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Activation functions: Softmax vs Sigmoid

I've been trying to build an image classifier with CNN. There are 2300 images in my dataset and two categories: men and women. Here's the model I used:
early_stopping = EarlyStopping(min_delta = 0.001, patience = 30, restore_best_weights = True)
model = tf.keras.Sequential()
model.add(tf.keras.layers.Conv2D(256, (3, 3), input_shape=X.shape[1:], activation = 'relu'))
model.add(tf.keras.layers.MaxPooling2D(pool_size=(2, 2)))
model.add(tf.keras.layers.BatchNormalization())
model.add(tf.keras.layers.Conv2D(256, (3, 3), input_shape=X.shape[1:], activation = 'relu'))
model.add(tf.keras.layers.MaxPooling2D(pool_size=(2, 2)))
model.add(tf.keras.layers.BatchNormalization())
model.add(tf.keras.layers.Flatten()) # this converts our 3D feature maps to 1D feature vectors
model.add(tf.keras.layers.Dense(64))
model.add(tf.keras.layers.Dense(1, activation='softmax'))
model.compile(loss='binary_crossentropy',
optimizer='adam',
metrics=['accuracy'])
h= model.fit(xtrain, ytrain, validation_data=(xval, yval), batch_size=32, epochs=30, callbacks = [early_stopping], verbose = 0)
Accuracy of this model is 0.501897 and loss 7.595693(the model is stuck on these numbers in every epoch) but if I replace Softmax activation with Sigmoid, accuracy is about 0.98 and loss 0.06. Why does such strange thing happen with Softmax? All info I could find was that these two activations are similar and softmax is even better but I couldn't find anything about such abnormality. I'll be glad if someone could explain what the problem is.

Summary of your results:
a) CNN with Softmax activation function -> accuracy ~ 0.50, loss ~ 7.60
b) CNN with Sigmoid activation function -> accuracy ~ 0.98, loss ~ 0.06
TLDR
Update:
Now that I also see you are using only 1 output neuron with Softmax, you will not be able to capture the second class in binary classification. With Softmax you need to define K neurons in the output layer - where K is the number of classes you want to predict. Whereas with Sigmoid: 1 output neuron is sufficient for binary classification.
so in short, this should change in your code when using softmax for 2 classes:
#use 2 neurons with softmax
model.add(tf.keras.layers.Dense(2, activation='softmax'))
Additionally:
When doing binary classification, a sigmoid function is more suitable as it is simply computationally more effective compared to the more generalized softmax function (which is normally being used for multi-class prediction when you have K>2 classes).
Further Reading:
Some attributes of selected activation functions
If the short answer above is not enough for you, I can share with you some things I've learned from my research about activation functions with NNs in short:
To begin with, let's be clear with the terms activation and activation function
activation (alpha): is the state of a neuron. The state of neurons in hidden or output layers will be quantified by the weighted sum of input signals from a previous layer
activation function f(alpha): Is a function that transforms an activation to a neuron signal. Usually a non-linear and differentiable function as for instance the sigmoid function. Many applications & research has been applied with the sigmoid function (see Bengio & Courville, 2016, p.67 ff.). Mostly the same activation function is being used throughout the neural network, but it is possible to use multiple (e.g. different ones in different layers).
Now to the effects of activation functions:
The choice of activation function can have an immense impact on learning of neural networks (as you have seen in your example). Historically it was common to use the sigmoid function, as it was a good function to depict a saturated neuron. Today, especially in CNNs other activation functions, also only partially linear activation functions (like relu) is being preferred over sigmoid function. There are many different functions, just to name some: sigmoid, tanh, relu, prelu, elu ,maxout, max, argmax, softmax etc.
Now let's only compare sigmoid, relu/maxout and softmax:
# pseudo code / formula
sigmoid = f(alpha) = 1 / (1 + exp(-alpha))
relu = f(alpha) = max(0,alpha)
maxout = f(alpha) = max(alpha1, alpha2)
softmax = f(alpha_j) = alpha_j / sum_K(alpha_k)
sigmoid:
in binary classification preferably used for output layer
values can range between [0,1], suitable for a probabilistic interpretation (+)
saturated neurons can eliminate gradient (-)
not zero centered (-)
exp() is computationally expensive (-)
relu:
no saturated neurons in positive regions (+)
computationally less expensive (+)
not zero centered (-)
saturated neurons in negative regions (-)
maxout:
positive attributes of relu (+)
doubles the number of parameters per neuron, normally requires an increased learning effort (-)
softmax:
can bee seen as a generalization of sigmoid function
mainly being used as output activation function in multi-class prediction problems
values range between [0,1], suitable for a probabilistic interpretation (+)
computationally more expensive because of exp() terms (-)
Some good references for further reading:
http://cs231n.stanford.edu/2020/syllabus
http://deeplearningbook.org (Bengio & Courtville)
https://arxiv.org/pdf/1811.03378.pdf
https://papers.nips.cc/paper/2018/file/6ecbdd6ec859d284dc13885a37ce8d81-Paper.pdf

