import numpy as np
import matplotlib.pyplot as plt
import scipy.io
import math
import sklearn
import sklearn.datasets
from 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 testCase 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'
#Batch Gradient Descent.
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 # number of layers in the neural networks
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
#Create mini-batches
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) # To make your "random" minibatches the same as ours
m = X.shape[1] # number of training examples
mini_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.
num_complete_minibatches = math.floor(m/mini_batch_size) # number of mini batches of size mini_batch_size in your partitionning
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)
# Handling the end case (last mini-batch < mini_batch_size)
if m % mini_batch_size != 0:
mini_batch_X = shuffled_X[:, math.floor(m/mini_batch_size)*mini_batch_size : m]
mini_batch_Y = shuffled_Y[:,math.floor(m/mini_batch_size)*mini_batch_size : m]
mini_batch = (mini_batch_X, mini_batch_Y)
mini_batches.append(mini_batch)
return mini_batches
#Initialize velocity
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 # number of layers in the neural networks
v = {}
# Initialize velocity
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
#implement the parameters update with momentum.
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 # number of layers in the neural networks
# Momentum update for each parameter
for l in range(L):
# compute velocities
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)]
return parameters, v
#Initialize the Adam variables v,sv,s which keep track of the past information.
def 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) // 2 # number of layers in the neural networks
v = {}
s = {}
# Initialize v, s. Input: "parameters". Outputs: "v, s".
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]))
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]))
return v, s
#update parameters with Adam
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 # number of layers in the neural networks
v_corrected = {} # Initializing first moment estimate, python dictionary
s_corrected = {} # Initializing second moment estimate, python dictionary
# Perform Adam update on all parameters
for l in range(L):
# Moving average of the gradients. Inputs: "v, grads, beta1". Output: "v".
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)]
# Compute bias-corrected first moment estimate. Inputs: "v, beta1, t". Output: "v_corrected".
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))
# Moving average of the squared gradients. Inputs: "s, grads, beta2". Output: "s".
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)*np.square(grads["db"+str(l+1)])
# Compute bias-corrected second raw moment estimate. Inputs: "s, beta2, t". Output: "s_corrected".
s_corrected["dW" + str(l+1)] = s["dW" + str(l+1)]/(1-np.power(beta1,t))
s_corrected["db" + str(l+1)] = s["db" + str(l+1)]/(1-np.power(beta1,t))
# Update parameters. Inputs: "parameters, learning_rate, v_corrected, s_corrected, epsilon". Output: "parameters".
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
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) # number of layers in the neural networks
costs = [] # to keep track of the cost
t = 0 # initializing the counter required for Adam update
seed = 10 # For grading purposes, so that your "random" minibatches are the same as ours
# Initialize parameters
parameters = initialize_parameters(layers_dims)
# Initialize the optimizer
if optimizer == "gd":
pass # no initialization required for gradient descent
elif optimizer == "momentum":
v = initialize_velocity(parameters)
elif optimizer == "adam":
v, s = initialize_adam(parameters)
# Optimization loop
for i in range(num_epochs):
# Define the random minibatches. We increment the seed to reshuffle differently the dataset after each epoch
seed = seed + 1
minibatches = random_mini_batches(X, Y, mini_batch_size, seed)
for minibatch in minibatches:
# Select a minibatch
(minibatch_X, minibatch_Y) = minibatch
# Forward propagation
a3, caches = forward_propagation(minibatch_X, parameters)
# Compute cost
cost = compute_cost(a3, minibatch_Y)
# Backward propagation
grads = backward_propagation(minibatch_X, minibatch_Y, caches)
# Update parameters
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 # Adam counter
parameters, v, s = update_parameters_with_adam(parameters, grads, v, s,
t, learning_rate, beta1, beta2, epsilon)
# Print the cost every 1000 epoch
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)
# plot the cost
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('epochs (per 100)')
plt.title("Learning rate = " + str(learning_rate))
plt.show()
return parametersOptimization Methods
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.
