5])axes.set_ylim([-1,1.5])plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)
运行结果如下：
Cost after epoch 0: 0.690736Cost after epoch 1000: 0.685273Cost after epoch 2000: 0.647072Cost after epoch 3000: 0.619525Cost after epoch 4000: 0.576584Cost after epoch 5000: 0.607243Cost after epoch 6000: 0.529403Cost after epoch 7000: 0.460768Cost after epoch 8000: 0.465586Cost after epoch 9000: 0.464518
Accuracy: 0.796666666667c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.maximum.reduce will be axis=0, not the current None, to match np.maximum.reduce. Explicitly pass 0 or None to silence this warning. return self.reduce(a)c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.minimum.reduce will be axis=0, not the current None, to match np.minimum.reduce. Explicitly pass 0 or None to silence this warning. return self.reduce(a)
5-2 Mini-batch gradient descent with momentum
代码如下：
# train 3-layer modellayers_dims = [train_X.shape[0], 5, 2, 1]parameters = model(train_X, train_Y, layers_dims, beta = 0.9, optimizer = "momentum")# Predictpredictions = predict(train_X, train_Y, parameters)# Plot decision boundaryplt.title("Model with Momentum optimization")axes = plt.gca()axes.set_xlim([-1.5,2.5])axes.set_ylim([-1,1.5])plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)
运行结果如下：
Cost after epoch 0: 0.690741Cost after epoch 1000: 0.685341Cost after epoch 2000: 0.647145Cost after epoch 3000: 0.619594Cost after epoch 4000: 0.576665Cost after epoch 5000: 0.607324Cost after epoch 6000: 0.529476Cost after epoch 7000: 0.460936Cost after epoch 8000: 0.465780Cost after epoch 9000: 0.464740
Accuracy: 0.796666666667c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.maximum.reduce will be axis=0, not the current None, to match np.maximum.reduce. Explicitly pass 0 or None to silence this warning. return self.reduce(a)c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.minimum.reduce will be axis=0, not the current None, to match np.minimum.reduce. Explicitly pass 0 or None to silence this warning. return self.reduce(a)
5-3 Mini-batch with Adam mode
代码如下：
# train 3-layer modellayers_dims = [train_X.shape[0], 5, 2, 1]parameters = model(train_X, train_Y, layers_dims, optimizer = "adam")# Predictpredictions = predict(train_X, train_Y, parameters)# Plot decision boundaryplt.title("Model with Adam optimization")axes = plt.gca()axes.set_xlim([-1.5,2.5])axes.set_ylim([-1,1.5])plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)
运行结果如下：
Cost after epoch 0: 0.690552Cost after epoch 1000: 0.185567Cost after epoch 2000: 0.150852Cost after epoch 3000: 0.074454Cost after epoch 4000: 0.125936Cost after epoch 5000: 0.104235Cost after epoch 6000: 0.100552Cost after epoch 7000: 0.031601Cost after epoch 8000: 0.111709Cost after epoch 9000: 0.197648
Accuracy: 0.94c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.maximum.reduce will be axis=0, not the current None, to match np.maximum.reduce. Explicitly pass 0 or None to silence this warning. return self.reduce(a)c:\users\jason\appdata\local\programs\python\python35\lib\site-packages\numpy\ma\core.py:6385: MaskedArrayFutureWarning: In the future the default for ma.minimum.reduce will be axis=0, not the current None, to match np.minimum.reduce. Explicitly pass 0 or None to silence this warning. return self.reduce(a)
5-4 总结：
optimization methodaccuracycost shapeGradient descent79.7%oscillationsMomentum79.7%oscillationsAdam94%smootherMomentum一般都是有助于提升速度，但是当学习率较小，数据集相对简单的时候，其性能的优越性没有太明显。我们在优化算法中看到的那些较大的震荡是由于一些minibatches 相对更加复杂所造成的。
从运行结果可以看出，Adam算法比mini-batch gradient descent 和 Momentum都要显得优越。对于model如果在简单数据集上，迭代次数更多的话，这三种优化算法都会产生较好的结果，但是我们也可以看出，Adam算法收敛得更快些。
Adam算法的优点:
- 内存要求低 (尽管比gradient descent 和 gradient descent with momentum要高些)
- 一般微调超参数就可以获得较好的结果(除了α)
