def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
### YOUR CODE HERE
grad = np.zeros(outputVectors.shape)
gradPred = np.zeros(predicted.shape)
# indices = [target]
# for k in xrange(K):
# newidx = dataset.sampleTokenIdx()
# while newidx == target:
# newidx = dataset.sampleTokenIdx()
# indices += [newidx]
#
# labels = np.array([1] + [-1 for k in xrange(K)])
# vecs = outputVectors[indices,:]
#
# t = sigmoid(vecs.dot(predicted) * labels)
# cost = -np.sum(np.log(t))
#
# delta = labels * (t - 1)
# gradPred = delta.reshape((1,K+1)).dot(vecs).flatten()
# gradtemp = delta.reshape((K+1,1)).dot(predicted.reshape(
# (1,predicted.shape[0])))
# for k in xrange(K+1):
# grad[indices[k]] += gradtemp[k,:]
t = sigmoid(predicted.dot(outputVectors[target,:]))
cost = -np.log(t)
delta = t - 1
gradPred += delta * outputVectors[target, :]
grad[target, :] += delta * predicted
for k in xrange(K):
idx = dataset.sampleTokenIdx()
t = sigmoid(-predicted.dot(outputVectors[idx,:]))
cost += -np.log(t)
delta = 1 - t
gradPred += delta * outputVectors[idx, :]
grad[idx, :] += delta * predicted
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors,
dataset, K = 10):
klist = []
for k in range(K):
randomId = dataset.sampleTokenIdx()
while randomId == target:
randomId = dataset.sampleTokenIdx()
klist.append(randomId)
u0 = outputVectors[target]
uks = -outputVectors[klist]
vc = predicted
U = np.vstack((u0, uks))
dot = U.dot(vc)
sigmoid_value = sigmoid(dot)
cost = -np.sum(np.log(sigmoid_value))
# print "cost is %f" % (cost, )
gradPred = np.zeros(predicted.shape)
grad = np.zeros(outputVectors.shape)
temp = sigmoid(u0.dot(vc)) - 1
intermediate = (sigmoid_value-1).reshape(-1, 1) * np.vstack((u0, -uks))
gradPred += intermediate[0] - np.sum(intermediate[1:,], axis=0)
grad[target] += temp * vc
counter_dictionary = Counter(klist)
unique_ks = list(counter_dictionary.keys())
frequency_count = np.array(list(counter_dictionary.values()))
grad[unique_ks] += (sigmoid(-outputVectors[unique_ks].dot(vc)) -1).reshape(-1,1) * -vc
grad[unique_ks] *= frequency_count.reshape(-1, 1)
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size.
#
# Input/Output Specifications: same as softmaxCostAndGradient
u_o = outputVectors[target,:]
sigmoid_o = sigmoid(np.dot(u_o, predicted))
cost = - np.log(sigmoid_o)
gradPred = -u_o*(1-sigmoid_o)
grad = np.zeros_like(outputVectors)
grad[target,:] = - predicted*(1-sigmoid_o)
for _ in range(K):
k = dataset.sampleTokenIdx()
while k == target:
k = dataset.sampleTokenIdx()
sigmoid_k = sigmoid(-np.dot(outputVectors[k,:],predicted))
cost += - np.log(sigmoid_k)
gradPred += outputVectors[k,:] * (1-sigmoid_k)
grad[k,:] += predicted * (1-sigmoid_k)
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
### YOUR CODE HERE
negativeSamples = [dataset.sampleTokenIdx() for i in range(K)]
sigmoidTargetPred = sigmoid(outputVectors[target,:].transpose().dot(predicted))
cost = -np.log(sigmoidTargetPred)
gradPred = (sigmoidTargetPred - 1.0)*outputVectors[target,:]
grad = np.zeros(outputVectors.shape)
grad[target,:] = predicted * (sigmoidTargetPred - 1.0)
for sample in negativeSamples:
sigmoidSamplePredicted = sigmoid(-outputVectors[sample,:].transpose().dot(predicted))
cost -= np.log(sigmoidSamplePredicted)
gradPred += (1.0 - sigmoidSamplePredicted)*outputVectors[sample,:]
grad[sample,:] += (1.0 - sigmoidSamplePredicted)*predicted.transpose()
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
### YOUR CODE HERE
grad = np.zeros(outputVectors.shape)
s = sigmoid(np.dot(outputVectors[target,:], predicted))
cost = -np.log(s)
gradPred = - sigmoid_grad(s)/s*outputVectors[target,:]
grad[target,:] = - sigmoid_grad(s)/s*predicted
for k in range(K):
i = dataset.sampleTokenIdx()
s = sigmoid( - np.dot(outputVectors[i,:], predicted))
cost -= np.log(s)
gradPred += sigmoid_grad(s)/s*outputVectors[i,:]
grad[i,:] += sigmoid_grad(s)/s*predicted
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
### YOUR CODE HERE
'''
Keep track of dims:
D - dim of word vector
V - number of words
predicted : (D, )
target : integer
outputVectors : (V, D)
cost : float
gradPred : (D, )
grad : (V, D)
'''
predicted = predicted.reshape(-1, 1) # (D ,1)
sampledIndices = [dataset.sampleTokenIdx() for i in xrange(K)]
sampledVectors = outputVectors[sampledIndices, :] # (K, D)
outputVec = outputVectors[target, :] # (D, )
prob_out = sigmoid(outputVec.dot(predicted)) # float
probs_negative = sigmoid(-sampledVectors.dot(predicted)) # (K, 1)
cost = - np.log(prob_out) - np.sum(np.log(probs_negative))
gradPred = (prob_out - 1) * outputVec - np.sum((probs_negative-1) * sampledVectors, axis=0) # (D, )
grad = np.zeros_like(outputVectors) # (V, D)
grad[target, :] = (prob_out - 1) * predicted.reshape(-1)
# Note that sampledIndices may have repeated indices, we may loop over all K samples.
# And target should not be appeared in sampledIndices, but it's ok in gradient check,
# because we use += to update each output vector, and no grads will be missed.
for i in xrange(K):
grad[sampledIndices[i], :] += (1 - probs_negative[i]) * predicted.reshape(-1)
### END YOUR CODE
return cost, gradPred, grad
def test_sigmoid_permutation_axis1(dim_1):
a1 = np.random.normal(size=(1,dim_1))
s1 = sigmoid(a1)
permutation = np.random.permutation(dim_1)
inverse_permutation = np.argsort(permutation)
s1_perm = sigmoid(a1.ravel()[permutation])
assert rel_error(s1_perm.ravel()[inverse_permutation], s1) <= 1e-8
def test_sigmoid_gradient(dim_1, dim_2):
a1 = np.random.normal(loc=0., scale=20., size=(dim_1,dim_2))
shift = np.random.uniform(low=1e-9, high=1e-5, size=(dim_1,dim_2))
ap = a1 + shift
am = a1 - shift
dsigmoid = (sigmoid(ap) - sigmoid(am)) / (2*shift)
assert np.abs(np.max(dsigmoid - sigmoid_grad(sigmoid(a1)))) <= 1e-7
assert np.abs(np.min(dsigmoid - sigmoid_grad(sigmoid(a1)))) <= 1e-7
def test_sigmoid_shape(dim):
testing_shape = []
for y in range(0,dim):
testing_shape.append(np.random.randint(3,8))
shape = tuple(testing_shape)
#z = np.random.randn(*testing_shape)
x = np.random.standard_normal(shape)
y = np.copy(x)
assert x.shape == sigmoid(y).shape
assert x.shape == sigmoid_grad(sigmoid(y)).shape
def test_sigmoid_permutation_axis0(dim_1, execution_number):
""" sigmoid needs to be applied element-wise;"""
a1 = np.random.normal(size=(dim_1,1))
s1 = sigmoid(a1)
permutation = np.random.permutation(dim_1)
inverse_permutation = np.argsort(permutation)
s1_perm = sigmoid(a1[permutation])
assert rel_error(s1_perm[inverse_permutation], s1) <= 1e-8
def sigmoid_forward(x):
"""
Computes the forward pass for a sigmoid activation.