Gradient descent goes "downhill" on a cost function JJ. Think of it as trying to do this:
At each step of the training, you update your parameters following a certain direction to try to get to the lowest possible point.
Notations: As usual, ∂J∂a=∂J∂a= da for any variable a.
To get started, run the following code to import the libraries you will need.
In [63]:
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) # set default size of plotsplt.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.
Warm-up exercise: Implement the gradient descent update rule. The gradient descent rule is, for l=1,...,Ll=1,...,L:
W[l]=W[l]−α dW[l](1)(1)W[l]=W[l]−α dW[l]
b[l]=b[l]−α db[l](2)(2)b[l]=b[l]−α db[l]
where L is the number of layers and αα is the learning rate. All parameters should be stored in the parameters dictionary. Note that the iterator l starts at 0 in the for loop while the first parameters are W[1]W[1] and b[1]b[1]. You need to shift l to l+1 when coding.
In [64]:
# GRADED FUNCTION: update_parameters_with_gddef 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 # 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 parameters
In [65]:
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"]))Expected Output:
| 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]] |
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 propagation a, 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 propagation a, caches = forward_propagation(X[:,j], parameters) # Compute cost cost = compute_cost(a, Y[:,j]) # Backward propagation grads = 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:
"+" denotes a minimum of the cost. SGD leads to many oscillations to reach convergence. But each step is a lot faster to compute for SGD than for GD, as it uses only one training example (vs. the whole batch for GD).
Note also that implementing SGD requires 3 for-loops in total:
- Over the number of iterations
- Over the mm training examples
- Over the layers (to update all parameters, from (W[1],b[1])(W[1],b[1]) to (W[L],b[L])(W[L],b[L]))
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.
"+" denotes a minimum of the cost. Using mini-batches in your optimization algorithm often leads to faster optimization.
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:
- Shuffle: Create a shuffled version of the training set (X, Y) as shown below. Each column of X and Y represents a training example. Note that the random shuffling is done synchronously between X and Y. Such that after the shuffling the ithith column of X is the example corresponding to the ithith label in Y. The shuffling step ensures that examples will be split randomly into different mini-batches.
- Partition: Partition the shuffled (X, Y) into mini-batches of size
mini_batch_size(here 64). Note that the number of training examples is not always divisible bymini_batch_size. The last mini batch might be smaller, but you don't need to worry about this. When the final mini-batch is smaller than the fullmini_batch_size, it will look like this:
Exercise: Implement random_mini_batches. We coded the shuffling part for you. To help you with the partitioning step, we give you the following code that selects the indexes for the 1st1st and 2nd2nd mini-batches:
first_mini_batch_X = shuffled_X[:, 0 : mini_batch_size]
second_mini_batch_X = shuffled_X[:, mini_batch_size : 2 * mini_batch_size] ... Note that the last mini-batch might end up smaller than mini_batch_size=64. Let ⌊s⌋⌊s⌋ represents ss rounded down to the nearest integer (this is math.floor(s) in Python). If the total number of examples is not a multiple of mini_batch_size=64 then there will be ⌊mmini_batch_size⌋⌊mmini_batch_size⌋ mini-batches with a full 64 examples, and the number of examples in the final mini-batch will be (m−mini_batch_size×⌊mmini_batch_size⌋m−mini_batch_size×⌊mmini_batch_size⌋).
In [66]:
# 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, integer Returns: mini_batches -- list of synchronous (mini_batch_X, mini_batch_Y) """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.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*mini_batch_size : (k+1)*mini_batch_size]mini_batch_Y =shuffled_Y[:, k*mini_batch_size : (k+1)*mini_batch_size]### 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[:, math.floor(m/mini_batch_size)*mini_batch_size : m]mini_batch_Y = shuffled_Y[:,math.floor(m/mini_batch_size)*mini_batch_size : m]### END CODE HERE ###mini_batch = (mini_batch_X, mini_batch_Y)mini_batches.append(mini_batch)return mini_batches
In [67]:
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]))Expected Output:
| 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 ] |
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.
