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When computing the loss between y_true and y_pred, the keras loss functions reduce the dimensionality by one. For example, when training a network on pairs of 64x64 greyscale images with batch size = 8, the shape of y_true and y_pred would be (8, 64, 64). The keras loss functions will produce a loss tensor with shape (8, 64), averaging over the last dimension.
I do not get why that would be necessary, all it does is average the loss over the rows of the image. Doesn't the network need the loss to be calculated individually for every output value (and therefore conserve the shape)? As far as I understand it, backpropagation looks at the individual loss of each output value compared to the target, and then updates previous weights accordingly. How can it do that, just knowing the averaged loss of each row, not every value individually? Here is a code snippet that shows the behaviour I described:
y_true = K.random_uniform([8,64,64])
y_pred = K.random_uniform([8,64,64])
c= mean_absolute_error(y_true,y_pred)
print(K.eval(tf.shape(c))) # (8,64)
I wondered the same thing. I believe, Keras assumes your data to have the following dimensions: [batch, W, H, n_classes] which means averaging over axis=-1 means averaging the loss over all different classes. However, in your case you do not have that dimension because you presumably do a binary classification in a grayscale image. So instead it ends up averaging the loss over the rows/columns. Interestingly enough, the model can still train and even improve in performance like this, which makes me believe that people in similar situations often just train their model without ever noticing.
You can avoid this by adding a dummy axis to your data.
This is how I got there:
From: https://keras.io/api/losses/
"(Note ondN-1: all loss functions reduce by 1 dimension, usually axis=-1.) “
Furthermore: "loss class instances feature a reduction constructor argument, which defaults to "sum_over_batch_size" (i.e. average). Allowable values are "sum_over_batch_size", "sum", and "none":
• "sum_over_batch_size" means the loss instance will return the average of the per-sample losses in the batch.
• "sum" means the loss instance will return the sum of the per-sample losses in the batch.
• "none" means the loss instance will return the full array of per-sample losses. "
From https://www.tensorflow.org/api_docs/python/tf/keras/losses/Reduction
"Caution: Verify the shape of the outputs when using Reduction.NONE. The builtin loss functions wrapped by the loss classes reduce one dimension (axis=-1, or axis if specified by loss function). Reduction.NONE just means that no additional reduction is applied by the class wrapper. For categorical losses with an example input shape of [batch, W, H, n_classes] the n_classes dimension is reduced. For pointwise losses you must include a dummy axis so that [batch, W, H, 1] is reduced to [batch, W, H]. Without the dummy axis [batch, W, H] will be incorrectly reduced to [batch, W]."
I'd like to use a neural network to predict a scalar value which is the sum of a function of the input values and a random value (I'm assuming gaussian distribution) whose variance also depends on the input values. Now I'd like to have a neural network that has two outputs - the first output should approximate the deterministic part - the function, and the second output should approximate the variance of the random part, depending on the input values. What loss function do I need to train such a network?
(It would be nice if there was an example with Python for Tensorflow, but I'm also interested in general answers. I'm also not quite clear how I could write something like in Python code - none of the examples I found so far show how to address individual outputs from the loss function.)
You can use dropout for that. With a dropout layer you can make several different predictions based on different settings of which nodes dropped out. Then you can simply count the outcomes and interpret the result as a measure for uncertainty.
For details, read:
Gal, Yarin, and Zoubin Ghahramani. "Dropout as a bayesian approximation: Representing model uncertainty in deep learning." international conference on machine learning. 2016.
Since I've found nothing simple to implement, I wrote something myself, that models that explicitly: here is a custom loss function that tries to predict mean and variance. It seems to work but I'm not quite sure how well that works out in practice, and I'd appreciate feedback. This is my loss function:
def meanAndVariance(y_true: tf.Tensor , y_pred: tf.Tensor) -> tf.Tensor :
"""Loss function that has the values of the last axis in y_true
approximate the mean and variance of each value in the last axis of y_pred."""
y_pred = tf.convert_to_tensor(y_pred)
y_true = math_ops.cast(y_true, y_pred.dtype)
mean = y_pred[..., 0::2]
variance = y_pred[..., 1::2]
res = K.square(mean - y_true) + K.square(variance - K.square(mean - y_true))
return K.mean(res, axis=-1)
The output dimension is twice the label dimension - mean and variance of each value in the label. The loss function consists of two parts: a mean squared error that has the mean approximate the mean of the label value, and the variance that approximates the difference of the value from the predicted mean.