Inputs:
- x: Input data, numpy array of arbitary shape;
Returns a tuple (out, cache)
- out: output of the same shape as x
- cache: identical to out; required for backpropagation
"""
return sigmoid(x), sigmoid(x)
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
### YOUR CODE HERE
v_c = predicted
u_o = outputVectors[target]
suv = sigmoid(u_o.dot(v_c))
pos = np.log(suv) # positive sample
# sample w/ dataset.sampleTokenIdx() method iteratively
n = [] # indexes of negative samples
while len(n) < K:
x = dataset.sampleTokenIdx()
if x != target:
n.append(x)
neg_samples = outputVectors[n]
skv = sigmoid((neg_samples.dot(v_c)))
neg = np.sum(np.log(1-skv))
cost = -pos - neg
# neg_samples: K x d, skv: 1 x K
gradPred = -(1 - suv) * u_o + (neg_samples.T * (skv)).sum(axis=1)
grad = np.zeros(outputVectors.shape)
grad[target] += -(1-suv) * v_c
negGrad = np.outer(skv, v_c)
# sum grads together when they have been sampled with replacement
for i,x in enumerate(n):
grad[x] += negGrad[i]
### END YOUR CODE
return cost, gradPred, grad
def test_sigmoidgrad():
""" Original sigmoid gradient test defined in q2_sigmoid.py; """
x = np.array([[1, 2], [-1, -2]])
f = sigmoid(x)
g = sigmoid_grad(f)
assert rel_error(g, np.array([[0.19661193, 0.10499359],
[0.19661193, 0.10499359]])) <= 1e-7
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
T = data.shape[0]
### YOUR CODE HERE: forward propagation
z1 = np.dot(data, W1) + b1 # MxH + 1xH = MxH
h = sigmoid(z1)
z2 = np.dot(h, W2) + b2 # MxDy + 1xDy = MxDy
y_ = softmax(z2) # MxDy
cost = -1*np.sum(np.log(y_)*labels)/T
#raise NotImplementedError
### END YOUR CODE
### YOUR CODE HERE: backward propagation
dz2 = (y_ - labels)/T # MxDy
db2 = np.sum(dz2, axis=0) # 1xDy
dh = np.dot(dz2, W2.T) # MxH
dW2 = np.dot(h.T, dz2) # HxDy
dz1 = h*(1-h)*dh # MxH
db1 = np.sum(dz1, axis=0) # 1xH
dW1 = np.dot(data.T, dz1) # Dx x H
gradb2 = db2
gradW2 = dW2
gradb1 = db1
gradW1 = dW1
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
#print "Cost: %f \t grad[0] %f, grad[1] %f" % (cost, grad[0], grad[1])
return cost, grad
def forward_backward_prop(X, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
the backward propagation for the gradients for all parameters.
Notice the gradients computed here are different from the gradients in
the assignment sheet: they are w.r.t. weights, not inputs.
Arguments:
X -- M x Dx matrix, where each row is a training example x.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
# Note: compute cost based on `sum` not `mean`.
### YOUR CODE HERE: forward propagation
h = sigmoid(np.dot(X, W1) + b1)
yhat = softmax(np.dot(h, W2) + b2)
cross_entropy = -np.log(yhat)[labels == 1]
cost = np.sum(cross_entropy) # / len(labels)
### END YOUR CODE
# Things look too good to be true... Tried twice and the gradient check passed.
### YOUR CODE HERE: backward propagation
dl_dyhat = (-1 / yhat)[labels == 1] # m x 1, m the number of points;
dyhat_dsoftmax = yhat * (1 - yhat) # m x n, n the number of classes;
dl_dsoftmax = dyhat_dsoftmax * np.reshape(dl_dyhat, (-1, 1))
gradW2 = np.reshape(np.sum(h, 0), [-1, 1]) * np.reshape(
np.sum(dl_dsoftmax, 0),
[1, -1]) #np.dot(h.T, dl_dsoftmax) # n x h, transpose shape of W2;
gradb2 = np.reshape(np.sum(dl_dsoftmax, axis=0), (1, -1)) # n x 1
dl_dh = np.dot(np.sum(dl_dsoftmax, axis=0), W2.T) # m x h, sum up all m
dh_dsigmoid = np.sum(sigmoid_grad(h), 0) # m x h, sumup all m
dl_dsigmoid = dl_dh * dh_dsigmoid
gradW1 = np.reshape(np.sum(X, 0), [-1, 1]) * np.reshape(
dl_dsigmoid, [1, -1])
gradb1 = np.reshape(dl_dsigmoid, (1, -1))
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def test_sigmoid():
""" Original sigmoid test defined in q2_sigmoid.py; """
x = np.array([[1, 2], [-1, -2]])
f = sigmoid(x)
assert rel_error(
f, np.array([[0.73105858, 0.88079708], [0.26894142, 0.11920292]
])) <= 1e-7
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
u_o, v_c = outputVectors[target], outputVectors[indices[1:]]
loss = -np.log(sigmoid(np.matmul(u_o, predicted)))
print(u_o)
print(v_c)
### END YOUR CODE
return cost, gradPred, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
layer1 = data.dot(W1) + b1
activation1 = sigmoid(layer1)
layer2 = activation1.dot(W2) + b2
predictions = softmax(layer2)
softmaxs = np.sum(predictions * labels, axis=1)
cost = -np.log(softmaxs)
num_train = data.shape[0]
cost = np.sum(cost) / num_train
### END YOUR CODE
### YOUR CODE HERE: backward propagation
dC = 1.0
dLayer2 = dC / num_train * predictions - dC / num_train * labels
gradb2 = np.sum(dLayer2, axis=0)
gradW2 = activation1.T.dot(dLayer2)
dActivation1 = dLayer2.dot(W2.T)
dLayer1 = dActivation1 * sigmoid_grad(activation1)
gradb1 = np.sum(dLayer1, axis=0)
gradW1 = data.T.dot(dLayer1)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def negSamplingCostAndGradient(predicted,
target,
outputVectors,
dataset,
K=10):
grad = np.zeros(outputVectors.shape)
gradPred = np.zeros(predicted.shape)
indices = [target]
for k in range(K):
newidx = dataset.sampleTokenIdx()
while newidx == target:
newidx = dataset.sampleTokenIdx()
indices += [newidx]
labels = np.array([1] + [-1 for k in range(K)])
vecs = outputVectors[indices, :]
t = sigmoid(vecs.dot(predicted) * labels)
cost = -np.sum(np.log(t))
delta = labels * (t - 1)
gradPred = delta.reshape((1, K + 1)).dot(vecs).flatten()
gradtemp = delta.reshape(
(K + 1, 1)).dot(predicted.reshape((1, predicted.shape[0])))
for k in range(K + 1):
grad[indices[k]] += gradtemp[k, :]
return cost, gradPred, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
N = data.shape[0]
a1 = data
z2 = np.matmul(a1, W1) + b1
a2 = sigmoid(z2)
z3 = np.matmul(a2, W2) + b2
a3 = softmax(z3)
ycap = a3
cost = -np.sum(labels * np.log(ycap))
#raise NotImplementedError
### END YOUR CODE
### YOUR CODE HERE: backward propagation
a3grad = a3 - labels # this is the grad for softmax
gradW2 = np.dot(a2.T, a3grad)
gradb2 = np.sum(a3grad, axis=0, keepdims=True)
t = np.dot(W2, a3grad.T) * sigmoid_grad(a2).T
gradW1 = np.dot(t, a1).T
gradb1 = np.sum(np.dot(a3grad, W2.T) * sigmoid_grad(a2),
axis=0,
keepdims=True)
#raise NotImplementedError
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
