先看下下载题目后的目录：
本题目在 Optimization Methods中。
Until now, you've always used Gradient Descent to update the parameters and minimize the cost. In this notebook, you will learn more advanced optimization methods that can speed up learning and perhaps even get you to a better final value for the cost function. Having a good optimization algorithm can be the difference between waiting days vs. just a few hours to get a good result.
To get started, run the following code to import the libraries you will need.
import numpy as np
import matplotlib.pyplot as plt
import scipy.io
import math
import sklearn
import sklearn.datasetsfrom opt_utils import load_params_and_grads, initialize_parameters, forward_propagation, backward_propagation
from opt_utils import compute_cost, predict, predict_dec, plot_decision_boundary, load_dataset
from testCases import *%matplotlib inline
plt.rcParams['figure.figsize'] = (7.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'1 - Gradient Descent
A simple optimization method in machine learning is gradient descent (GD). When you take gradient steps with respect to all mm examples on each step, it is also called Batch Gradient Descent.
# GRADED FUNCTION: update_parameters_with_gddef update_parameters_with_gd(parameters, grads, learning_rate):"""Update parameters using one step of gradient descentArguments:parameters -- python dictionary containing your parameters to be updated:parameters['W' + str(l)] = Wlparameters['b' + str(l)] = blgrads -- python dictionary containing your gradients to update each parameters:grads['dW' + str(l)] = dWlgrads['db' + str(l)] = dbllearning_rate -- the learning rate, scalar.Returns:parameters -- python dictionary containing your updated parameters """L = len(parameters) // 2 # number of layers in the neural networks# Update rule for each parameterfor l in range(L):### START CODE HERE ### (approx. 2 lines)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)]### END CODE HERE ###return parametersparameters, 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"]))
Expected Output:
A variant of this is Stochastic Gradient Descent (SGD), which is equivalent to mini-batch gradient descent where each mini-batch has just 1 example. The update rule that you have just implemented does not change. What changes is that you would be computing gradients on just one training example at a time, rather than on the whole training set. The code examples below illustrate the difference between stochastic gradient descent and (batch) gradient descent.
- (Batch) Gradient Descent:
-
X = data_input Y = labels parameters = initialize_parameters(layers_dims) for i in range(0, num_iterations):# Forward propagationa, caches = forward_propagation(X, parameters)# Compute cost.cost = compute_cost(a, Y)# Backward propagation.grads = backward_propagation(a, caches, parameters)# Update parameters.parameters = update_parameters(parameters, grads) - Stochastic Gradient Descent:
X = data_input Y = labels parameters = initialize_parameters(layers_dims) for i in range(0, num_iterations):for j in range(0, m):# Forward propagationa, caches = forward_propagation(X[:,j], parameters)# Compute costcost = compute_cost(a, Y[:,j])# Backward propagationgrads = backward_propagation(a, caches, parameters)# Update parameters.parameters = update_parameters(parameters, grads)In Stochastic Gradient Descent, you use only 1 training example before updating the gradients. When the training set is large, SGD can be faster. But the parameters will "oscillate" toward the minimum rather than converge smoothly. Here is an illustration of this:
Note also that implementing SGD requires 3 for-loops in total:
In practice, you'll often get faster results if you do not use neither the whole training set, nor only one training example, to perform each update. Mini-batch gradient descent uses an intermediate number of examples for each step. With mini-batch gradient descent, you loop over the mini-batches instead of looping over individual training examples.
What you should remember:
- The difference between gradient descent, mini-batch gradient descent and stochastic gradient descent is the number of examples you use to perform one update step.
- You have to tune a learning rate hyperparameter αα.
- With a well-turned mini-batch size, usually it outperforms either gradient descent or stochastic gradient descent (particularly when the training set is large).
2 - Mini-Batch Gradient descent
Let's learn how to build mini-batches from the training set (X, Y).
