I have tensor t with shape (Batch_Size x Dims) and another tensor v with shape (Vocab_Size x Dims). I'd like to produce a tensor d with shape (Batch_Size x Vocab_Size), such that d[i,j] = norm(t[i] - v[j]).
Doing this for a single tensor (no batches) is trivial: d = torch.norm(v - t), since t would be broadcast. How can I do this when the tensors have batches?
Insert unitary dimensions into v and t to make them (1 x Vocab_Size x Dims) and (Batch_Size x 1 x Dims) respectively. Next, take the broadcasted difference to get a tensor of shape (Batch_Size x Vocab_Size x Dims). Pass that to torch.norm along with the optional dim=2 argument so that the norm is taken along the last dimension. This will result in the desired (Batch_Size x Vocab_Size) tensor of norms.
d = torch.norm(v.unsqueeze(0) - t.unsqueeze(1), dim=2)
Edit: As pointed out by #KonstantinosKokos in the comments, due to the broadcasting rules used by numpy and pytorch, the leading unitary dimension on v does not need to be explicit. I.e. you can use
d = torch.norm(v - t.unsqueeze(1), dim=2)
I have a list of embeddings. The list has N lists with M embedding (tensors) each.
list_embd = [[M embeddings], [M embeddings], ...]
(Each embedding is a tensor with size (1,512))
What I want to do is create a tensor size (N, M), where each "cell" is one embedding.
Tried this for numpy array.
array = np.zeros(n,m)
for i in range(n):
for j in range(m):
array[i, j] = list_embd[i][j]
But still got errors.
In pytorch tried to concat all M embeddings into one tensor size (1, M), and then concat all rows. But when I concat along dim 1 two of those M embeddings, I get a tensor shaped (1, 1028) instead (1, 2).
final = torch.tensor([])
for i in range(n):
interm = torch.tensor([])
for j in range(m):
interm = torch.cat((interm, list_embd[i][j]), 0)
final = = torch.cat((final, interm), 1)
Any ideas or suggestions?
I need a matrix with the embeddings in each cell.
You can use torch.cat and torch.stack to create a final 3D tensor of shape (N, M, 512):
final = torch.stack([torch.cat(sub_list, dim=0) for sub_list in list_embd], dim=0)
First, you use torch.cat to create a list of N 2D tensors of shape (M, 512) from each list of M embeddings. Then torch.stack is used to stack these N 2D matrices into a single 3D tensor final.
Using PyTorch, I would like to calculate the Hessian vector product, where the Hessian is the second-derivative matrix of the loss function of some neural net, and the vector will be the vector of gradients of that loss function.
I know how to calculate the Hessian vector product for a regular function thanks to this post. However, I am running into trouble when the function is the loss function of a neural network. This is because the parameters are packaged into a module, accessible via nn.parameters(), and not a torch tensor.
I want to do something like this (doesn't work):
### a simple neural network
linear = nn.Linear(10, 20)
x = torch.randn(1, 10)
y = linear(x).sum()
### compute the gradient and make a copy that is detached from the graph
grad = torch.autograd.grad(y, linear.parameters(),create_graph=True)
v = grad.clone().detach()
### compute the Hessian vector product
z = grad # v
z.backward()
In analogy this this (does work):
x = Variable(torch.Tensor([1, 1]), requires_grad=True)
f = 3*x[0]**2 + 4*x[0]*x[1] + x[1]**2
grad, = torch.autograd.grad(f, x, create_graph=True)
v = grad.clone().detach()
z = grad # v
z.backward()
This post addresses a similar (possibly the same?) issue, but I don't understand the solution.
You are saying it doesn't work but do not show what error you get, this is why you haven't got any answers
torch.autograd.grad(outputs, inputs, grad_outputs=None, retain_graph=None, create_graph=False, only_inputs=True, allow_unused=False)
outputs and inputs are expected to be sequences of tensors. But you
use just a tensor as outputs.
What this is saying is that you should pass a sequence, so pass [y] instead of y
I'm working on a project where I have to predict the future states of a 1D vector with y entries. I'm trying to do this using an ANN setup with LSTM units in combination with a convolution layer. The method I'm using is based on the method they used in a (pre-release paper). The suggested setup is as follows:
In the picture c is the 1D vector with y entries. The ANN gets the n previous states as an input and produces o next states as an output.
Currently, my ANN setup looks like this:
inputLayer = Input(shape = (n, y))
encoder = LSTM(200)(inputLayer)
x = RepeatVector(1)(encoder)
decoder = LSTM(200, return_sequences=True)(x)
x = Conv1D(y, 4, activation = 'linear', padding = 'same')(decoder)
model = Model(inputLayer, x)
Here n is the length of the input sequences and y is the length of the state array. As can be seen I'm repeating the d vector only 1 time, as I'm trying to predict only 1 time step in the future. Is this the way to setup the above mentioned network?