Momentum takes into account the past gradients to smooth out the update. We will store the 'direction' of the previous gradients in the variable vv. Formally, this will be the exponentially weighted average of the gradient on previous steps. You can also think of vv as the "velocity" of a ball rolling downhill, building up speed (and momentum) according to the direction of the gradient/slope of the hill.
Figure 3 : The red arrows shows the direction taken by one step of mini-batch gradient descent with momentum. The blue points show the direction of the gradient (with respect to the current mini-batch) on each step. Rather than just following the gradient, we let the gradient influence vv and then take a step in the direction of vv.
Exercise: Initialize the velocity. The velocity, vv, is a python dictionary that needs to be initialized with arrays of zeros. Its keys are the same as those in the grads dictionary, that is: for l=1,...,Ll=1,...,L:
v["dW" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["W" + str(l+1)]) v["db" + str(l+1)] = ... #(numpy array of zeros with the same shape as parameters["b" + str(l+1)]) Note that the iterator l starts at 0 in the for loop while the first parameters are v["dW1"] and v["db1"] (that's a "one" on the superscript). This is why we are shifting l to l+1 in the for loop.
In [68]:
# 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)] = 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 # 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]))### END CODE HERE ###return v
In [69]:
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"]))Expected Output:
| 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.]] |
Exercise: Now, implement the parameters update with momentum. The momentum update rule is, for l=1,...,Ll=1,...,L:
{vdW[l]=βvdW[l]+(1−β)dW[l]W[l]=W[l]−αvdW[l](3)(3){vdW[l]=βvdW[l]+(1−β)dW[l]W[l]=W[l]−αvdW[l]
{vdb[l]=βvdb[l]+(1−β)db[l]b[l]=b[l]−αvdb[l](4)(4){vdb[l]=βvdb[l]+(1−β)db[l]b[l]=b[l]−αvdb[l]
where L is the number of layers, ββ is the momentum and αα is the learning rate. All parameters should be stored in the parameters dictionary. Note that the iterator l starts at 0 in the for loop while the first parameters are W[1]W[1] and b[1]b[1] (that's a "one" on the superscript). So you will need to shift l to l+1 when coding.
In [70]:
# GRADED FUNCTION: update_parameters_with_momentumdef 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 # number of layers in the neural networks# Momentum update for each parameterfor l in range(L):### START CODE HERE ### (approx. 4 lines)# compute velocitiesv["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 parametersparameters["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, v
In [71]:
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"]))Expected Output:
| 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]] |
Note that:
- The velocity is initialized with zeros. So the algorithm will take a few iterations to "build up" velocity and start to take bigger steps.
- If β=0β=0, then this just becomes standard gradient descent without momentum.
How do you choose ββ?
- The larger the momentum ββ is, the smoother the update because the more we take the past gradients into account. But if ββ is too big, it could also smooth out the updates too much.
- Common values for ββ range from 0.8 to 0.999. If you don't feel inclined to tune this, β=0.9β=0.9 is often a reasonable default.
- Tuning the optimal ββ for your model might need trying several values to see what works best in term of reducing the value of the cost function JJ.
What you should remember:
- Momentum takes past gradients into account to smooth out the steps of gradient descent. It can be applied with batch gradient descent, mini-batch gradient descent or stochastic gradient descent.
- You have to tune a momentum hyperparameter ββ and a learning rate αα.
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?
- It calculates an exponentially weighted average of past gradients, and stores it in variables vv (before bias correction) and vcorrectedvcorrected (with bias correction).
- It calculates an exponentially weighted average of the squares of the past gradients, and stores it in variables ss (before bias correction) and scorrectedscorrected(with bias correction).
- It updates parameters in a direction based on combining information from "1" and "2".
The update rule is, for l=1,...,Ll=1,...,L:
vdW[l]=β1vdW[l]+(1−β1)∂∂W[l]vcorrecteddW[l]=vdW[l]1−(β1)tsdW[l]=β2sdW[l]+(1−β2)(∂∂W[l])2scorrecteddW[l]=sdW[l]1−(β1)tW[l]=W[l]−αvcor