There are two steps:
# GRADED FUNCTION: random_mini_batchesdef 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, integerReturns:mini_batches -- list of synchronous (mini_batch_X, mini_batch_Y) 同步列表"""np.random.seed(seed) # To make your "random" minibatches the same as oursm = X.shape[1] # number of training examplesmini_batches = []# Step 1: Shuffle (X, Y)permutation = list(np.random.permutation(m))shuffled_X = X[:, permutation]shuffled_Y = Y[:, permutation].reshape((1,m))# Step 2: Partition (shuffled_X, shuffled_Y). Minus the end case.#Math.floor() 返回小于或等于一个给定数字的最大整数。num_complete_minibatches = math.floor(m/mini_batch_size) # number of mini batches of size mini_batch_size in your partitionningfor k in range(0, num_complete_minibatches):### START CODE HERE ### (approx. 2 lines)mini_batch_X = shuffled_X[:,k*64:k*64+64]mini_batch_Y = shuffled_Y[:,k*64:k*64+64]### END CODE HERE ###mini_batch = (mini_batch_X, mini_batch_Y)mini_batches.append(mini_batch)# Handling the end case (last mini-batch < mini_batch_size)if m % mini_batch_size != 0:### START CODE HERE ### (approx. 2 lines)mini_batch_X = shuffled_X[:,num_complete_minibatches*64:m]mini_batch_Y = shuffled_Y[:,num_complete_minibatches*64:m]### END CODE HERE ###mini_batch = (mini_batch_X, mini_batch_Y)mini_batches.append(mini_batch)return mini_batchesX_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]))
Expected Output:
What you should remember:
- Shuffling and Partitioning are the two steps required to build mini-batches
- Powers of two are often chosen to be the mini-batch size, e.g., 16, 32, 64, 128.
3 - Momentum
Because mini-batch gradient descent makes a parameter update after seeing just a subset of examples, the direction of the update has some variance, and so the path taken by mini-batch gradient descent will "oscillate" toward convergence. Using momentum can reduce these oscillations.
由于小批量梯度下降法仅在看到一个示例子集后就进行参数更新，所以更新的方向有一定的方差，因此小批量梯度下降法所走的路径将“振荡”向收敛方向。利用动量可以减少这些振荡。
动量考虑了过去的梯度来平滑更新。我们将在变量v中存储前面梯度的“方向”。形式上，这是之前步骤梯度的指数加权平均值。你也可以把v想象成一个滚下山的球的“速度”，根据山的坡度/坡度的方向增加速度(和动量)。 红色箭头表示的方向，采取了一个步骤的小批量梯度下降与动量。蓝色的点表示每一步上梯度的方向(相对于当前的小批处理)。不只是沿着梯度，我们让梯度影响v，然后沿着v的方向走一步。
# GRADED FUNCTION: initialize_velocitydef 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)] = Wlparameters['b' + str(l)] = blReturns:v -- python dictionary containing the current velocity.v['dW' + str(l)] = velocity of dWlv['db' + str(l)] = velocity of dbl"""L = len(parameters) // 2 # number of layers in the neural networksv = {}# Initialize velocityfor l in range(L):### START CODE HERE ### (approx. 2 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]))v["dW" + str(l+1)] = np.zeros_like(parameters['W'+str(l+1)])v["db" + str(l+1)] = np.zeros_like(parameters['b'+ str(l+1)])### END CODE HERE ###return vparameters = initialize_velocity_test_case()
print(parameters["W1"])
print(parameters["W2"])
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"])) 
Expected Output:
# GRADED FUNCTION: update_parameters_with_momentumdef update_parameters_with_momentum(parameters, grads, v, beta, learning_rate):"""Update parameters using MomentumArguments:parameters -- python dictionary containing your parameters:parameters['W' + str(l)] = Wlparameters['b' + str(l)] = blgrads -- python dictionary containing your gradients for each parameters:grads['dW' + str(l)] = dWlgrads['db' + str(l)] = dblv -- python dictionary containing the current velocity:v['dW' + str(l)] = ...v['db' + str(l)] = ...beta -- the momentum hyperparameter, scalarlearning_rate -- the learning rate, scalarReturns:parameters -- python dictionary containing your updated parameters v -- python dictionary containing your updated velocities"""L = len(parameters) // 2 # number of layers in the neural networks# Momentum update for each parameterfor l in range(L):### START CODE HERE ### (approx. 4 lines)# compute velocities#v["dW" + str(l+1)] = beta * v["dW" + str(l+1)]+(1- beta)*grads["W"+ l+1]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)]# update parameters 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)] ### END CODE HERE ###return parameters, vparameters, 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"]))
Expected Output:
Note that:
4 - Adam
Adam is one of the most effective optimization algorithms for training neural networks. It combines ideas from RMSProp (described in lecture) and Momentum.
How does Adam work?