Furthermore, I have a numpy array (data) with a shape of (Sequences, Time Steps, State Variables) to train with. I was trying to divide this in randomly selected batches with a generator like this:
def BatchGenerator(batch_size, n, y, data):
# Infinite loop.
while True:
# Allocate a new array for the batch of input-signals.
x_shape = (batch_size, n, y)
x_batch = np.zeros(shape=x_shape, dtype=np.float16)
# Allocate a new array for the batch of output-signals.
y_shape = (batch_size, 1, y)
y_batch = np.zeros(shape=y_shape, dtype=np.float16)
# Fill the batch with random sequences of data.
for i in range(batch_size):
# Select a random sequence
seq_idx = np.random.randint(data.shape[0])
# Get a random start-index.
# This points somewhere into the training-data.
start_idx = np.random.randint(data.shape[1] - n)
# Copy the sequences of data starting at this
# Each batch inside x_batch has a shape of [n, y]
x_batch[i,:,:] = data[seq_idx, start_idx:start_idx+n, :]
# Each batch inside y_batch has a shape of [1, y] (as we predict only 1 time step in advance)
y_batch[i,:,:] = data[seq_idx, start_idx+n, :]
yield (x_batch, y_batch)
The problem is that it gives an error if I'm using a batch_size of more than 1. Could anyone help me to set this data up in a way that it can be used optimally to train my neural network?
The model is now trained using:
generator = BatchGenerator(batch_size, n, y, data)
model.fit_generator(generator = generator, steps_per_epoch = steps_per_epoch, epochs = epochs)
Thanks in advance!
So, I'm using Michael Nielson's machine learning book as a reference for my code (it is basically identical): http://neuralnetworksanddeeplearning.com/chap1.html
The code in question:
def backpropagate(self, image, image_value) :
# declare two new numpy arrays for the updated weights & biases
new_biases = [np.zeros(bias.shape) for bias in self.biases]
new_weights = [np.zeros(weight_matrix.shape) for weight_matrix in self.weights]
# -------- feed forward --------
# store all the activations in a list
activations = [image]
# declare empty list that will contain all the z vectors
zs = []
for bias, weight in zip(self.biases, self.weights) :
print(bias.shape)
print(weight.shape)
print(image.shape)
z = np.dot(weight, image) + bias
zs.append(z)
activation = sigmoid(z)
activations.append(activation)
# -------- backward pass --------
# transpose() returns the numpy array with the rows as columns and columns as rows
delta = self.cost_derivative(activations[-1], image_value) * sigmoid_prime(zs[-1])
new_biases[-1] = delta
new_weights[-1] = np.dot(delta, activations[-2].transpose())
# l = 1 means the last layer of neurons, l = 2 is the second-last, etc.
# this takes advantage of Python's ability to use negative indices in lists
for l in range(2, self.num_layers) :
z = zs[-1]
sp = sigmoid_prime(z)
delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
new_biases[-l] = delta
new_weights[-l] = np.dot(delta, activations[-l-1].transpose())
return (new_biases, new_weights)
My algorithm can only get to the first round backpropagation before this error occurs:
File "D:/Programming/Python/DPUDS/DPUDS_Projects/Fall_2017/MNIST/network.py", line 97, in stochastic_gradient_descent
self.update_mini_batch(mini_batch, learning_rate)
File "D:/Programming/Python/DPUDS/DPUDS_Projects/Fall_2017/MNIST/network.py", line 117, in update_mini_batch
delta_biases, delta_weights = self.backpropagate(image, image_value)
File "D:/Programming/Python/DPUDS/DPUDS_Projects/Fall_2017/MNIST/network.py", line 160, in backpropagate
z = np.dot(weight, activation) + bias
ValueError: shapes (30,50000) and (784,1) not aligned: 50000 (dim 1) != 784 (dim 0)
I get why it's an error. The number of columns in weights doesn't match the number of rows in the pixel image, so I can't do matrix multiplication. Here's where I'm confused -- there are 30 neurons used in the backpropagation, each with 50,000 images being evaluated. My understanding is that each of the 50,000 should have 784 weights attached, one for each pixel. But when I modify the code accordingly:
count = 0
for bias, weight in zip(self.biases, self.weights) :
print(bias.shape)
print(weight[count].shape)
print(image.shape)
z = np.dot(weight[count], image) + bias
zs.append(z)
activation = sigmoid(z)
activations.append(activation)
count += 1
I still get a similar error:
ValueError: shapes (50000,) and (784,1) not aligned: 50000 (dim 0) != 784 (dim 0)
I'm just really confuzzled by all the linear algebra involved and I think I'm just missing something about the structure of the weight matrix. Any help at all would be greatly appreciated.
