While trying to implement neural network training algorithms, I came across different concepts, including that of Gradient descent which tries to mimic a ball rolling down a hill, and of velocity and momentum to better model the rolling ball.
I initialized my weights, weight_deltas, and weight_velocities thus:
sizes = [2, 3, 1]
momentum_coefficient = 0.5
weights = [ 2 * np.random.random((a, b)) - 1 for a, b in zip(sizes[:-1], sizes[1:]) ]
weight_velocities = [ np.ones(w.shape) for w in weights ]
weight_deltas = [ np.zeros(w.shape) for w in weights ]
After calculating the deltas (derivative of the cost function with respect to the weights, I updated the weights thus:
for l in xrange(sizes - 1):
weight_velocities[l] = (momentum_factor * weight_velocities[l]) - weight_deltas[l]
weights[l] += weight_velocities[l]
I used np.zeros to initialise my velocities, and I was able to get up to 80% accuracy (for a particular dataset). But when I initialised with np.ones, I could not get up to 20% accuracy. I've been using ones, but I can't figure out why zeros does work. And there's also the random method from numpy.
What's the recommended approach to initialise the weight_velocities?
Notice that I intentionally excluded the biases units, and the learning rate, and I'm importing numpy as np.
Related
I am new babie to the Deep Learning field, and I am use log-likelihood method to compare the MSE metrics.Could anyone be able to show how to calculate the following 2 predicted output examples with 3 outputs neurons each. Thanks
yt = [ [1,0,0],[0,0,1]]
yp = [ [0.9, 0.2,0.2], [0.2,0.8,0.3] ]
MSE or Mean Squared Error is simply the expected value of the squared difference between the predicted and the ground truth labels, represented as
\text{MSE}(\hat{\theta}) = E\left[(\hat{\theta} - \theta)^2\right]
where theta is the ground truth labels and theta^hat is the predicted labels
I am not sure what are you referring to exactly, like a theoretical question or a part of code
As a Python implementation
def mean_squared_error(A, B):
return np.square(np.subtract(A,B)).mean()
yt = [[1,0,0],[0,0,1]]
yp = [[0.9, 0.2,0.2], [0.2,0.8,0.3]]
mse = mean_squared_error(yt, yp)
print(mse)
This will give a value of 0.21
If you are using one of the DL frameworks like TensorFlow, then they are already providing the function which calculates the mse loss between tensors
tf.losses.mean_squared_error
where
tf.losses.mean_squared_error(
labels,
predictions,
weights=1.0,
scope=None,
loss_collection=tf.GraphKeys.LOSSES,
reduction=Reduction.SUM_BY_NONZERO_WEIGHTS
)
Args:
labels: The ground truth output tensor, same dimensions as 'predictions'.
predictions: The predicted outputs.
weights: Optional Tensor whose rank is either 0, or the same rank as labels, and must be broadcastable to labels (i.e., all dimensions
must be either 1, or the same as the corresponding losses dimension).
scope: The scope for the operations performed in computing the loss.
loss_collection: collection to which the loss will be added.
reduction: Type of reduction to apply to loss.
Returns:
Weighted loss float Tensor. If reduction is NONE, this has the same
shape as labels; otherwise, it is scalar.
I just implemented the generalised dice loss (multi-class version of dice loss) in keras, as described in ref :
(my targets are defined as: (batch_size, image_dim1, image_dim2, image_dim3, nb_of_classes))
def generalized_dice_loss_w(y_true, y_pred):
# Compute weights: "the contribution of each label is corrected by the inverse of its volume"
Ncl = y_pred.shape[-1]
w = np.zeros((Ncl,))
for l in range(0,Ncl): w[l] = np.sum( np.asarray(y_true[:,:,:,:,l]==1,np.int8) )
w = 1/(w**2+0.00001)
# Compute gen dice coef:
numerator = y_true*y_pred
numerator = w*K.sum(numerator,(0,1,2,3))
numerator = K.sum(numerator)
denominator = y_true+y_pred
denominator = w*K.sum(denominator,(0,1,2,3))
denominator = K.sum(denominator)
gen_dice_coef = numerator/denominator
return 1-2*gen_dice_coef
But something must be wrong. I'm working with 3D images that I have to segment for 4 classes (1 background class and 3 object classes, I have a imbalanced dataset). First odd thing: while my train loss and accuracy improve during training (and converge really fast), my validation loss/accuracy are constant trough epochs (see image). Second, when predicting on test data, only the background class is predicted: I get a constant volume.