prod = sigmoid(np.dot(outputVectors[target], predicted))
cost = -np.log(prod) - \
sum([np.log(sigmoid(-np.dot(outputVectors[i], predicted))) for i in indices[1:]])
gradPred = (prod - 1) * outputVectors[target] - \
sum([(sigmoid(-np.dot(outputVectors[i], predicted)) - 1) * outputVectors[i] for i in indices[1:]])
grad = np.zeros(outputVectors.shape)
grad[target] = (prod - 1) * predicted
### IMPORTANT: GRADIENTS FOR SAMPLED WORDS SHOULD BE ACCUMULATED BECAUSE THEY CAN APPEAR SEVERAL TIMES => -=
for i in indices[1:]:
grad[i] -= (sigmoid(-np.dot(outputVectors[i], predicted)) - 1) * predicted
assert gradPred.shape == predicted.shape
assert grad.shape == outputVectors.shape
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted,
target,
outputVectors,
dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
gradPred = np.zeros_like(predicted)
grad = np.zeros_like(outputVectors)
### YOUR CODE HERE
V = outputVectors.shape[0] #vocabulary size
y = np.zeros(V)
y[target] = 1
similarity = np.dot(predicted, outputVectors[target].T) #Ut . Vc
probability = sigmoid(similarity)
# cost = -np.log(probability)
cost = -np.sum(y * np.log(probability))
# context_word_vec = np.zeros((len(indices), outputVectors.shape[1]))
gradPred = (probability - 1) * outputVectors[target]
grad[target] = np.dot(probability - 1, predicted)
i = 0
for i in indices[1:]:
neg_similarity = np.dot(predicted, outputVectors[i].T)
neg_probability = sigmoid(-neg_similarity)
# cost += -np.log(neg_probability)
cost += -np.sum(y * np.log(neg_probability))
gradPred += (1 - neg_probability) * outputVectors[i]
grad[i] += np.dot((1 - neg_probability), predicted)
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted,
target,
outputVectors,
dataset,
K=10):
""" Negative sampling cost function for word2vec models
"""
### YOUR CODE HERE
V, D = outputVectors.shape
# get the k random indices
k_indicies = []
for i in range(K):
rand_index = dataset.sampleTokenIdx() # 有没有可能随机到正确的样本？有，但概率很小
k_indicies.append(rand_index)
# loss function
neg_sample_vector = outputVectors[k_indicies, :] # KxD
assert neg_sample_vector.shape == (K, D)
sigm_neg = sigmoid(-1.0 *
np.dot(neg_sample_vector, predicted.reshape(
(D, 1)))) # KxD Dx1 = Kx1
cost_neg = np.sum(np.log(sigm_neg), axis=0)
sigm_cor = sigmoid(np.dot(outputVectors[target], predicted.reshape(
(D, 1))))
cost = -1.0 * np.log(sigm_cor) - cost_neg
# gradient on output vectors
grad = np.zeros(outputVectors.shape) # V, D
grad[target] = predicted * (sigm_cor - 1.0) # 1xD
for k in k_indicies:
grad[k, :] += -1.0 * predicted.reshape(
(D, )) * (sigmoid(np.dot(-1.0 * predicted, outputVectors[k])) -
1.0)
# gradient on input vector
# 这里第一项减一跟公式有点不一样啊。。但是结果是对的。。1 - sigm_neg.reshape((1,K))
gradPred_neg = np.dot(1 - sigm_neg.reshape((1, K)),
neg_sample_vector).reshape((1, D)) # 1xK KxD = 1xD
gradPred_cor = (sigm_cor - 1) * outputVectors[target].reshape((1, D))
gradPred = gradPred_neg + gradPred_cor
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
#initialize variables
cost = 0
gradPred = np.zeros(predicted.shape)
grad = np.zeros(outputVectors.shape)
output_word = outputVectors[target]
target_sigmoid = sigmoid(np.dot(output_word, predicted))
cost = -np.log(target_sigmoid)
gradPred = (target_sigmoid - 1.0) * output_word
grad[target] = (target_sigmoid - 1.0) * predicted
for index in indices:
word = outputVectors[index]
k_sigmoid = sigmoid(np.dot(-word, predicted))
cost -= np.log(k_sigmoid)
gradPred += ((1.0 - k_sigmoid) * word)
grad[index] += -((k_sigmoid - 1.0) * predicted)
assert predicted.shape == gradPred.shape
assert outputVectors.shape == grad.shape
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted,
target,
outputVectors,
dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
# indices[0]=target, indices[1--K] = not target
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
u_o = outputVectors[indices[0]]
v_c = predicted
# [dim]
z = sigmoid(np.dot(u_o.T, v_c))
cost = -np.log(z)
gradPred = np.zeros(np.shape(predicted))
gradPred += (z - 1.0) * u_o
grad = np.zeros(np.shape(outputVectors))
# 按位乘 [1,2,3]*[1,2,3] = [1,4,9]
# 对u_o求偏导
grad[target] += (z - 1.0) * v_c
# for negative samples
for k in range(K):
u_k = outputVectors[indices[k + 1]]
z = sigmoid(np.dot(u_k.T, v_c))
cost -= np.log(1.0 - z)
gradPred += z * u_k
grad[indices[k + 1]] += z * v_c
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted,
target,
outputVectors,
dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
grad = np.zeros(outputVectors.shape)
gradPred = np.zeros(predicted.shape)
cost = 0
z = sigmoid(np.dot(outputVectors[target], predicted))
cost -= np.log(z)
grad[target] += predicted * (z - 1.0)
gradPred += outputVectors[target] * (z - 1.0)
grad = np.zeros(outputVectors.shape)
grad[target] = predicted * (sigmoid(outputVectors[target].dot(predicted)) -
1)
gradPred = outputVectors[target] * (
sigmoid(outputVectors[target].dot(predicted)) - 1)
cost = -np.log(sigmoid(outputVectors[target].dot(predicted)))
for i in xrange(1, K + 1):
negtive_index = indices[i]
sig = sigmoid((outputVectors[negtive_index]).dot(predicted))
cost += -np.log(1 - sig)
grad[negtive_index] += predicted * sig
gradPred += outputVectors[negtive_index] * sig
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted,
target,
outputVectors,
dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
### YOUR CODE HERE
grad = np.zeros_like(outputVectors)
gradPred = np.zeros_like(predicted)
activate = sigmoid(np.dot(predicted.reshape(-1), outputVectors[target].T))
cost = 0
cost -= np.log(activate)
grad[target:target + 1] = (activate - 1) * predicted
gradPred += (activate - 1) * outputVectors[target]
neg_samples = []
for i in range(K):
idx = dataset.sampleTokenIdx()
if (idx == target) or (idx in neg_samples):
i -= 1
continue
neg_samples.append(idx)
neg_activate = sigmoid(
-np.dot(predicted.reshape(-1), outputVectors[idx].T))
cost -= np.log(neg_activate)
grad[idx:idx + 1] = -(neg_activate - 1) * predicted
gradPred -= (neg_activate - 1) * outputVectors[idx]
### END YOUR CODE
return cost, gradPred, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data [input] -- M x Dx matrix, where each row is a training example.