# 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.值:与相应的梯度/参数形状相同的零的numpy数组。Arguments:parameters -- python dictionary containing your parameters.parameters["W" + str(l)] = Wlparameters["b" + str(l)] = blReturns: 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) // 2 # number of layers in the neural networksv = {}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_like(parameters['b'+ str(l+1)])### END CODE HERE ###return v, s
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"]))
Expected Output:
Exercise:
# GRADED FUNCTION: update_parameters_with_adamdef 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 AdamArguments:parameters -- python dictionary containing your parameters:parameters['W' + str(l)] = Wlparameters['b' + str(l)] = blgrads -- python dictionary containing your gradients for each parameters:grads['dW' + str(l)] = dWlgrads['db' + str(l)] = dblv -- Adam variable, moving average of the first gradient, python dictionary Adam变量，移动平均的第一个梯度，python字典s -- Adam variable, moving average of the squared gradient, python dictionary Adam变量，移动平均的第一个梯度，python字典learning_rate -- the learning rate, scalar.beta1 -- Exponential decay hyperparameter for the first moment estimates 第一个矩的指数衰减超参数估计beta2 -- Exponential decay hyperparameter for the second moment estimates epsilon -- hyperparameter preventing division by zero in Adam updatesReturns:parameters -- python dictionary containing your updated parameters v -- Adam variable, moving average of the first gradient, python dictionarys -- Adam variable, moving average of the squared gradient, python dictionary"""L = len(parameters) // 2 # number of layers in the neural networksv_corrected = {} # Initializing first moment estimate, python dictionarys_corrected = {} # Initializing second moment estimate, python dictionary# Perform Adam update on all parametersfor l in range(L):# Moving average of the gradients. Inputs: "v, grads, beta1". Output: "v".### START CODE HERE ### (approx. 2 lines)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)]### END CODE HERE #### Compute bias-corrected first moment estimate. Inputs: "v, beta1, t". Output: "v_corrected".### START CODE HERE ### (approx. 2 lines)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 - math.pow(beta1,l)) 错啦v_corrected["db" + str(l+1)] = v["db" + str(l+1)] / (1 - np.power(beta1,t))### END CODE HERE #### Moving average of the squared gradients. Inputs: "s, grads, beta2". Output: "s".### START CODE HERE ### (approx. 2 lines)s["dW" + str(l+1)] = beta2 * s["dW" + str(l+1)] + (1-beta2)* np.square(grads["dW" + str(l+1)])#s["db" + str(l+1)] = beta2 * s["db" + str(l+1)] + (1-beta2)* math.pow(grads["db" + str(l+1)],2) 错啦s["db" + str(l+1)] = beta2 * s["db" + str(l+1)] + (1-beta2)* np.square(grads["db" + str(l+1)])### END CODE HERE #### Compute bias-corrected second raw moment estimate. Inputs: "s, beta2, t". Output: "s_corrected".### START CODE HERE ### (approx. 2 lines)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))### END CODE HERE #### Update parameters. Inputs: "parameters, learning_rate, v_corrected, s_corrected, epsilon". Output: "parameters".### START CODE HERE ### (approx. 2 lines)#parameters["W" + str(l+1)] = parameters["W" + str(l+1)] - learning_rate * (v_corrected["dW" + str(l+1)] / (math.pow( s_corrected["dW" + str(l+1)], 1/2))+epsilon) 错啦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))### END CODE HERE ###return parameters, v, sparameters, 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"]))
Expected Output:
You now have three working optimization algorithms (mini-batch gradient descent, Momentum, Adam). Let's implement a model with each of these optimizers and observe the difference.
总结一下的东西：
1、TypeError: data type not understood
如何使用Python产生一个数组，数组的长度为1024，数组的元素全为0？
很简单啊， 使用zeros(1024) 即可实现！
如何产生一个2×1024的全0矩阵呢？是否是zeros(2,1024) ?
若是上述这种写法就会出现 TypeError: data type not understood 这种错误；
正确的写法是 zeros((2,1024))，python的二维数据表示要用二层括号来进行表示。
三维数据是否使用三层括号？试一试，果然可以正确输出！试猜一猜， 下述三层括号中的数字分别代表什么含义？
In [9]: zeros(((2,2,3)))
Out[9]:
array([[[ 0., 0., 0.],
[ 0., 0., 0.]],
[[ 0., 0., 0.],
[ 0., 0., 0.]]])
---------------------
2、python之numpy.power()数组元素求n次方 与 math.pow()的区别呢
-
numpy.power(x1, x2)
-
数组的元素分别求n次方。x2可以是数字，也可以是数组，但是x1和x2的列数要相同。
3、平方根！ 平方
np.square(grads["dW" + str(l + 1)])
np.sqrt(s_corrected["dW" + str(l + 1)] + epsilon) 注意这里的epsilon是放在根号里面的呀！