I used the exact same data and script but with categorical cross-entropy loss and get plausible results (object classes are segmented). Which means something is wrong with my implementation. Any idea what it could be?
Plus I believe it would be usefull to the keras community to have a generalised dice loss implementation, as it seems to be used in most of recent semantic segmentation tasks (at least in the medical image community).
PS: it seems odd to me how the weights are defined; I get values around 10^-10. Anyone else has tried to implement this? I also tested my function without the weights but get same problems.
I think the problem here are your weights. Imagine you are trying to solve a multiclass segmentation problem, but in each image only a few classes are ever present. A toy example of this (and the one which led me to this problem) is to create a segmentation dataset from mnist in the following way.
x = 28x28 image and y = 28x28x11 where each pixel is classified as background if it is below a normalised grayscale value of 0.4, and otherwise is classified as the digit which is the original class of x. So if you see a picture of the number one, you will have a bunch of pixels classified as one, and the background.
Now in this dataset you will only ever have two classes present in the image. This means that, following your dice loss, 9 of the weights will be
1./(0. + eps) = large
and so for every image we are strongly penalising all 9 non-present classes. An evidently strong local minima the network wants to find in this situation is to predict everything as a background class.
We do want to penalise any incorrectly predicted classes which are not in the image, but not so strongly. So we just need to modify the weights. This is how I did it:
def gen_dice(y_true, y_pred, eps=1e-6):
"""both tensors are [b, h, w, classes] and y_pred is in logit form"""
# [b, h, w, classes]
pred_tensor = tf.nn.softmax(y_pred)
y_true_shape = tf.shape(y_true)
# [b, h*w, classes]
y_true = tf.reshape(y_true, [-1, y_true_shape[1]*y_true_shape[2], y_true_shape[3]])
y_pred = tf.reshape(pred_tensor, [-1, y_true_shape[1]*y_true_shape[2], y_true_shape[3]])
# [b, classes]
# count how many of each class are present in
# each image, if there are zero, then assign
# them a fixed weight of eps
counts = tf.reduce_sum(y_true, axis=1)
weights = 1. / (counts ** 2)
weights = tf.where(tf.math.is_finite(weights), weights, eps)
multed = tf.reduce_sum(y_true * y_pred, axis=1)
summed = tf.reduce_sum(y_true + y_pred, axis=1)
# [b]
numerators = tf.reduce_sum(weights*multed, axis=-1)
denom = tf.reduce_sum(weights*summed, axis=-1)
dices = 1. - 2. * numerators / denom
dices = tf.where(tf.math.is_finite(dices), dices, tf.zeros_like(dices))
return tf.reduce_mean(dices)
When we train neural networks, we typically use gradient descent, which relies on a continuous, differentiable real-valued cost function. The final cost function might, for example, take the mean squared error. Or put another way, gradient descent implicitly assumes the end goal is regression - to minimize a real-valued error measure.
Sometimes what we want a neural network to do is perform classification - given an input, classify it into two or more discrete categories. In this case, the end goal the user cares about is classification accuracy - the percentage of cases classified correctly.
But when we are using a neural network for classification, though our goal is classification accuracy, that is not what the neural network is trying to optimize. The neural network is still trying to optimize the real-valued cost function. Sometimes these point in the same direction, but sometimes they don't. In particular, I've been running into cases where a neural network trained to correctly minimize the cost function, has a classification accuracy worse than a simple hand-coded threshold comparison.