labels [expected O/p] -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
# M,Dx * (Dx,H) + H = M * H
pre_h = data.dot(W1) + b1
h = sigmoid(pre_h)
# M,H * H,Dy + Dy = M * Dy
pre_Y = np.dot(h, W2) + b2
Y = softmax(pre_Y)
# //cross entorpy cost
cost = -np.sum(labels * np.log(Y)) # M * Dy
### END YOUR CODE
### YOUR CODE HERE: backward propagation
#first grad of cost function
delta_cost = Y - labels
gradW2 = h.T.dot(delta_cost)
gradb2 = np.sum(delta_cost, axis=0)
delta_h = delta_cost.dot(W2.T) * sigmoid_grad(h)
gradW1 = data.T.dot(delta_h)
gradb1 = np.sum(delta_h, axis=0)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(X, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
the backward propagation for the gradients for all parameters.
Notice the gradients computed here are different from the gradients in
the assignment sheet: they are w.r.t. weights, not inputs.
Arguments:
X -- M x Dx matrix, where each row is a training example x.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
# Note: compute cost based on `sum` not `mean`.
### YOUR CODE HERE: forward propagation
#raise NotImplementedError
### END YOUR CODE
h = sigmoid(X.dot(W1) + b1)
y_h = softmax(h.dot(W2) + b2)
### YOUR CODE HERE: backward propagation
cost = np.sum(-np.log(y_h[labels == 1])) / X.shape[0]
delta1 = (y_h - labels) / X.shape[0]
delta2 = delta1.dot(W2.transpose())
delta3 = sigmoid_grad(h) * delta2
#calculate gradient
gradW1 = X.transpose().dot(delta3)
gradb1 = np.sum(delta3, 0)
gradW2 = h.transpose().dot(delta1)
gradb2 = np.sum(delta1, 0, keepdims = True)
#raise NotImplementedError
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(X, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
the backward propagation for the gradients for all parameters.
Notice the gradients computed here are different from the gradients in
the assignment sheet: they are w.r.t. weights, not inputs.
Arguments:
X -- M x Dx matrix, where each row is a training example x.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
# Note: compute cost based on `sum` not `mean`.
### YOUR CODE HERE: forward propagation
z1 = np.dot(X,W1)+b1 # (M x H)
h = sigmoid(z1) # (M x H)
z2 = np.dot(h,W2)+b2 # (M x Dy)
y_dash = softmax(z2) # (M x Dy)
cost = - np.sum(labels* np.log(y_dash))
### END YOUR CODE
### YOUR CODE HERE: backward propagation
cost_theta = y_dash - labels
gradW2 = np.dot(h.T, cost_theta)
gradb2 = np.reshape(np.sum(cost_theta, axis = 0), b2.shape)
cost_h = np.dot(cost_theta, W2.T) # (M x H)
cost_z1 = sigmoid_grad(h) * cost_h # (M x H)
gradW1 = np.dot(X.T, cost_z1) # (Dx x M)*(M x H) = (Dx x M)
gradb1 = np.reshape(np.sum(cost_z1, axis = 0), b1.shape)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
h = sigmoid(
np.dot(data, W1) + b1
) # when adding bias +b1 with dimentions(1, H), then we add each element of b1 to its correspond position in dot(data, W1)
y_pred = softmax(np.dot(h, W2) + b2)
### END YOUR CODE
### YOUR CODE HERE: backward propagation
cost = np.sum(-np.log(y_pred[labels == 1])) / data.shape[
0] # cross entropy cost - average sum of all elements in y_pred where the correspondent in labels (y) is 1 but not 0 (label: one-hot vector)
delta_3 = (y_pred - labels) / data.shape[
0] # QUESTION: why dividing by data.shape[0] (number of examples)
delta_2 = sigmoid_grad(h) * np.dot(
delta_3, W2.T
) # compute delta using the dot product between the error (delta_3) and weights of second layer and the Hadamard product with the derivative of the activations
gradW2 = np.dot(h.T, delta_3)
gradb2 = np.sum(
delta_3, 0,
keepdims=True) # QUESTION: why summing the values of delta_3
gradW1 = np.dot(data.T, delta_2)
gradb1 = np.sum(
delta_2, 0,
keepdims=True) # QUESTION: why summing the values of delta_2
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(X, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
the backward propagation for the gradients for all parameters.
Notice the gradients computed here are different from the gradients in
the assignment sheet: they are w.r.t. weights, not inputs.
Arguments:
X -- M x Dx matrix, where each row is a training example x.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
# Note: compute cost based on `sum` not `mean`.
### YOUR CODE HERE: forward propagation
hidden = sigmoid(np.dot(X, W1) + b1) # R: M*Dx * Dx*H + 1*H = M*H
output = softmax(np.dot(hidden, W2) + b2) # R: M*H * H*Dy + 1*Dy = M*Dy
cost = -np.sum(labels * np.log(output))
### YOUR CODE HERE: backward propagation
gradb2 = -labels + output # R: M*Dy - M*Dy = M*Dy
gradW2 = hidden[:, :, np.
newaxis] * gradb2[:, np.
newaxis, :] # R: M*H*1 * M*1*Dy = M*H*Dy
gradb1 = np.sum(gradb2[:, np.newaxis, :] * W2[np.newaxis, :, :],
axis=2) * sigmoid_grad(
hidden) # R: = sum(M*1*Dy * M*Dy*H) * M*H= M*H
gradW1 = X[:, :,
np.newaxis] * gradb1[:,
np.newaxis, :] #R: M*H*1 * M*1*Dx = M*Dx*H
gradb2 = np.sum(gradb2, axis=0) # sum by column
gradb1 = np.sum(gradb1, axis=0)
gradW1 = np.sum(gradW1, axis=0)
gradW2 = np.sum(gradW2, axis=0)
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
### YOUR CODE HERE
indices = [target]
for k in xrange(K):
sampleTokenIdx = dataset.sampleTokenIdx()
while sampleTokenIdx == target:
sampleTokenIdx = dataset.sampleTokenIdx()
indices += [sampleTokenIdx]
signs = np.array([1] + [-1 for k in xrange(K)])
vecs = outputVectors[indices, :]
t = sigmoid(vecs.dot(predicted) * signs)
delta = (t - 1) * signs
cost = np.sum(-np.log(t))
gradPred = delta.reshape(1, K + 1).dot(vecs).flatten()
grad = np.zeros(outputVectors.shape)
gradtemp = delta.reshape((K+1,1)).dot(predicted.reshape(
(1,predicted.shape[0])))
for k in xrange(K+1):
grad[indices[k]] += gradtemp[k,:]