When using dropout to estimate the uncertainty (or any other stochastic regularization method), make sure to also checkout our recent work on providing a sampling-free approximation of Monte-Carlo dropout.
https://arxiv.org/pdf/1908.00598.pdf
We essentially follow ur idea. Treat the activations as random variables and then propagate mean and variance using error propagation to the output layer. Consequently, we obtain two outputs - the mean and the variance.
I am attempting to replicate an deep convolution neural network from a research paper. I have implemented the architecture, but after 10 epochs, my cross entropy loss suddenly increases to infinity. This can be seen in the chart below. You can ignore what happens to the accuracy after the problem occurs.
Here is the github repository with a picture of the architecture
After doing some research I think using an AdamOptimizer or relu might be a problem.
x = tf.placeholder(tf.float32, shape=[None, 7168])
y_ = tf.placeholder(tf.float32, shape=[None, 7168, 3])
#Many Convolutions and Relus omitted
final = tf.reshape(final, [-1, 7168])
keep_prob = tf.placeholder(tf.float32)
W_final = weight_variable([7168,7168,3])
b_final = bias_variable([7168,3])
final_conv = tf.tensordot(final, W_final, axes=[[1], [1]]) + b_final
cross_entropy = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(labels=y_, logits=final_conv))
train_step = tf.train.AdamOptimizer(1e-5).minimize(cross_entropy)
correct_prediction = tf.equal(tf.argmax(final_conv, 2), tf.argmax(y_, 2))
accuracy = tf.reduce_mean(tf.cast(correct_prediction, tf.float32))
EDIT
If anyone is interested, the solution was that I was basically feeding in incorrect data.
Solution: Control the solution space. This might mean using smaller datasets when training, it might mean using less hidden nodes, it might mean initializing your wb differently. Your model is reaching a point where the loss is undefined, which might be due to the gradient being undefined, or the final_conv signal.
Why: Sometimes no matter what, a numerical instability is reached. Eventually adding a machine epsilon to prevent dividing by zero (cross entropy loss here) just won't help because even then the number cannot be accurately represented by the precision you are using. (Ref: https://en.wikipedia.org/wiki/Round-off_error and https://floating-point-gui.de/basic/)
Considerations:
1) When tweaking epsilons, be sure to be consistent with your data type (Use the machine epsilon of the precision you are using, in your case float32 is 1e-6 ref: https://en.wikipedia.org/wiki/Machine_epsilon and python numpy machine epsilon.
2) Just in-case others reading this are confused: The value in the constructor for Adamoptimizer is the learning rate, but you can set the epsilon value (ref: How does paramater epsilon affects AdamOptimizer? and https://www.tensorflow.org/api_docs/python/tf/train/AdamOptimizer)
3) Numerical instability of tensorflow is there and its difficult to get around. Yes there is tf.nn.softmax_with_cross_entropy but this is too specific (what if you don't want a softmax?). Refer to Vahid Kazemi's 'Effective Tensorflow' for an insightful explanation: https://github.com/vahidk/EffectiveTensorflow#entropy
that jump in your loss graph is very weird...
I would like you to focus on few points :
if your images are not normalized between 0 and 1 then normalize them
if you have normalized your values between -1 and 1 then use a sigmoid layer instead of softmax because softmax squashes the values between 0 and 1
before using softmax add a sigmoid layer to squash your values (Highly Recommended)
other things you can do is add dropouts for every layer
also I would suggest you to use tf.clip so that your gradients does not explode and implode
you can also use L2 regularization
and experiment with the learning rate and epsilon of AdamOptimizer
I would also suggest you to use tensor-board to keep track of the weights so that way you will come to know where the weights are exploding
You can also use tensor-board for keeping track of loss and accuracy
See The softmax formula below:
Probably that e to power of x, the x is being a very large number because of which softmax is giving infinity and hence the loss is infinity
Heavily use tensorboard to debug and print the values of the softmax so that you can figure out where you are going wrong
One more thing I noticed you are not using any kind of activation functions after the convolution layers... I would suggest you to leaky relu after every convolution layer
Your network is a humongous network and it is important to use leaky relu as activation function so that it adds non-linearity and hence improves the performance
You may want to use a different value for epsilon in the Adam optimizer (e.g. 0.1 -- 1.0).This is mentioned in the documentation:
The default value of 1e-8 for epsilon might not be a good default in general. For example, when training an Inception network on ImageNet a current good choice is 1.0 or 0.1.