I've boiled this down to a minimal test case using TensorFlow. It sets up a perceptron (neural network with no hidden layers), trains it on an absolutely minimal dataset (one input variable, one binary output variable) assesses the classification accuracy of the result, then compares it to the classification accuracy of a simple hand-coded threshold comparison; the results are 60% and 80% respectively. Intuitively, this is because a single outlier with a large input value, generates a correspondingly large output value, so the way to minimize the cost function is to try extra hard to accommodate that one case, in the process misclassifying two more ordinary cases. The perceptron is correctly doing what it was told to do; it's just that this does not match what we actually want of a classifier. But the classification accuracy is not a continuous differentiable function, so we can't use it as the target for gradient descent.
How can we train a neural network so that it ends up maximizing classification accuracy?
import numpy as np
import tensorflow as tf
sess = tf.InteractiveSession()
tf.set_random_seed(1)
# Parameters
epochs = 10000
learning_rate = 0.01
# Data
train_X = [
[0],
[0],
[2],
[2],
[9],
]
train_Y = [
0,
0,
1,
1,
0,
]
rows = np.shape(train_X)[0]
cols = np.shape(train_X)[1]
# Inputs and outputs
X = tf.placeholder(tf.float32)
Y = tf.placeholder(tf.float32)
# Weights
W = tf.Variable(tf.random_normal([cols]))
b = tf.Variable(tf.random_normal([]))
# Model
pred = tf.tensordot(X, W, 1) + b
cost = tf.reduce_sum((pred-Y)**2/rows)
optimizer = tf.train.GradientDescentOptimizer(learning_rate).minimize(cost)
tf.global_variables_initializer().run()
# Train
for epoch in range(epochs):
# Print update at successive doublings of time
if epoch&(epoch-1) == 0 or epoch == epochs-1:
print('{} {} {} {}'.format(
epoch,
cost.eval({X: train_X, Y: train_Y}),
W.eval(),
b.eval(),
))
optimizer.run({X: train_X, Y: train_Y})
# Classification accuracy of perceptron
classifications = [pred.eval({X: x}) > 0.5 for x in train_X]
correct = sum([p == y for (p, y) in zip(classifications, train_Y)])
print('{}/{} = perceptron accuracy'.format(correct, rows))
# Classification accuracy of hand-coded threshold comparison
classifications = [x[0] > 1.0 for x in train_X]
correct = sum([p == y for (p, y) in zip(classifications, train_Y)])
print('{}/{} = threshold accuracy'.format(correct, rows))
How can we train a neural network so that it ends up maximizing classification accuracy?
I'm asking for a way to get a continuous proxy function that's closer to the accuracy
To start with, the loss function used today for classification tasks in (deep) neural nets was not invented with them, but it goes back several decades, and it actually comes from the early days of logistic regression. Here is the equation for the simple case of binary classification:
The idea behind it was exactly to come up with a continuous & differentiable function, so that we would be able to exploit the (vast, and still expanding) arsenal of convex optimization for classification problems.
It is safe to say that the above loss function is the best we have so far, given the desired mathematical constraints mentioned above.
Should we consider this problem (i.e. better approximating the accuracy) solved and finished? At least in principle, no. I am old enough to remember an era when the only activation functions practically available were tanh and sigmoid; then came ReLU and gave a real boost to the field. Similarly, someone may eventually come up with a better loss function, but arguably this is going to happen in a research paper, and not as an answer to a SO question...
That said, the very fact that the current loss function comes from very elementary considerations of probability and information theory (fields that, in sharp contrast with the current field of deep learning, stand upon firm theoretical foundations) creates at least some doubt as to if a better proposal for the loss may be just around the corner.
There is another subtle point on the relation between loss and accuracy, which makes the latter something qualitatively different than the former, and is frequently lost in such discussions. Let me elaborate a little...
All the classifiers related to this discussion (i.e. neural nets, logistic regression etc) are probabilistic ones; that is, they do not return hard class memberships (0/1) but class probabilities (continuous real numbers in [0, 1]).
Limiting the discussion for simplicity to the binary case, when converting a class probability to a (hard) class membership, we are implicitly involving a threshold, usually equal to 0.5, such as if p[i] > 0.5, then class[i] = "1". Now, we can find many cases whet this naive default choice of threshold will not work (heavily imbalanced datasets are the first to come to mind), and we'll have to choose a different one. But the important point for our discussion here is that this threshold selection, while being of central importance to the accuracy, is completely external to the mathematical optimization problem of minimizing the loss, and serves as a further "insulation layer" between them, compromising the simplistic view that loss is just a proxy for accuracy (it is not). As nicely put in the answer of this Cross Validated thread:
the statistical component of your exercise ends when you output a probability for each class of your new sample. Choosing a threshold beyond which you classify a new observation as 1 vs. 0 is not part of the statistics any more. It is part of the decision component.
Enlarging somewhat an already broad discussion: Can we possibly move completely away from the (very) limiting constraint of mathematical optimization of continuous & differentiable functions? In other words, can we do away with back-propagation and gradient descend?
Well, we are actually doing so already, at least in the sub-field of reinforcement learning: 2017 was the year when new research from OpenAI on something called Evolution Strategies made headlines. And as an extra bonus, here is an ultra-fresh (Dec 2017) paper by Uber on the subject, again generating much enthusiasm in the community.
I think you are forgetting to pass your output through a simgoid. Fixed below:
import numpy as np
import tensorflow as tf
sess = tf.InteractiveSession()
tf.set_random_seed(1)
# Parameters
epochs = 10000
learning_rate = 0.01
# Data
train_X = [
[0],
[0],
[2],
[2],
[9],
]
train_Y = [
0,
0,
1,
1,
0,
]
rows = np.shape(train_X)[0]
cols = np.shape(train_X)[1]
# Inputs and outputs
X = tf.placeholder(tf.float32)
Y = tf.placeholder(tf.float32)
# Weights
W = tf.Variable(tf.random_normal([cols]))
b = tf.Variable(tf.random_normal([]))
# Model
# CHANGE HERE: Remember, you need an activation function!
pred = tf.nn.sigmoid(tf.tensordot(X, W, 1) + b)
cost = tf.reduce_sum((pred-Y)**2/rows)
optimizer = tf.train.GradientDescentOptimizer(learning_rate).minimize(cost)
tf.global_variables_initializer().run()
# Train
for epoch in range(epochs):
# Print update at successive doublings of time
if epoch&(epoch-1) == 0 or epoch == epochs-1:
print('{} {} {} {}'.format(
epoch,
cost.eval({X: train_X, Y: train_Y}),
W.eval(),
b.eval(),
))
optimizer.run({X: train_X, Y: train_Y})
# Classification accuracy of perceptron
classifications = [pred.eval({X: x}) > 0.5 for x in train_X]
correct = sum([p == y for (p, y) in zip(classifications, train_Y)])
print('{}/{} = perceptron accuracy'.format(correct, rows))
# Classification accuracy of hand-coded threshold comparison
classifications = [x[0] > 1.0 for x in train_X]
correct = sum([p == y for (p, y) in zip(classifications, train_Y)])
print('{}/{} = threshold accuracy'.format(correct, rows))
The output:
0 0.28319069743156433 [ 0.75648874] -0.9745011329650879
1 0.28302448987960815 [ 0.75775659] -0.9742625951766968
2 0.28285878896713257 [ 0.75902224] -0.9740257859230042
4 0.28252947330474854 [ 0.76154679] -0.97355717420578
8 0.28187844157218933 [ 0.76656926] -0.9726400971412659
16 0.28060704469680786 [ 0.77650583] -0.970885694026947
32 0.27818527817726135 [ 0.79593837] -0.9676888585090637
64 0.2738055884838104 [ 0.83302218] -0.9624817967414856
128 0.26666420698165894 [ 0.90031379] -0.9562843441963196
256 0.25691407918930054 [ 1.01172411] -0.9567816257476807
512 0.2461051195859909 [ 1.17413962] -0.9872989654541016
1024 0.23519910871982574 [ 1.38549554] -1.088881492614746
2048 0.2241383194923401 [ 1.64616168] -1.298340916633606
4096 0.21433120965957642 [ 1.95981205] -1.6126530170440674
8192 0.2075471431016922 [ 2.31746769] -1.989408016204834
9999 0.20618653297424316 [ 2.42539024] -2.1028473377227783
4/5 = perceptron accuracy
4/5 = threshold accuracy
I am messing around with LSTMs and have a conceptual question. I created a matrix of bogus data on the following rules:
For each 1-D list in the matrix:
If previous element is less than 10, then this next element is the previous one plus 1.