# naive implementation but not efficient cause it makes |V| computation.
# uv = outputVectors.dot(predicted)
# negSamplesCost = 0
# negSampleGradPred = np.zeros(predicted.shape[0])
# grad = np.zeros(outputVectors.shape)
#
# for i in xrange(K):
# sampleTokenIdx = dataset.sampleTokenIdx()
# while sampleTokenIdx == target:
# sampleTokenIdx = dataset.sampleTokenIdx()
# negSamplesCost += np.log(sigmoid(-uv[sampleTokenIdx]))
# negSampleGradPred += (sigmoid(-uv[sampleTokenIdx]) - 1) * outputVectors[sampleTokenIdx, :]
# grad[sampleTokenIdx] += -(sigmoid(-uv[sampleTokenIdx]) - 1) * predicted
#
# cost = -np.log(sigmoid(uv[target])) - negSamplesCost
# gradPred = (sigmoid(uv[target]) - 1) * outputVectors[target, :] - negSampleGradPred
# grad[target] = (sigmoid(uv[target]) - 1) * predicted
### END YOUR CODE
return cost, gradPred, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
# data: N x Dx, W1: Dx x H, b: 1 x H
a = data.dot(W1) + b1
h = sigmoid(a)
# h: N x H, W2: H x Dy, b2: 1 x Dy
t = h.dot(W2) + b2
y_hat = softmax(t)
# y_hat: N x Dy, labels: N x Dy (as int)
probs = labels * y_hat
cost = np.sum(-np.log(probs.sum(axis=1)))
### END YOUR CODE
### YOUR CODE HERE: backward propagation
# obtain the softmax gradient
dJdt = (y_hat - labels) # N x Dy
# b2 grad is sum along each index of the Dy vectors
gradb2 = np.sum(dJdt, 0)
# h: N x H, dJdt: N x Dy
gradW2 = h.T.dot(dJdt) # H x Dy
# dJdt: N x Dy, W2: H x Dy
dJdh = dJdt.dot(W2.T)
# h: N x H
dhda = sigmoid_grad(h)
# data: N x Dx, dhda: N x H, DJdh: N x H
gradW1 = data.T.dot(dhda * dJdh)
# dhda: N x H, DJdh: N x H
gradb1 = np.sum(dhda * dJdh, 0)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
z = np.dot(data, W1) + b1
h = sigmoid(z)
scores = np.dot(h, W2) + b2
probs = softmax(scores)
cost = - np.sum(labels * np.log(probs))
### END YOUR CODE
### YOUR CODE HERE: backward propagation
dscores = probs - labels
gradW2 = np.dot(h.T, dscores)
gradb2 = np.sum(dscores, axis=0, keepdims=True)
dh = np.dot(dscores, W2.T)
dz = sigmoid_grad(h) * dh
gradW1 = np.dot(data.T, dz)
gradb1 = np.sum(dz, axis=0, keepdims=True)
assert(np.all(gradW2.shape == W2.shape))
assert(np.all(gradb2.shape == b2.shape))
assert(np.all(gradW1.shape == W1.shape))
assert(np.all(gradb1.shape == b1.shape))
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(X, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
the backward propagation for the gradients for all parameters.
Notice the gradients computed here are different from the gradients in
the assignment sheet: they are w.r.t. weights, not inputs.
Arguments:
X -- M x Dx matrix, where each row is a training example x.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
# Note: compute cost based on `sum` not `mean`.
### YOUR CODE HERE: forward propagation
z1 = np.matmul(X, W1) + b1
a1 = sigmoid(z1)
a2 = softmax(np.matmul(a1, W2) + b2)
cost = (-np.log(a2) * labels).sum()
### END YOUR CODE
### YOUR CODE HERE: backward propagation
dz2 = a2 - labels
dw2 = np.matmul(a1.T, dz2)
assert dw2.shape == (H, Dy)
db2 = dz2.sum(0)
assert db2.shape == (Dy, )
dz1 = np.matmul(dz2, W2.T) * sigmoid_grad(a1)
dw1 = np.matmul(X.T, dz1)
assert dw1.shape == (Dx, H)
db1 = dz1.sum(0)
assert db1.shape == (H, )
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate(
(dw1.flatten(), db1.flatten(), dw2.flatten(), db2.flatten()))
return cost, grad
def forward_backward_prop(X, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
the backward propagation for the gradients for all parameters.
Notice the gradients computed here are different from the gradients in
the assignment sheet: they are w.r.t. weights, not inputs.
Arguments:
X -- M x Dx matrix, where each row is a training example x.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H)) # (10, 5)
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H)) # (1, 5)
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy)) # (5, 10)
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy)) # (1, 10)
### YOUR CODE HERE: forward propagation
z1 = np.dot(X, W1) + b1 # (20, 5)
a1 = sigmoid(z1) # (20, 5)
scores = np.dot(a1, W2) + b2 # (20, 10)
y_pred = softmax(scores)
cost = -np.sum(labels * np.log(y_pred))
# ### END YOUR CODE
# ### YOUR CODE HERE: backward propagation
dscores = y_pred - labels # (20, 10)
gradW2 = np.dot(a1.T, dscores) # (5, 10)
# gradW2 bp step 1: back to d(softmax(theta))/d(theta), this is equal to y-bar - y, which is dscores defined in line 52
# gradW2 bp step 2: back to d(W2*X2)/d(W2), this is equal to X2
# so combine 2 steps together, gradW2 = X2.T * (y-bar - y)
gradb2 = np.sum(dscores, axis=0) # (1, 10)
da1 = np.dot(dscores, W2.T)
dz1 = sigmoid_grad(a1)*da1
gradW1 = np.dot(X.T, dz1)
gradb1 = np.sum(dz1, axis=0)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad # cost is a single number, grad is a nparray
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### 前向运算
N = Dx
# 第一个隐层做内积
a1 = sigmoid(data.dot(W1) + b1)
# 第二个隐层做内积
a2 = softmax(a1.dot(W2) + b2)
cost = - np.sum(np.log(a2[labels == 1]))/N
### 反向传播
# Calculate analytic gradient for the cross entropy loss function
grad_a2 = ( a2 - labels ) / N
# Backpropagate through the second latent layer
gradW2 = np.dot( a1.T, grad_a2 )
gradb2 = np.sum( grad_a2, axis=0, keepdims=True )
# Backpropagate through the first latent layer
grad_a1 = np.dot( grad_a2, W2.T ) * sigmoid_grad(a1)
gradW1 = np.dot( data.T, grad_a1 )
gradb1 = np.sum( grad_a1, axis=0, keepdims=True )
# if verbose: # Verbose mode for logging information
# print ("W1 shape: {}".format( str(W1.shape) ))
# print ("W1 gradient shape: {}".format( str(gradW1.shape) ))
# print ("b1 shape: {}".format( str(b1.shape) ))
# print ("b1 gradient shape: {}".format( str(gradb1.shape) ))
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
h = sigmoid(data.dot(W1) + b1)
y_hat = softmax(h.dot(W2) + b2)
# cost = - np.sum(np.log(y_hat).dot(labels.transpose()))
cost = -np.sum(labels * np.log(y_hat)) / data.shape[0]
### END YOUR CODE
### YOUR CODE HERE: backward propagation
# gradb2 = y_hat - labels
# gradW2 = np.matmul((y_hat - labels), sigmoid(np.matmul(data, W1) + b1))
# gradb1 = np.matmul(np.matmul((y_hat - labels), W2.transpose()), np.matmul(sigmoid(np.matmul(data, W1), + b1), (1- sigmoid(np.matmul(data, W1), + b1))))
#
# gradb1 = (y_hat - labels) * W2 * sigmoid(data * W1 + b1) * (1 - sigmoid(data * W1 + b1))
# gradW1 = (y_hat - labels) * W2 * sigmoid(data * W1 + b1) * (1 - sigmoid(data * W1 + b1)) * data
gradZ2 = (y_hat - labels) / data.shape[0]
gradb2 = np.sum(gradZ2, axis=0, keepdims=True)
gradW2 = (h.T).dot(gradZ2)
gradH = gradZ2.dot(W2.T)
gradZ1 = gradH * sigmoid_grad(h) #相同坐标的元素相乘
gradb1 = np.sum(gradZ1, axis=0, keepdims=True)
gradW1 = (data.T).dot(gradZ1)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
#z1 = data.dot(W1) + b1
#hidden = sigmoid(z1)
#z2 = hidden.dot(W2) + b2
#print 'z2.shape: ', z2.shape
#prediction = softmax(z2)
### END YOUR CODE
hidden = sigmoid(data.dot(W1) + b1)
prediction = softmax(hidden.dot(W2) + b2)
cost = -np.sum(np.log(prediction) * labels)
### YOUR CODE HERE: backward propagation
#print 'NN: ', Dx, H, Dy
#print 'b1.shape: ', b1.shape
#print 'prediction.shape: ', prediction.shape
#print 'labels.shape : ', labels.shape
#print 'W2.shape: ', W2.shape
#print 'hidden.shape: ', hidden.shape
#print 'hidden.T.shape: ', hidden.T.shape
#print 'delta.shape: ', delta.shape
#print 'W1.shape: ', W1.shape
#print 'data.shape: ', data.shape
#gradW2 = delta * hidden
#print 'sigmoid_grad(hidden).shape: ', sigmoid_grad(hidden).shape
delta = prediction - labels
gradW2 = hidden.T.dot(delta)
gradb2 = np.sum(delta, axis = 0)
hidden_delta = delta.dot(W2.T) * sigmoid_grad(hidden)
gradW1 = data.T.dot(hidden_delta)
gradb1 = np.sum(hidden_delta, axis = 0)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(X, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
the backward propagation for the gradients for all parameters.