The reason why you see those different results is the size of your output layer - it is 1 neuron.
Softmax by definition requires more than 1 output neuron to make sense. 1 Softmax neuron will always output 1 (lookup the formula and think about it). That is why you see ~50% accuracy, since your network always predicts class 1.
Sigmoid doesn't have this problem and can output anything, that's why it trains.
If you want to test softmax, you have to make an output neuron for each class and then "one-hot encode" your ytrain and yval (look up one-hot encoding for more explanations). In your case this means: label 0 -> [1, 0], label 1 -> [0, 1]. You can see, the index of the one encodes the class. I'm not sure but in that case I believe you'd use the categorical cross entropy. I was not able to tell conclusively from the docs, but it seems to me that binary cross entropy expects 1 output neuron that's either 0 or 1 (where Sigmoid is the correct activation to use) whereas the categorical cross entropy expects one output neuron for each class, where Softmax makes sense. You could use Sigmoid even for the multioutput case, but it's not common.
So in short, it seems to me that binary xentropy expects the class encoded by the value of the 1 neuron, whereas categorical xentropy expects the class encoded by which output neuron is the most active. (in simplifying terms)

tf.keras.losses.categorical_crossentropy returning wrong value

I have
y_true = 16
and
y_pred = array([1.1868494e-08, 1.8747659e-09, 1.2777099e-11, 3.6140797e-08,
6.5852622e-11, 2.2888577e-10, 1.4515833e-09, 2.8392664e-09,
4.7054605e-10, 9.5605066e-11, 9.3647139e-13, 2.6149302e-10,
2.5338919e-14, 4.8815413e-10, 3.9381631e-14, 2.1434269e-06,
9.9999785e-01, 3.0857247e-08, 1.3536775e-09, 4.6811921e-10,
3.0638234e-10, 2.0818169e-09, 2.9950772e-10, 1.0457132e-10,
3.2959850e-11, 3.4232595e-10, 5.1689473e-12], dtype=float32)
When I use tf.keras.losses.categorical_crossentropy(to_categorical(y_true,num_classes=27),y_pred,from_logits=True)
The loss value I get is 2.3575358.
But if I use the formula for categorical cross entropy to get the loss value
-np.sum(to_categorical(gtp_out_true[0],num_classes=27)*np.log(gtp_pred[0]))
according to the formula
I get the value 2.1457695e-06
Now, my question is, why does the function tf.keras.losses.categorical_crossentropy give different value.
The strange thing is that, my model gives 100% accuracy even though the loss is stuck at 2.3575.
Below is the image of the plot of accuracy and losses during training.
What formula does Tensorflow use to calculate categorical cross-entropy?

Found where the problem is
I used softmax activation in my last layer
output = Dense(NUM_CLASSES, activation='softmax')(x)
But I used from_logits=True in tf.keras.losses.categorical_crossentropy, which resulted in softmax being applied again on the output of the last layer (which was already softmax(logits)). So, the output argument that I was passing to the loss function was softmax(softmax(logits)).
Hence, the anomaly in the values of loss.
When using softmax as activation in the last layer, we should use from_logits=False

y_pred as a probability vector so you should not use from_logits=True. Set it to False and you get:
>>> print(categorical_crossentropy(to_categorical(16, num_classes = 27),
y_pred, from_logits = False).numpy())
2.264979e-06
The reason it is not equal to the expected 2.1457695e-06 is, I belive, because y_pred[16] is very close to 1.0 and categorical_crossentropy adds some smoothing.
See the answer here for a discussion on logits: What is the meaning of the word logits in TensorFlow?
You can also use the sparse version of the function if each input value can only have one label:
print(sparse_categorical_crossentropy(16, y_pred))

Tensorflow 2.0 - do these model predictions represent probabilities?

I have a very simple Tensorflow 2 Keras model to do penalized logistic regression on some data. I was hoping to get the probabilties of each class, instead of just the predicted values of [0 or 1].
I think I got what I wanted, but just wanted to make sure that these numbers are what I think they are. I used the model.predict_on_batch() function from Tensorflow.keras, but the documentation just says that this provides a numpy array of predictions. However I believe I am getting probabilities, but I was hoping someone could confirm.
The model code looks like this:
feature_layer = tf.keras.layers.DenseFeatures(features)
model = tf.keras.Sequential([
feature_layer,
layers.Dense(1, activation='sigmoid', kernel_regularizer=tf.keras.regularizers.l1(0.01))
])
model.compile(optimizer='adam',
loss='binary_crossentropy',
metrics=['accuracy'])
predictions = model.predict_on_batch(validation_dataset)
print('Predictions for a single batch.')
print(predictions)
So the predictions I am getting look like:
Predictions for a single batch.
tf.Tensor(
[[0.10916319]
[0.14546806]
[0.13057315]
[0.11713684]
[0.16197902]
[0.19613355]
[0.1388464 ]
[0.14122346]
[0.26149303]
[0.12516734]
[0.1388464 ]
[0.14595506]
[0.14595506]]
Now for predictions in a logistic regression that would be an array of either 0 or 1. But since I am getting floating point values. However, I am just getting a single value when there is actually a probability that the example is a 0 and the probability that the example is a 1. So I would imagine an array of 2 probabilities for each row or example. Of course, the Probability(Y = 0) + Probability(Y = 1) = 1, so this might just be some concise representation.
So again, do the values in the array below represent probabilities that the example or Y = 1, or something else?