Else, this element is sin(previous element)
This way, it is a sequence that is pretty simply based on the previous information. I set up an LSTM to learn the recurrence and ran it to train on the lists one at a time. I have an LSTM layer followed by a fully connected feed-forward layer. It learns the +1 step very easily, but has trouble with the sin step. It will seemingly pick a random number between -1 and 1 when making the next element when the previous one was greater than 10. My question is this: is the training only modifying the variables in my fully connected feed forward layer? Is that why it can't learn the non-linear sin function?
Here's the code snippet in question:
lstm = rnn_cell.LSTMCell(lstmSize)
y_ = tf.placeholder(tf.float32, [None, OS])
outputs, state = rnn.rnn(lstm, x, dtype=tf.float32)
outputs = tf.transpose(outputs, [1, 0, 2])
last = tf.gather(outputs, int(outputs.get_shape()[0]) - 1)
weights = tf.Variable(tf.truncated_normal([lstmSize, OS]))
bias = tf.Variable(tf.constant(0.1, shape=[OS]))
y = tf.nn.elu(tf.matmul(last, weights) + bias)
error = tf.reduce_mean(tf.square(tf.sub(y_, y)))
train_step = tf.train.AdamOptimizer(learning_rate=1e-3).minimize(error)
The error and shape organization seems to be correct, at least in the sense that it does learn the +1 step quickly without crashing. Shouldn't the LSTM be able to handle the non-linear sin function? It seems almost trivially easy, so my guess is that I set something up wrong and the LSTM isn't learning anything.
I am trying to understand a simple implementation of Softmax classifier from this link - CS231n - Convolutional Neural Networks for Visual Recognition. Here they implemented a simple softmax classifier. In the example of Softmax Classifier on the link, there are random 300 points on a 2D space and a label associated with them. The softmax classifier will learn which point belong to which class.
Here is the full code of the softmax classifier. Or you can see the link I have provided.
# initialize parameters randomly
W = 0.01 * np.random.randn(D,K)
b = np.zeros((1,K))
# some hyperparameters
step_size = 1e-0
reg = 1e-3 # regularization strength
# gradient descent loop
num_examples = X.shape[0]
for i in xrange(200):
# evaluate class scores, [N x K]
scores = np.dot(X, W) + b
# compute the class probabilities
exp_scores = np.exp(scores)
probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True) # [N x K]
# compute the loss: average cross-entropy loss and regularization
corect_logprobs = -np.log(probs[range(num_examples),y])
data_loss = np.sum(corect_logprobs)/num_examples
reg_loss = 0.5*reg*np.sum(W*W)
loss = data_loss + reg_loss
if i % 10 == 0:
print "iteration %d: loss %f" % (i, loss)
# compute the gradient on scores
dscores = probs
dscores[range(num_examples),y] -= 1
dscores /= num_examples
# backpropate the gradient to the parameters (W,b)
dW = np.dot(X.T, dscores)
db = np.sum(dscores, axis=0, keepdims=True)
dW += reg*W # regularization gradient
# perform a parameter update
W += -step_size * dW
b += -step_size * db
I cant understand how they computed the gradient here. I assume that they computed the gradient here -
dW = np.dot(X.T, dscores)
db = np.sum(dscores, axis=0, keepdims=True)
dW += reg*W # regularization gradient
But How? I mean Why gradient of dW is np.dot(X.T, dscores)? And Why the gradient of db is np.sum(dscores, axis=0, keepdims=True)?? So how they computed the gradient on weight and bias? Also why they computed the regularization gradient?