Notice the gradients computed here are different from the gradients in
the assignment sheet: they are w.r.t. weights, not inputs.
Arguments:
X -- M x Dx matrix, where each row is a training example x.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
# Note: compute cost based on `sum` not `mean`.
### YOUR CODE HERE: forward propagation
f1 = np.dot(X, W1) + b1
h = sigmoid(f1)
f2 = np.dot(h, W2) + b2
y_hat = softmax(f2)
cost = -np.sum(labels * np.log(y_hat))
### END YOUR CODE
### YOUR CODE HERE: backward propagation
sigma1 = y_hat - labels
sigma2 = W2.transpose()
sigma3 = sigmoid_grad(h)
gradb2 = np.sum(sigma1, axis=0)
gradW2 = np.dot(h.transpose(), sigma1)
sigma4 = sigma3 * np.dot(sigma1, sigma2)
gradb1 = np.sum(sigma4, axis=0)
gradW1 = np.dot(X.transpose(), sigma4)
# print(gradW1.shape)
# print(gradb1.shape)
# print(gradW2.shape)
# print(gradb2.shape)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
# print(grad.shape)
return cost, grad
def forward_backward_prop(X, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
the backward propagation for the gradients for all parameters.
Notice the gradients computed here are different from the gradients in
the assignment sheet: they are w.r.t. weights, not inputs.
Arguments:
X -- M x Dx matrix, where each row is a training example x.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
# Note: compute cost based on `sum` not `mean`.
### YOUR CODE HERE: forward propagation
z1 = np.dot(X, W1) + b1
g1 = sigmoid(z1) # M , H
z2 = np.dot(g1, W2) + b2
final_scores = softmax(z2) # M , Dy
cost = -np.sum(labels * np.log(final_scores))
### END YOUR CODE
### YOUR CODE HERE: backward propagation
gradSoft = final_scores - labels # M , Dy
gradW2 = g1.T.dot(gradSoft)
gradb2 = np.sum(gradSoft, axis=0) # 1 , Dy
gradz1 = gradSoft.dot(W2.T) # M , H
gradSig = gradz1 * sigmoid_grad(g1) # M , H
gradW1 = X.T.dot(gradSig) # Dx , H
gradb1 = np.sum(gradSig, axis=0) # 1 , H
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
# data: N x Dx, W1: Dx x H, b: 1 x H
a = data.dot(W1) + b1
h = sigmoid(a)
# h: N x H, W2: H x Dy, b2: 1 x Dy
t = h.dot(W2) + b2
y_hat = softmax(t)
# y_hat: N x Dy, labels: N x Dy (as int)
probs = labels * y_hat
cost = np.sum(-np.log(probs.sum(axis=1)))
### END YOUR CODE
### YOUR CODE HERE: backward propagation
# obtain the softmax gradient
dJdt = (y_hat - labels) # N x Dy
# b2 grad is sum along each index of the Dy vectors
gradb2 = np.sum(dJdt, 0)
# h: N x H, dJdt: N x Dy
gradW2 = h.T.dot(dJdt) # H x Dy
# dJdt: N x Dy, W2: H x Dy
dJdh = dJdt.dot(W2.T)
# h: N x H
dhda = sigmoid_grad(h)
# data: N x Dx, dhda: N x H, DJdh: N x H
gradW1 = data.T.dot(dhda * dJdh)
# dhda: N x H, DJdh: N x H
gradb1 = np.sum(dhda * dJdh, 0)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
# refactorization of the first draft below.
V,D = outputVectors.shape
output = outputVectors[indices] # the 0th is the center
uv = np.dot(output, predicted)
uv[0] = -uv[0]
sig = sigmoid(-uv)
cost = -np.sum(np.log(sig))
gradTheta = 1 - sig
gradTheta[0] = - gradTheta[0]
gradPred = np.dot(output.T, gradTheta) # 1 x D array
samples = np.reshape(gradTheta, (-1, 1)) * predicted
grad = np.zeros([V, D])
for i in range(len(indices)):
grad[indices[i]] += samples[i]
########################## First draft ###########################
### !!!this is super slow !!! bottle neck should be the 'for' loop
# uov = np.dot(predicted, outputVectors[target])
# sigmoid_uov = sigmoid(uov)
# ukv = np.dot(outputVectors[indices[1:]], predicted) # exclude the target
# sigmoid_ukv = sigmoid(-ukv) # 1 x K
# cost = -np.log(sigmoid_uov) - np.sum(np.log(sigmoid_ukv))
# gradTheta1 = -(1 - sigmoid_uov) # a scalar
# gradTheta2 = (1 - sigmoid_ukv) # K x 1 array
# gradPred = gradTheta1 * outputVectors[target] + np.sum(outputVectors[indices[1:]] * np.reshape(gradTheta2, (-1,1)), axis = 0) # 1 x D array
# gradOutput = np.zeros([V,D])
#
# # only K none-zero rows indicates the K negative samples, same numble of samples but might contain duplicated words
# # Can be parallelized further.
# samples = np.reshape(gradTheta2, (-1, 1)) * predicted
# for i in range(V):
# gradOutput[i] = np.sum(samples[np.where(np.array(indices[1:]) == i)], axis=0)
# # the positive sample
# gradOutput[indices[0]] = np.reshape(gradTheta1, (-1, 1)) * predicted
# grad = gradOutput
### END YOUR CODE
return cost, gradPred, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
#raise NotImplementedError
z1 = np.matmul(data, W1) + b1
h = sigmoid(z1)
z2 = np.matmul(h, W2) + b2
y_hat = softmax(z2)
cost = np.sum(-np.log(y_hat[labels == 1])) / (data.shape[0])
### END YOUR CODE
### YOUR CODE HERE: backward propagation
#raise NotImplementedError
d1 = (y_hat - labels) / (data.shape[0])
gradW2 = np.matmul(np.transpose(h), d1)
gradb2 = np.sum(d1, axis=0, keepdims=True)
d2 = np.matmul(d1, np.transpose(W2))
d3 = d2 * sigmoid_grad(h)
gradW1 = np.matmul(np.transpose(data), d3)
gradb1 = np.sum(d3, axis=0, keepdims=True)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
### YOUR CODE HERE
K_set = []
while (len(K_set) < K):
candidateIndex = dataset.sampleTokenIdx()
if (candidateIndex != target):
K_set += [candidateIndex]
cost = 0
grad = np.zeros(outputVectors.shape)
gradPred = np.zeros(predicted.shape)
for k in K_set:
score = sigmoid(- outputVectors[k].dot(predicted))
cost += - np.log(score)
grad[k,:] += - (score - 1) * predicted
gradPred += - (score - 1) * outputVectors[k]
score = sigmoid(outputVectors[target].dot(predicted))
cost += - np.log(score)
grad[target,:] = (score - 1) * predicted
gradPred += (score - 1) * outputVectors[target]
# ugly fix ...
gradPred = gradPred[np.newaxis, :]
#raise NotImplementedError
### END YOUR CODE
return cost, gradPred, grad
def test_sigmoid(dim_1, dim_2):
a1 = np.random.normal(loc=0., scale=20., size=(dim_1,dim_2))
a1_copy = a1.copy()
s_a1 = sigmoid(a1)
s_sol_a1 = sigmoid_sol(a1_copy)
assert rel_error(sigmoid_grad(s_a1), sigmoid_grad_sol(s_sol_a1)) <= 1e-10
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
# data : N * Dx
# W1 : Dx * H
# b1 : 1 * H
# W2 : H * Dy
# b2 : 1 * Dy
N = data.shape[0]
z1 = data.dot(W1) + b1
a1 = sigmoid(z1) # N * H
z2 = a1.dot(W2) + b2
a2 = softmax(z2) # N * Dy
cost = np.sum(-np.log(a2[labels == 1])) / N
### END YOUR CODE
### YOUR CODE HERE: backward propagation
delta_score = a2 - labels # 1 * Dy
delta_score /= N
gradW2 = np.dot(a1.T, delta_score) # H * 1 * 1 * Dy = H * Dy
gradb2 = np.sum(delta_score, axis=0)
grad_h = np.dot(delta_score, W2.T) # 1 * Dy * Dy * H = 1 * H
grad_h = sigmoid_grad(a1) * grad_h
gradW1 = np.dot(data.T, grad_h)
gradb1 = np.sum(grad_h, axis=0)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
#raise NotImplementedError
U = outputVectors
u_o = U[target]
v_c = predicted
N = U.shape[0]
sig_o_c = sigmoid(np.dot(u_o, v_c))
cost = -np.log(sig_o_c)
gradPred = (sig_o_c - 1)*u_o
grad = np.zeros((N, len(v_c)))
grad[target] += (sig_o_c - 1)*v_c
for k in indices:
u_k = U[k]
sig_k_c = sigmoid(-np.dot(u_k, v_c))
cost += -np.log(sig_k_c)
gradPred += -(sig_k_c - 1)*u_k
grad[k] += -(sig_k_c - 1)*v_c
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
### YOUR CODE HERE
W,D = outputVectors.shape
UK = np.zeros((K+1, D))
indices = [target]
for i in xrange(K):
k = dataset.sampleTokenIdx()
while k == target:
k = dataset.sampleTokenIdx()
indices.append(k)
for i,ix in enumerate(indices):
UK[i] = outputVectors[ix]
u_o = outputVectors[target] # (D,)
cost = - np.log(sigmoid(np.dot(u_o, predicted))) - np.sum(np.log(sigmoid(-np.dot(UK[1:], predicted))))
gradPred = (sigmoid(np.dot(u_o,predicted))-1) * u_o + np.dot(UK[1:].T,sigmoid(np.dot(UK[1:], predicted))) # dJ/dV_c, (D,)
y = np.zeros(K+1); y[0] = 1.0 #
grad = np.zeros(outputVectors.shape)
gradK = np.outer(sigmoid(np.dot(UK, predicted)) - y, predicted)
for i,ix in enumerate(indices):
grad[ix] += gradK[i]
### END YOUR CODE
return cost, gradPred, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs + Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
fc_out = np.dot(data, W1) + b1 # shape (M, H)
fc_sigmoid_out = sigmoid(fc_out) # shape (M, H)
scores = np.dot(fc_sigmoid_out, W2) + b2 # shape (M, Dy)
y_hat = softmax(scores) # shape (M, Dy)
# M = data.shape[0]
cost = -np.sum(labels * np.log(y_hat)) # / M
### END YOUR CODE
### YOUR CODE HERE: backward propagation
dscores = y_hat - labels # / M # shape (M, Dy)
gradW2 = np.dot(fc_sigmoid_out.T, dscores) # shape (H, Dy)
gradb2 = np.sum(dscores, axis=0) # shape (Dy,)
dfc_sigmoid_out = np.dot(dscores, W2.T) # shape (M, H)
dfc_out = dfc_sigmoid_out * sigmoid_grad(fc_sigmoid_out) # shape (M, H)
gradW1 = np.dot(data.T, dfc_out) # shape (Dx, H)
gradb1 = np.sum(dfc_out, axis=0) # shape (H,)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
"""
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
d = predicted.shape[0]
v = outputVectors.shape[0]
u_0 = outputVectors[target].reshape(1,d) # row
v_c = predicted.reshape(d,1) # column
predict = sigmoid(np.dot(u_0, v_c)) # reuse
cost = - np.log(predict)
grad = np.zeros((v,d))+0.
grad[target] = ((predict- 1.0) *(v_c.T)).reshape((d,))
gradPred = (predict - 1.0) *u_0
for k in indices[1:]:
u_k = outputVectors[k]
u_k = u_k.reshape((1,d)) # row
predict = sigmoid(np.dot(u_k, v_c))
cost -= np.log(1-predict)
gradPred += predict * u_k
grad[k] +=( predict * v_c.T).reshape((d,))
gradPred = gradPred.reshape((d,))
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models """