The values represented here:
tf.Tensor(
[[0.10916319]
[0.14546806]
[0.13057315]
[0.11713684]
[0.16197902]
[0.19613355]
[0.1388464 ]
[0.14122346]
[0.26149303]
[0.12516734]
[0.1388464 ]
[0.14595506]
[0.14595506]]
Are the probabilities corresponding to each one of your classes.
Since you used sigmoid activation on your last layer, these will
be in the range [0, 1].
Your model is very shallow (few layers) and thus these prediction probabilities are very close between classes. I suggest you add more layers.
Conclusion
To answer your question, these are probabilities but only due to your activation function selection (sigmoid). If you used tanh activation these would be in range [-1,1].
Note that these probabilities are "binary" for each class due to the use of binary_crossentropy loss - aka 10.92% that class 1 is present and 89.08% that it is not, and so on for other classes. If you want the predictions to follow probabilistic rules (sum = 1) then you should consider categorical_crossentropy.

Tensorflow: Sigmoid cross entropy loss does not force network outputs to be 0 or 1

I would like to learn image segmentation in TensorFlow with values in {0.0,1.0}. I have two images, ground_truth and prediction and each have shape (120,160). The ground_truth image pixels only contain values that are either 0.0 or 1.0.
The prediction image is the output of a decoder and the last two layers of it are a tf.layers.conv2d_transpose and tf.layers.conv2d like so:
transforms (?,120,160,30) -> (?,120,160,15)
outputs = tf.layers.conv2d_transpose(outputs, filters=15, kernel_size=1, strides=1, padding='same')
# ReLU
outputs = activation(outputs)
# transforms (?,120,160,15) -> (?,120,160,1)
outputs = tf.layers.conv2d(outputs, filters=1, kernel_size=1, strides=1, padding='same')
The last layer does not carry an activation function and thus it's output is unbounded. I use the following loss function:
logits = tf.reshape(predicted, [-1, predicted.get_shape()[1] * predicted.get_shape()[2]])
labels = tf.reshape(ground_truth, [-1, ground_truth.get_shape()[1] * ground_truth.get_shape()[2]])
loss = 0.5 * tf.reduce_mean(tf.nn.sigmoid_cross_entropy_with_logits(labels=labels,logits=logits))
This setup converges nicely. However, I have realized that the outputs of my last NN layer at validation time seem to be in [-inf, inf]. If I visualize the output I can see that the segmented object is not segmented since almost all pixels are "activated". The distributions of values for a single output of the last conv2d layer looks like this:
Question:
Do I have to post-process the outputs (crop negative values or run output trough a sigmoid activation etc.)? What do I need to do to enforce my output values to be {0,1}?

Solved it. The problem was that the tf.nn.sigmoid_cross_entropy_with_logits runs the logits through a sigmoid which is of course not used at validation time since the loss operation is only called during train time. The solution therefore is:
make sure to run the network outputs through a tf.nn.sigmoid at validation/test time like this:
return output if is_training else tf.nn.sigmoid(output)

TensorFlow MLP example outputs binary instead of decimal

I'm trying to train a multilayer perseptron to classify between true or false, based on the given input.
So far I'm using the example:
https://github.com/aymericdamien/TensorFlow-Examples/blob/master/examples/3_NeuralNetworks/multilayer_perceptron.py
But this gives me the output as a binary value and I rather have a decimal or percentage based output.
What I've tried:
I've tried to change the optimizer for the other available ones with no success.
optimizer =
tf.train.AdamOptimizer(learning_rate=learning_rate).minimize(cost)

The optimizer will not change the output that is actually given by the layers.
The provided example uses ReLu for the layers, which is good for classification but to model probability it wouldn't work. You would be better off with a sigmoid function instead.
The sigmoid function can be used to model probability, whereas ReLu can be used to model positive real number.
In order to make it work for the provided example, change the multilayer_perceptron function to:
def multilayer_perceptron(_X, _weights, _biases):
layer_1 = tf.sigmoid(tf.add(tf.matmul(_X, _weights['h1']), _biases['b1']), name="sigmoid_l1") #Hidden layer with sigmoid activation
layer_2 = tf.sigmoid(tf.add(tf.matmul(layer_1, _weights['h2']), _biases['b2']), name="sigmoid_l2") #Hidden layer with sigmoid activation
return tf.matmul(layer_2, _weights['out'], name="matmul_lout") + _biases['out']
It basically replaces the ReLu activation for a sigmoid one.
Then, for the evaluation, use softmax as follows:
output1 = tf.nn.softmax((multilayer_perceptron(x, weights, biases)), name="output")
avd = sess.run(output1, feed_dict={x: features_t})
It will provide you a range between 0 and 1 for each class. Also, you'll probably have to increase the number of epochs for this to work.