It looks like the issue is in your changes to the original code.
I’be downloaded example from the link you provided and it works without any errors:
Here is full source code I used:
import cPickle
import gzip
import numpy as np
import random
def load_data():
"""Return the MNIST data as a tuple containing the training data,
the validation data, and the test data.
The ``training_data`` is returned as a tuple with two entries.
The first entry contains the actual training images. This is a
numpy ndarray with 50,000 entries. Each entry is, in turn, a
numpy ndarray with 784 values, representing the 28 * 28 = 784
pixels in a single MNIST image.
The second entry in the ``training_data`` tuple is a numpy ndarray
containing 50,000 entries. Those entries are just the digit
values (0...9) for the corresponding images contained in the first
entry of the tuple.
The ``validation_data`` and ``test_data`` are similar, except
each contains only 10,000 images.
This is a nice data format, but for use in neural networks it's
helpful to modify the format of the ``training_data`` a little.
That's done in the wrapper function ``load_data_wrapper()``, see
below.
"""
f = gzip.open('../data/mnist.pkl.gz', 'rb')
training_data, validation_data, test_data = cPickle.load(f)
f.close()
return (training_data, validation_data, test_data)
def load_data_wrapper():
"""Return a tuple containing ``(training_data, validation_data,
test_data)``. Based on ``load_data``, but the format is more
convenient for use in our implementation of neural networks.
In particular, ``training_data`` is a list containing 50,000
2-tuples ``(x, y)``. ``x`` is a 784-dimensional numpy.ndarray
containing the input image. ``y`` is a 10-dimensional
numpy.ndarray representing the unit vector corresponding to the
correct digit for ``x``.
``validation_data`` and ``test_data`` are lists containing 10,000
2-tuples ``(x, y)``. In each case, ``x`` is a 784-dimensional
numpy.ndarry containing the input image, and ``y`` is the
corresponding classification, i.e., the digit values (integers)
corresponding to ``x``.
Obviously, this means we're using slightly different formats for
the training data and the validation / test data. These formats
turn out to be the most convenient for use in our neural network
code."""
tr_d, va_d, te_d = load_data()
training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]
training_results = [vectorized_result(y) for y in tr_d[1]]
training_data = zip(training_inputs, training_results)
validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]
validation_data = zip(validation_inputs, va_d[1])
test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]
test_data = zip(test_inputs, te_d[1])
return (training_data, validation_data, test_data)
def vectorized_result(j):
"""Return a 10-dimensional unit vector with a 1.0 in the jth
position and zeroes elsewhere. This is used to convert a digit
(0...9) into a corresponding desired output from the neural
network."""
e = np.zeros((10, 1))
e[j] = 1.0
return e
class Network(object):
def __init__(self, sizes):
"""The list ``sizes`` contains the number of neurons in the
respective layers of the network. For example, if the list
was [2, 3, 1] then it would be a three-layer network, with the
first layer containing 2 neurons, the second layer 3 neurons,
and the third layer 1 neuron. The biases and weights for the
network are initialized randomly, using a Gaussian
distribution with mean 0, and variance 1. Note that the first
layer is assumed to be an input layer, and by convention we
won't set any biases for those neurons, since biases are only
ever used in computing the outputs from later layers."""
self.num_layers = len(sizes)
self.sizes = sizes
self.biases = [np.random.randn(y, 1) for y in sizes[1:]]
self.weights = [np.random.randn(y, x)
for x, y in zip(sizes[:-1], sizes[1:])]
def feedforward(self, a):
"""Return the output of the network if ``a`` is input."""
for b, w in zip(self.biases, self.weights):
a = sigmoid(np.dot(w, a)+b)
return a
def SGD(self, training_data, epochs, mini_batch_size, eta,
test_data=None):
"""Train the neural network using mini-batch stochastic
gradient descent. The ``training_data`` is a list of tuples
``(x, y)`` representing the training inputs and the desired
outputs. The other non-optional parameters are
self-explanatory. If ``test_data`` is provided then the
network will be evaluated against the test data after each
epoch, and partial progress printed out. This is useful for
tracking progress, but slows things down substantially."""
if test_data: n_test = len(test_data)
n = len(training_data)
for j in xrange(epochs):
random.shuffle(training_data)
mini_batches = [
training_data[k:k+mini_batch_size]
for k in xrange(0, n, mini_batch_size)]
for mini_batch in mini_batches:
self.update_mini_batch(mini_batch, eta)
if test_data:
print "Epoch {0}: {1} / {2}".format(
j, self.evaluate(test_data), n_test)
else:
print "Epoch {0} complete".format(j)
def update_mini_batch(self, mini_batch, eta):
"""Update the network's weights and biases by applying
gradient descent using backpropagation to a single mini batch.
The ``mini_batch`` is a list of tuples ``(x, y)``, and ``eta``
is the learning rate."""
nabla_b = [np.zeros(b.shape) for b in self.biases]
nabla_w = [np.zeros(w.shape) for w in self.weights]
for x, y in mini_batch:
delta_nabla_b, delta_nabla_w = self.backprop(x, y)
nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]
self.weights = [w-(eta/len(mini_batch))*nw
for w, nw in zip(self.weights, nabla_w)]
self.biases = [b-(eta/len(mini_batch))*nb
for b, nb in zip(self.biases, nabla_b)]
def backprop(self, x, y):
"""Return a tuple ``(nabla_b, nabla_w)`` representing the
gradient for the cost function C_x. ``nabla_b`` and
``nabla_w`` are layer-by-layer lists of numpy arrays, similar
to ``self.biases`` and ``self.weights``."""
nabla_b = [np.zeros(b.shape) for b in self.biases]
nabla_w = [np.zeros(w.shape) for w in self.weights]
# feedforward
activation = x
activations = [x] # list to store all the activations, layer by layer
zs = [] # list to store all the z vectors, layer by layer
for b, w in zip(self.biases, self.weights):
z = np.dot(w, activation)+b
zs.append(z)
activation = sigmoid(z)
activations.append(activation)
# backward pass
delta = self.cost_derivative(activations[-1], y) * \
sigmoid_prime(zs[-1])
nabla_b[-1] = delta
nabla_w[-1] = np.dot(delta, activations[-2].transpose())
# Note that the variable l in the loop below is used a little
# differently to the notation in Chapter 2 of the book. Here,
# l = 1 means the last layer of neurons, l = 2 is the
# second-last layer, and so on. It's a renumbering of the
# scheme in the book, used here to take advantage of the fact
# that Python can use negative indices in lists.
for l in xrange(2, self.num_layers):
z = zs[-l]
sp = sigmoid_prime(z)
delta = np.dot(self.weights[-l+1].transpose(), delta) * sp
nabla_b[-l] = delta
nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())
return (nabla_b, nabla_w)
def evaluate(self, test_data):
"""Return the number of test inputs for which the neural
network outputs the correct result. Note that the neural
network's output is assumed to be the index of whichever
neuron in the final layer has the highest activation."""
test_results = [(np.argmax(self.feedforward(x)), y)
for (x, y) in test_data]
return sum(int(x == y) for (x, y) in test_results)
def cost_derivative(self, output_activations, y):
"""Return the vector of partial derivatives \partial C_x /
\partial a for the output activations."""
return (output_activations-y)
#### Miscellaneous functions
def sigmoid(z):
"""The sigmoid function."""
return 1.0/(1.0+np.exp(-z))
def sigmoid_prime(z):
"""Derivative of the sigmoid function."""
return sigmoid(z)*(1-sigmoid(z))
training_data, validation_data, test_data = load_data_wrapper()
net = Network([784, 30, 10])
net.SGD(training_data, 30, 10, 3.0, test_data=test_data)
Additional info:
However, I would recommend using one of existing frameworks, for example - Keras to don't reinvent the wheel
Also, it was checked with python 3.6:
Kudos on digging into Nielsen's code. It's a great resource to develop thorough understanding of NN principles. Too many people leap ahead to Keras without knowing what goes on under the hood.
Each training example doesn't get its own weights. Each of the 784 features does. If each example got its own weights then each weight set would overfit to its corresponding training example. Also, if you later used your trained network to run inference on a single test example, what would it do with 50,000 sets of weights when presented with just one handwritten digit? Instead, each of the 30 neurons in your hidden layer learns a set of 784 weights, one for each pixel, that offers high predictive accuracy when generalized to any handwritten digit.
Import network.py and instantiate a Network class like this without modifying any code:
net = network.Network([784, 30, 10])
..which gives you a network with 784 input neurons, 30 hidden neurons and 10 output neurons. Your weight matrices will have dimensions [30, 784] and [10, 30], respectively. When you feed the network an input array of dimensions [784, 1] the matrix multiplication that gave you an error is valid because dim 1 of the weight matrix equals dim 0 of the input array (both 784).
Your problem is not implementation of backprop but rather setting up a network architecture appropriate for the shape of your input data. If memory serves Nielsen leaves backprop as a black box in chapter 1 and doesn't dive into it until chapter 2. Keep at it, and good luck!