# Implement the cost and gradients for one predicted word vector
# and one target word vector as a building block for word2vec
# models, using the negative sampling technique. K is the sample
# size. You might want to use dataset.sampleTokenIdx() to sample
# a random word index.
#
# Note: See test_word2vec below for dataset's initialization.
#
# Input/Output Specifications: same as softmaxCostAndGradient
# We will not provide starter code for this function, but feel
# free to reference the code you previously wrote for this
# assignment!
cost = -np.log(sigmoid(np.dot(predicted,outputVectors[target])))
num_rand = 0
grad = np.zeros(outputVectors.shape)
gradPred = -outputVectors[target]*(1-sigmoid(np.dot(predicted,outputVectors[target])))
grad[target] = -predicted*(1-sigmoid(np.dot(predicted,outputVectors[target])))
while num_rand < K:
rand = dataset.sampleTokenIdx()
if rand == target:
continue
num_rand += 1
cost -= np.log(sigmoid(-np.dot(predicted,outputVectors[rand])))
grad[rand] += predicted*(1-sigmoid(-np.dot(predicted,outputVectors[rand])))
gradPred += outputVectors[rand]*(1-sigmoid(-np.dot(predicted,outputVectors[rand])))
return cost, gradPred, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### forward propagation
N = data.shape[0]
l1 = data.dot(W1) + b1
h = sigmoid(l1)
l2 = h.dot(W2) + b2
y_hat = softmax(l2)
cost = -np.sum(labels * np.log(y_hat)) / N # cross entropy
### backward propagation
dl2 = y_hat - labels
dW2 = np.dot(h.T, dl2)
db2 = np.sum(dl2, axis=0)
dh = np.dot(dl2, W2.T)
dl1 = dh * sigmoid_grad(h)
dW1 = np.dot(data.T, dl1)
db1 = np.sum(dl1, axis=0)
gradW2 = dW2/N
gradb2 = db2/N
gradW1 = dW1/N
gradb1 = db1/N
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
p1 = np.dot(data, W1) + b1
h = sigmoid(p1) #(M,H)
p2 = np.dot(h, W2) + b2
y_pred = softmax(p2) #(M, Dy)
cost = np.mean(np.sum(-1 * np.multiply(labels, np.log(y_pred)), axis=1))
### END YOUR CODE
### YOUR CODE HERE: backward propagation
gradp2 = (y_pred - labels) / np.shape(data)[0] #(M,Dy)
gradW2 = np.dot(h.T, gradp2) #(H, Dy)
gradb2 = np.sum(gradp2, axis=0).reshape((1,-1)) #(1, Dy)
gradh = np.dot(gradp2, W2.T) #(M,H)
gradp1 = np.multiply(gradh, h * (1 - h)) #(M,H) element wise multiplication
gradW1 = np.dot(data.T, gradp1) # (Dx,H)
gradb1 = np.sum(gradp1, axis=0).reshape((1,-1)) #(1,H)
### END YOUR CODE
### Stack gradients (do not modify)[0].reshape((1,-1)) #(1, Dy)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
Arguments:
data -- M x Dx matrix, where each row is a training example.
labels -- M x Dy matrix, where each row is a one-hot vector.
params -- Model parameters, these are unpacked for you.
dimensions -- A tuple of input dimension, number of hidden units
and output dimension
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
h = sigmoid(np.dot(data,W1) + b1)
yhat = softmax(np.dot(h,W2) + b2)
### END YOUR CODE
### YOUR CODE HERE: backward propagation
cost = np.sum(-np.log(yhat[labels==1])) / data.shape[0]
d3 = (yhat - labels) / data.shape[0]
gradW2 = np.dot(h.T, d3)
gradb2 = np.sum(d3,0,keepdims=True)
dh = np.dot(d3,W2.T)
grad_h = sigmoid_grad(h) * dh
gradW1 = np.dot(data.T,grad_h)
gradb1 = np.sum(grad_h,0)
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
cost = 0.
gradPred = np.zeros(predicted.shape) # shape (N,)
grad = np.zeros(outputVectors.shape) # shape (W, N)
vc = predicted # shape (N,)
uo = outputVectors[target] # shape (N,)
yo = sigmoid(np.dot(uo, vc)) # scaler
cost += -np.log(yo)
gradPred += (yo - 1) * uo # shape (N,)
grad[target] += (yo - 1) * vc
for k in indices[1:]:
uk = outputVectors[k]
y_neg_k = sigmoid(-np.dot(uk, vc))
cost += -np.log(y_neg_k)
gradPred += -(y_neg_k - 1) * uk # shape (N,)
grad[k] += -(y_neg_k - 1) * vc
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
grad = np.zeros(outputVectors.shape)
gradPred = np.zeros(predicted.shape)
cost = 0
z = sigmoid(np.dot(outputVectors[target], predicted))
cost -= np.log(z)
grad[target] += predicted * (z - 1.0)
gradPred += outputVectors[target] * (z - 1.0)
for k in xrange(K):
samp = indices[k + 1]
z = sigmoid(np.dot(outputVectors[samp], predicted))
cost -= np.log(1.0 - z)
grad[samp] += predicted * z
gradPred += outputVectors[samp] * z
### END YOUR CODE
return cost, gradPred, grad
def negSamplingCostAndGradient(predicted, target, outputVectors, dataset,
K=10):
""" Negative sampling cost function for word2vec models
Implement the cost and gradients for one predicted word vector
and one target word vector as a building block for word2vec
models, using the negative sampling technique. K is the sample
size.
Note: See test_word2vec below for dataset's initialization.
Arguments/Return Specifications: same as softmaxCostAndGradient
"""
# Sampling of indices is done for you. Do not modify this if you
# wish to match the autograder and receive points!
indices = [target]
indices.extend(getNegativeSamples(target, dataset, K))
### YOUR CODE HERE
gradPred = np.zeros_like(predicted)
grad = np.zeros_like(outputVectors)
cost = 0.0
product = np.dot(outputVectors[target], predicted) #intermediate value
cost = -np.log(sigmoid(product))
gradPred = (sigmoid(product) - 1) * outputVectors[target]
grad[target] = (sigmoid(product) - 1) * predicted
for index in indices:
neg_sig = sigmoid (-1 * np.dot(outputVectors[index], predicted))
cost += -np.log(neg_sig)
gradPred += -(neg_sig -1) * outputVectors[index]
grad[index] += -(neg_sig - 1) * predicted
### END YOUR CODE
return cost, gradPred, grad
def forward_backward_prop(data, labels, params, dimensions):
"""
Forward and backward propagation for a two-layer sigmoidal network
Compute the forward propagation and for the cross entropy cost,
and backward propagation for the gradients for all parameters.
"""
### Unpack network parameters (do not modify)
ofs = 0
Dx, H, Dy = (dimensions[0], dimensions[1], dimensions[2])
W1 = np.reshape(params[ofs:ofs+ Dx * H], (Dx, H))
ofs += Dx * H
b1 = np.reshape(params[ofs:ofs + H], (1, H))
ofs += H
W2 = np.reshape(params[ofs:ofs + H * Dy], (H, Dy))
ofs += H * Dy
b2 = np.reshape(params[ofs:ofs + Dy], (1, Dy))
### YOUR CODE HERE: forward propagation
N = data.shape[0]
Z1 = data.dot(W1) + b1 # (N, H)
A1 = sigmoid(Z1) # (N, H)
scores = A1.dot(W2) + b2 # (N, Dy)
probs = softmax(scores) # (N, Dy)
cost = -np.sum(np.log(probs[labels==1])) / N
### END YOUR CODE
### YOUR CODE HERE: backward propagation
dscores = (probs - labels) / N
dW2 = A1.T.dot(dscores)
db2 = np.sum(dscores, axis=0)
dA1 = dscores.dot(W2.T)
dZ1 = sigmoid_grad(A1) * dA1
dW1 = data.T.dot(dZ1)
db1 = np.sum(dZ1, axis=0)
gradW1 = dW1
gradW2 = dW2
gradb1 = db1
gradb2 = db2
### END YOUR CODE
### Stack gradients (do not modify)
grad = np.concatenate((gradW1.flatten(), gradb1.flatten(),
gradW2.flatten(), gradb2.flatten()))
return cost, grad
