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.
Figure 1 : Minimizing the cost is like finding the lowest point in a hilly landscape At each step of the training, you update your parameters following a certain direction to try to get to the lowest possible point.
To get started, run the following code to import the libraries you will need.
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
# GRADED FUNCTION: update_parameters_with_gddefupdate_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 inrange(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
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.
Figure 1 : SGD vs GD "+" 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).
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.
Figure 2SGD vs Mini-Batch GD "+" 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.
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:
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 by mini_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 full mini_batch_size, it will look like this:
# GRADED FUNCTION: random_mini_batchesdefrandom_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 inrange(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].reshape((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 ].reshape((1, m - math.floor(m/mini_batch_size)*mini_batch_size))
### END CODE HERE ### mini_batch = (mini_batch_X, mini_batch_Y) mini_batches.append(mini_batch)return mini_batches
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]))
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: [ 0.90085595-0.76120690.2344157 ]
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.
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.
# GRADED FUNCTION: initialize_velocitydefinitialize_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 velocityfor l inrange(L):### START CODE HERE ### (approx. 2 lines) v["dW"+str(l+1)]= np.zeros(parameters['W'+str(l +1)].shape) v["db"+str(l+1)]= np.zeros(parameters['b'+str(l +1)].shape)### END CODE HERE ###return v
The velocity is initialized with zeros. So the algorithm will take a few iterations to "build up" velocity and start to take bigger steps.
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.
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.
where:
t counts the number of steps taken of Adam
L is the number of layers
As usual, we will store all parameters in the parameters dictionary
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)])s["dW"+str(l+1)]= ... #(numpy array of zeros with the same shape as parameters["W" + str(l+1)])s["db"+str(l+1)]= ... #(numpy array of zeros with the same shape as parameters["b" + str(l+1)])
# GRADED FUNCTION: initialize_adamdefinitialize_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 inrange(L):### START CODE HERE ### (approx. 4 lines) v["dW"+str(l+1)]= np.zeros(parameters['W'+str(l+1)].shape) v["db"+str(l+1)]= np.zeros(parameters['b'+str(l+1)].shape) s["dW"+str(l+1)]= np.zeros(parameters['W'+str(l+1)].shape) s["db"+str(l+1)]= np.zeros(parameters['b'+str(l+1)].shape)### END CODE HERE ###return v, s
# GRADED FUNCTION: update_parameters_with_adamdefupdate_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 parametersfor l inrange(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- np.power(beta1, t))# 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.power(grads['dW'+str(l+1)], 2) s["db"+str(l+1)]= beta2*s["db"+str(l+1)]+ (1- beta2)*(grads['db'+str(l+1)]*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)]/(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, s
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.
5 - Model with different optimization algorithms
Lets use the following "moons" dataset to test the different optimization methods. (The dataset is named "moons" because the data from each of the two classes looks a bit like a crescent-shaped moon.)
train_X, train_Y =load_dataset()
We have already implemented a 3-layer neural network. You will train it with:
Mini-batch Gradient Descent: it will call your function:
update_parameters_with_gd()
Mini-batch Momentum: it will call your functions:
initialize_velocity() and update_parameters_with_momentum()
Mini-batch Adam: it will call your functions:
initialize_adam() and update_parameters_with_adam()
defmodel(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 optimizerif optimizer =="gd":pass# no initialization required for gradient descentelif optimizer =="momentum": v =initialize_velocity(parameters)elif optimizer =="adam": v, s =initialize_adam(parameters)# Optimization loopfor i inrange(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 parametersif 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 epochif 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 parameters
You will now run this 3 layer neural network with each of the 3 optimization methods.
5.1 - Mini-batch Gradient descent
Run the following code to see how the model does with mini-batch gradient descent.
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:0.796666666667
5.2 - Mini-batch gradient descent with momentum
Run the following code to see how the model does with momentum. Because this example is relatively simple, the gains from using momemtum are small; but for more complex problems you might see bigger gains.
Cost after epoch 0:0.690741 Cost after epoch 1000:0.685341 Cost after epoch 2000:0.647145 Cost after epoch 3000:0.619594 Cost after epoch 4000:0.576665 Cost after epoch 5000:0.607324 Cost after epoch 6000:0.529476 Cost after epoch 7000:0.460936 Cost after epoch 8000:0.465780 Cost after epoch 9000:0.464740
Accuracy:0.796666666667
5.3 - Mini-batch with Adam mode
Run the following code to see how the model does with Adam.
Cost after epoch 0:0.690552 Cost after epoch 1000:0.185567 Cost after epoch 2000:0.150852 Cost after epoch 3000:0.074454 Cost after epoch 4000:0.125936 Cost after epoch 5000:0.104235 Cost after epoch 6000:0.100552 Cost after epoch 7000:0.031601 Cost after epoch 8000:0.111709 Cost after epoch 9000:0.197648
Accuracy:0.94
5.4 - Summary
Momentum usually helps, but given the small learning rate and the simplistic dataset, its impact is almost negligeable. Also, the huge oscillations you see in the cost come from the fact that some minibatches are more difficult thans others for the optimization algorithm.
Adam on the other hand, clearly outperforms mini-batch gradient descent and Momentum. If you run the model for more epochs on this simple dataset, all three methods will lead to very good results. However, you've seen that Adam converges a lot faster.
Some advantages of Adam include:
Relatively low memory requirements (though higher than gradient descent and gradient descent with momentum)
Gradient descent goes "downhill" on a cost function J. Think of it as trying to do this:
Notations: As usual, ∂a∂J=da for any variable a.
A simple optimization method in machine learning is gradient descent (GD). When you take gradient steps with respect to all m 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,...,L: