Develop Reference

Is there a difference between multiple times loss.backward() and loss.backward() after multiplying loss by n in Pytorch?

Is there a difference between these two codes?
1
Loss.backward(retain_graph=True)
Loss.backward(retain_graph=True)
Loss.backward()
optimizer.step
2
Loss = 3 * Loss
Loss.backward()
optimizer.step
When I checked the gradient of the parameter after the last backward(), there was no difference between the two codes. However, there is a little difference in test accuracy after training.
I know this is not a common case, but it is related to the research I'm doing.

To me, it looks very different.
Computing the loss three time won't do anything (first code snippet). You just hold on to the gradient you have previously calculated. (Check on your leaf tensors the value of the .grad() attribute).
However, the second code snippet with just multiply the gradients by three, thus speeding up Gradient Descent. For a standard Gradient descent optimizer, it would be like mutliplying the learning rate by 3.
Hope this helps.

In option 1, every time you call .backward(), gradients are computed. After 3 calls, when you perform optimizer.step, the gradients are added and then weights are updated accordingly.
In option 2, you multiply the loss with a constant, so the gradients will be multiplied with that constant too.
So, adding a gradient value 3 times and multiplying the gradient value by 3 would result in the same parameter update.
Please note, I assume there is no loss due to floating point precision (as noted in the comments).

How can I predict the expected value and the variance simultaneously with a neural network?

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.

Incorporate side conditions into Keras neural network

I want to train my neural network (in Keras) with an additional condition on the output elements.
An example:
Minimize my loss function MSE between network output y_pred and y_true.
Additionally, ensure that the norm of y_pred is less or equal 1.
Without the condition, the task is straightforward.
Note: The condition is not necessarily the vector norm of y_pred.
How can I implement the additional condition/restriction in a Keras (or maybe Tensorflow) model?

In principle, tensorflow (and keras) don't allow you to add hard constraints to your model.
You have to convert your invarient (norm <= 1) to a penalty function, which is added to the loss. This could look like this:
y_norm = tf.norm(y_pred)
norm_loss = tf.where(y_norm > 1, y_norm, 0)
total_loss = mse + norm_loss
Look at the docs of where. If your prediction has a norm bigger than one, backpropagation tries to minimize the norm. If it is less than or equal, this part of the loss is simply 0. No gradient is produced.
But this can be very hard to optimize. Your predictions could oscillate around a norm of 1. It is also possible to add a factor: total_loss = mse + 1000* norm_loss. Be very careful with this, it makes optimization even harder.
In the example above, the norm above one contributes linearly to the loss. This is called l1-regularization. You could also square it, which would become l2-regularization.
In your specific case, you could get creative. Why not normalize your predictions and the targets to one (just a suggestion, might be a bad idea)?
loss = mse(y_pred / tf.norm(y_pred), y_target / np.linalg.norm(y_target)

Multi-Task Learning: Train a neural network to have different loss functions for the two classes?

I have a neural net with two loss functions, one is binary cross entropy for the 2 classes, and another is a regression. Now I want the regression loss to be evaluated only for class_2, and return 0 for class_1, because the regressed feature is meaningless for class_1.
How can I implement such an algorithm in Keras?
Training it separately on only class_1 data doesn't work because I get nan loss. There are more elegant ways to define the loss to be 0 for one half of the dataset and mean_square_loss for another half?

This is a question that's important in multi-task learning where you have multiple loss functions, a shared neural network structure in the middle, and inputs that may not all be valid for all loss functions.
You can pass in a binary mask which are 1 or 0 for each of your loss functions, in the same way that you pass in the labels. Then multiply each loss by its corresponding mask. The derivative of 1x is just dx, and the derivative of 0x is 0. You end up zeroing out the gradient in the appropriate loss functions. Virtually all optimizers are additive optimizers, meaning you're summing the gradient, adding a zero is a null operation. Your final loss function should be the sum of all your other losses.
I don't know much about Keras. Another solution is to change your loss function to use the labels only: L = cross_entropy * (label / (label + 1e-6)). That term will be almost 0 and almost 1. Close enough for government work and neural networks at least. This is what I actually used the first time before I realized it was as simple as multiplying by an array of mask values.
Another solution to this problem is to us tf.where and tf.gather_nd to select only the subset of labels and outputs that you want to compare and then pass that subset to the appropriate loss function. I've actually switched to using this method rather than multiplying by a mask. But both work.

Ridge regression using stochastic gradient descent in Python

I am trying to implement a solution to Ridge regression in Python using Stochastic gradient descent as the solver. My code for SGD is as follows:
def fit(self, X, Y):
# Convert to data frame in case X is numpy matrix
X = pd.DataFrame(X)
# Define a function to calculate the error given a weight vector beta and a training example xi, yi
# Prepend a column of 1s to the data for the intercept
X.insert(0, 'intercept', np.array([1.0]*X.shape[0]))
# Find dimensions of train
m, d = X.shape
# Initialize weights to random
beta = self.initializeRandomWeights(d)
beta_prev = None
epochs = 0
prev_error = None
while (beta_prev is None or epochs < self.nb_epochs):
print("## Epoch: " + str(epochs))
indices = range(0, m)
shuffle(indices)
for i in indices: # Pick a training example from a randomly shuffled set
beta_prev = beta
xi = X.iloc[i]
errori = sum(beta*xi) - Y[i] # Error[i] = sum(beta*x) - y = error of ith training example
gradient_vector = xi*errori + self.l*beta_prev
beta = beta_prev - self.alpha*gradient_vector
epochs += 1
The data I'm testing this on is not normalized and my implementation always ends up with all the weights being Infinity, even though I initialize the weights vector to low values. Only when I set the learning rate alpha to a very small value ~1e-8, the algorithm ends up with valid values of the weights vector.
My understanding is that normalizing/scaling input features only helps reduce convergence time. But the algorithm should not fail to converge as a whole if the features are not normalized. Is my understanding correct?

You can check from scikit-learn's Stochastic Gradient Descent documentation that one of the disadvantages of the algorithm is that it is sensitive to feature scaling. In general, gradient based optimization algorithms converge faster on normalized data.
Also, normalization is advantageous for regression methods.
The updates to the coefficients during each step will depend on the ranges of each feature. Also, the regularization term will be affected heavily by large feature values.
SGD may converge without data normalization, but that is subjective to the data at hand. Therefore, your assumption is not correct.

Your assumption is not correct.
It's hard to answer this, because there are so many different methods/environments but i will try to mention some points.
Normalization
When some method is not scale-invariant (i think every linear-regression is not) you really should normalize your data
I take it that you are just ignoring this because of debugging / analyzing
Normalizing your data is not only relevant for convergence-time, the results will differ too (think about the effect within the loss-function; big values might effect in much more loss to small ones)!
Convergence
There is probably much to tell about convergence of many methods on normalized/non-normalized data, but your case is special:
SGD's convergence theory only guarantees convergence to some local-minimum (= global-minimum in your convex-opt problem) for some chosings of hyper-parameters (learning-rate and learning-schedule/decay)
Even optimizing normalized data can fail with SGD when those params are bad!
This is one of the most important downsides of SGD; dependence on hyper-parameters
As SGD is based on gradients and step-sizes, non-normalized data has a possibly huge effect on not achieving this convergence!

In order for sgd to converge in linear regression the step size should be smaller than 2/s where s is the largest singular value of the matrix (see the Convergence and stability in the mean section in https://en.m.wikipedia.org/wiki/Least_mean_squares_filter), in the case of ridge regression it should be less than 2*(1+p/s^2)/s where p is the ridge penalty.
Normalizing rows of the matrix (or gradients) changes the loss function to give each sample an equal weight and it changes the singular values of the matrix such that you can choose a step size near 1 (see the NLMS section in https://en.m.wikipedia.org/wiki/Least_mean_squares_filter). Depending on your data it might require smaller step sizes or allow for larger step sizes. It all depends on whether or not the normalization increases or deacreses the largest singular value of the matrix.
Note that when deciding whether or not to normalize the rows you shouldn't just think about the convergence rate (which is determined by the ratio between the largest and smallest singular values) or stability in the mean, but also about how it changes the loss function and whether or not it fits your needs because of that, sometimes it makes sense to normalize but sometimes (for example when you want to give different importance for different samples or when you think that a larger energy for the signal means better snr) it doesn't make sense to normalize.