I am trying to train an autoencoder on some simulated data where an input is basically a vector with Gaussian noise applied. The code is almost exactly the same as in this example: https://github.com/aymericdamien/TensorFlow-Examples/blob/master/examples/3_NeuralNetworks/autoencoder.py
The only differences are I changed the network parameters and the cost function:
n_hidden_1 = 32 # 1st layer num features
n_hidden_2 = 16 # 2nd layer num features
n_input = 149 # LunaH-Map data input (number of counts per orbit)
cost = tf.reduce_mean(-tf.reduce_sum(y_true * tf.log(y_pred), reduction_indices=[1]))
During training, the error steadily decreases down to 0.00015, but the predicted and true values are very different, e.g.
as shown in this image. In fact, the predicted y vector is almost all ones.
How is it possible to get decreasing error with very wrong predictions? Is it possible that my network is just trying to move the weights closer to log(1) so as to minimize the cross entropy cost? If so, how do I combat this?
Yes, the network simply learns to to predict 1 which reduces the loss. The cross-entropy loss you are using is categorical which is used when y_true is one-hot code (example: [0,0,1,0]) and final layer is softmax (ensures sum of all output is 1). So when y_true[idx] is 0, the loss don't care while when the y_true[idx] is 1 and y_pred[idx] is 0 there is infinite(high) loss but if its 1 then loss is again 0.
Now categorical cross-entropy loss is not suitable for autoencoders. For real valued inputs and hence outputs its mean-squared-error, which is what is used in example that you cited. But there the final activation layer is sigmoid, implicitly saying that each element of x is 0/1. So either you need to convert your data to support the same or have the last layer of decoder linear.
If you do want to use cross-entropy loss you can use binary cross-entropy
For inputs with 0,1 binary cross-entropy: tf.reduce_mean(y_true * tf.log(y_pred) + (1-y_true) * tf.log(1-y_pred)). If you work it out in both misprediction case 0-1, 1-0, the network gets infinite loss. Note again here the final layer should be softmax and elements of x should be between 0 and 1
In the tensorflow API docs they use a keyword called logits. What is it? A lot of methods are written like:
tf.nn.softmax(logits, name=None)
If logits is just a generic Tensor input, why is it named logits?
Secondly, what is the difference between the following two methods?
tf.nn.softmax(logits, name=None)
tf.nn.softmax_cross_entropy_with_logits(logits, labels, name=None)
I know what tf.nn.softmax does, but not the other. An example would be really helpful.
The softmax+logits simply means that the function operates on the unscaled output of earlier layers and that the relative scale to understand the units is linear. It means, in particular, the sum of the inputs may not equal 1, that the values are not probabilities (you might have an input of 5). Internally, it first applies softmax to the unscaled output, and then and then computes the cross entropy of those values vs. what they "should" be as defined by the labels.
tf.nn.softmax produces the result of applying the softmax function to an input tensor. The softmax "squishes" the inputs so that sum(input) = 1, and it does the mapping by interpreting the inputs as log-probabilities (logits) and then converting them back into raw probabilities between 0 and 1. The shape of output of a softmax is the same as the input:
a = tf.constant(np.array([[.1, .3, .5, .9]]))
print s.run(tf.nn.softmax(a))
[[ 0.16838508 0.205666 0.25120102 0.37474789]]
See this answer for more about why softmax is used extensively in DNNs.
tf.nn.softmax_cross_entropy_with_logits combines the softmax step with the calculation of the cross-entropy loss after applying the softmax function, but it does it all together in a more mathematically careful way. It's similar to the result of:
sm = tf.nn.softmax(x)
ce = cross_entropy(sm)
The cross entropy is a summary metric: it sums across the elements. The output of tf.nn.softmax_cross_entropy_with_logits on a shape [2,5] tensor is of shape [2,1] (the first dimension is treated as the batch).
If you want to do optimization to minimize the cross entropy AND you're softmaxing after your last layer, you should use tf.nn.softmax_cross_entropy_with_logits instead of doing it yourself, because it covers numerically unstable corner cases in the mathematically right way. Otherwise, you'll end up hacking it by adding little epsilons here and there.