I am just starting to learn about convolutional neural networks and deep learning. And I heard that CS231n - Convolutional Neural Networks for Visual Recognition is a good starting place for that. I did not know where to place deep learning related post. So, i placed them on stackoverflow. If there is any place to post questions related to deep learning please let me know.
The gradients start being computed here:
# compute the gradient on scores
dscores = probs
dscores[range(num_examples),y] -= 1
dscores /= num_examples
First, this sets dscores equal to the probabilities computed by the softmax function. Then, it subtracts 1 from the probabilities computed for the correct classes in the second line, and then it divides by the number of training samples in the third line.
Why does it subtract 1? Because you want the probabilities of the correct labels to be 1, ideally. So it subtracts what it should predict from what it actually predicts: if it predicts something close to 1, the subtraction will be a large negative number (close to zero), so the gradient will be small, because you're close to a solution. Otherwise, it will be a small negative number (far from zero), so the gradient will be bigger, and you'll take larger steps towards the solution.
Your activation function is simply w*x + b. Its derivative with respect to w is x, which is why dW is the dot product between x and the gradient of the scores / output layer.
The derivative of w*x + b with respect to b is 1, which is why you simply sum dscores when backpropagating.
Gradient Descent
Backpropagation is to reduce the cost J of the entire system (softmax classifier here) and it is a problem to optimize the weight parameter W to minimize the cost. Providing the cost function J = f(W) is convex, the gradient descent W = W - α * f'(W) will result in the Wmin which minimizes J. The hyperparameter α is called learning rate which we need to optimize too, but not in this answer.
Y should be read as J in the diagram. Imagine you are on the surface of a place whose shape is defined as J = f(W) and you need to reach the point Wmin. There is no gravity so you do not know which way is toward the bottom but you know the function and your coordinate. How do you know which way you should go? You can find the direction from the derivative f'(W) and move to a new coordinate by W = W - α * f'(W). By repeating this, you can get closer and closer to the point Wmin.
Back propagation at Affin Layer
At the node where multiply or dot operation happens (affin), the function is J = f(W) = X * W. Suppose there are m number of fixed two dimensional coordinates represented as X. How can we find the hyper-plane which minimizes J = f(W) = X * W and its vector W?
We can get closer to the optimal W by repeating the gradient descent W += -α * X if α is appropriate.
Chain Rule
When there are layers after the Affine layer such as the softmax layer and the log loss layer in the softmax classifier, we can calculate the gradient with the chain rule. In the diagram, replace sigmoid with softmax.
As stated in Computing the Analytic Gradient with Backpropagation in the cs321 page, the gradient contribution from the softmax layer and the log loss layer is the dscore part. See the Note section below too.
By applying the gradient to that of the affine layer via the chain rule, the code is derived where α is replaced with step_size. In reality, the step_size needs to be learned as well.
dW = np.dot(X.T, dscores)
W += -step_size * dW
The bias gradient can be derived by applying the chain rule towards the bias b with the gradients (dscore) from the post layers.
db = np.sum(dscores, axis=0, keepdims=True)
Regularization
As stated in Regularization of the cs231 page, the cost function (objective) is adjusted by adding the regularization, which is reg_loss in the code. It is to reduce the over-fitting. The intuition is, in my understanding, if specific feature(s) cause overfitting, we can reduce it by inflating the cost with their weight parameters W, because the gradient descent will work to reduce the cost contributions from the weights. Since we do not know which ones, use all W. The reason of 0.5 * W*W is because it gives simple derivative W.
reg_loss = 0.5*reg*np.sum(W*W)
The gradient contribution reg*W is from the derivative of reg_loss. The reg is a hyper parameter to be learned in the real training.
reg_loss/dw -> 0.5 * reg * 2 * W
It is added to the gradient from the layers after the affin.
dW += reg*W # regularization gradient
The process to get the derivative from the cost including the regularization is omitted in the cs231 page referenced in the post, probably because it is a common practice to just put the gradient of the regularization, but confusing for those who are learning. See Coursera Machine Learning Week 3 Cost Function by Andrew Ng for the regularization.
Note
The bias parameter b is substituted with X0 as the bias can be omitted by shifting to the base.