Edited 2016-02-07:
If you have single-class labels, where an object can only belong to one class, you might now consider using tf.nn.sparse_softmax_cross_entropy_with_logits so that you don't have to convert your labels to a dense one-hot array. This function was added after release 0.6.0.
Short version:
Suppose you have two tensors, where y_hat contains computed scores for each class (for example, from y = W*x +b) and y_true contains one-hot encoded true labels.
y_hat = ... # Predicted label, e.g. y = tf.matmul(X, W) + b
y_true = ... # True label, one-hot encoded
If you interpret the scores in y_hat as unnormalized log probabilities, then they are logits.
Additionally, the total cross-entropy loss computed in this manner:
y_hat_softmax = tf.nn.softmax(y_hat)
total_loss = tf.reduce_mean(-tf.reduce_sum(y_true * tf.log(y_hat_softmax), [1]))
is essentially equivalent to the total cross-entropy loss computed with the function softmax_cross_entropy_with_logits():
total_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(y_hat, y_true))
Long version:
In the output layer of your neural network, you will probably compute an array that contains the class scores for each of your training instances, such as from a computation y_hat = W*x + b. To serve as an example, below I've created a y_hat as a 2 x 3 array, where the rows correspond to the training instances and the columns correspond to classes. So here there are 2 training instances and 3 classes.
import tensorflow as tf
import numpy as np
sess = tf.Session()
# Create example y_hat.
y_hat = tf.convert_to_tensor(np.array([[0.5, 1.5, 0.1],[2.2, 1.3, 1.7]]))
sess.run(y_hat)
# array([[ 0.5, 1.5, 0.1],
# [ 2.2, 1.3, 1.7]])
Note that the values are not normalized (i.e. the rows don't add up to 1). In order to normalize them, we can apply the softmax function, which interprets the input as unnormalized log probabilities (aka logits) and outputs normalized linear probabilities.
y_hat_softmax = tf.nn.softmax(y_hat)
sess.run(y_hat_softmax)
# array([[ 0.227863 , 0.61939586, 0.15274114],
# [ 0.49674623, 0.20196195, 0.30129182]])
It's important to fully understand what the softmax output is saying. Below I've shown a table that more clearly represents the output above. It can be seen that, for example, the probability of training instance 1 being "Class 2" is 0.619. The class probabilities for each training instance are normalized, so the sum of each row is 1.0.
Pr(Class 1) Pr(Class 2) Pr(Class 3)
,--------------------------------------
Training instance 1 | 0.227863 | 0.61939586 | 0.15274114
Training instance 2 | 0.49674623 | 0.20196195 | 0.30129182
So now we have class probabilities for each training instance, where we can take the argmax() of each row to generate a final classification. From above, we may generate that training instance 1 belongs to "Class 2" and training instance 2 belongs to "Class 1".
Are these classifications correct? We need to measure against the true labels from the training set. You will need a one-hot encoded y_true array, where again the rows are training instances and columns are classes. Below I've created an example y_true one-hot array where the true label for training instance 1 is "Class 2" and the true label for training instance 2 is "Class 3".
y_true = tf.convert_to_tensor(np.array([[0.0, 1.0, 0.0],[0.0, 0.0, 1.0]]))
sess.run(y_true)
# array([[ 0., 1., 0.],
# [ 0., 0., 1.]])
Is the probability distribution in y_hat_softmax close to the probability distribution in y_true? We can use cross-entropy loss to measure the error.
We can compute the cross-entropy loss on a row-wise basis and see the results. Below we can see that training instance 1 has a loss of 0.479, while training instance 2 has a higher loss of 1.200. This result makes sense because in our example above, y_hat_softmax showed that training instance 1's highest probability was for "Class 2", which matches training instance 1 in y_true; however, the prediction for training instance 2 showed a highest probability for "Class 1", which does not match the true class "Class 3".
loss_per_instance_1 = -tf.reduce_sum(y_true * tf.log(y_hat_softmax), reduction_indices=[1])
sess.run(loss_per_instance_1)
# array([ 0.4790107 , 1.19967598])
What we really want is the total loss over all the training instances. So we can compute:
total_loss_1 = tf.reduce_mean(-tf.reduce_sum(y_true * tf.log(y_hat_softmax), reduction_indices=[1]))
sess.run(total_loss_1)
# 0.83934333897877944
Using softmax_cross_entropy_with_logits()
We can instead compute the total cross entropy loss using the tf.nn.softmax_cross_entropy_with_logits() function, as shown below.
loss_per_instance_2 = tf.nn.softmax_cross_entropy_with_logits(y_hat, y_true)
sess.run(loss_per_instance_2)
# array([ 0.4790107 , 1.19967598])
total_loss_2 = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(y_hat, y_true))
sess.run(total_loss_2)
# 0.83934333897877922
Note that total_loss_1 and total_loss_2 produce essentially equivalent results with some small differences in the very final digits. However, you might as well use the second approach: it takes one less line of code and accumulates less numerical error because the softmax is done for you inside of softmax_cross_entropy_with_logits().
tf.nn.softmax computes the forward propagation through a softmax layer. You use it during evaluation of the model when you compute the probabilities that the model outputs.
tf.nn.softmax_cross_entropy_with_logits computes the cost for a softmax layer. It is only used during training.
The logits are the unnormalized log probabilities output the model (the values output before the softmax normalization is applied to them).
Mathematical motivation for term
When we wish to constrain an output between 0 and 1, but our model architecture outputs unconstrained values, we can add a normalisation layer to enforce this.
A common choice is a sigmoid function.1 In binary classification this is typically the logistic function, and in multi-class tasks the multinomial logistic function (a.k.a softmax).2
If we want to interpret the outputs of our new final layer as 'probabilities', then (by implication) the unconstrained inputs to our sigmoid must be inverse-sigmoid(probabilities). In the logistic case this is equivalent to the log-odds of our probability (i.e. the log of the odds) a.k.a. logit:
That is why the arguments to softmax is called logits in Tensorflow - because under the assumption that softmax is the final layer in the model, and the output p is interpreted as a probability, the input x to this layer is interpretable as a logit:
Generalised term
In Machine Learning there is a propensity to generalise terminology borrowed from maths/stats/computer science, hence in Tensorflow logit (by analogy) is used as a synonym for the input to many normalisation functions.
While it has nice properties such as being easily diferentiable, and the aforementioned probabilistic interpretation, it is somewhat arbitrary.
softmax might be more accurately called softargmax, as it is a smooth approximation of the argmax function.
Above answers have enough description for the asked question.
Adding to that, Tensorflow has optimised the operation of applying the activation function then calculating cost using its own activation followed by cost functions. Hence it is a good practice to use: tf.nn.softmax_cross_entropy() over tf.nn.softmax(); tf.nn.cross_entropy()
You can find prominent difference between them in a resource intensive model.
Tensorflow 2.0 Compatible Answer: The explanations of dga and stackoverflowuser2010 are very detailed about Logits and the related Functions.
All those functions, when used in Tensorflow 1.x will work fine, but if you migrate your code from 1.x (1.14, 1.15, etc) to 2.x (2.0, 2.1, etc..), using those functions result in error.
Hence, specifying the 2.0 Compatible Calls for all the functions, we discussed above, if we migrate from 1.x to 2.x, for the benefit of the community.
Functions in 1.x:
tf.nn.softmax
tf.nn.softmax_cross_entropy_with_logits
tf.nn.sparse_softmax_cross_entropy_with_logits
Respective Functions when Migrated from 1.x to 2.x:
tf.compat.v2.nn.softmax
tf.compat.v2.nn.softmax_cross_entropy_with_logits
tf.compat.v2.nn.sparse_softmax_cross_entropy_with_logits
For more information about migration from 1.x to 2.x, please refer this Migration Guide.
One more thing that I would definitely like to highlight as logit is just a raw output, generally the output of last layer. This can be a negative value as well. If we use it as it's for "cross entropy" evaluation as mentioned below:
-tf.reduce_sum(y_true * tf.log(logits))
then it wont work. As log of -ve is not defined.
So using o softmax activation, will overcome this problem.
This is my understanding, please correct me if Im wrong.
Logits are the unnormalized outputs of a neural network. Softmax is a normalization function that squashes the outputs of a neural network so that they are all between 0 and 1 and sum to 1. Softmax_cross_entropy_with_logits is a loss function that takes in the outputs of a neural network (after they have been squashed by softmax) and the true labels for those outputs, and returns a loss value.