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I am currently reading on normalization/standardization techniques as well as batch normalization in deep learning and I don't really understand why normal distributions are so important inside deep learning models. If it's true that neural networks can approximate any function, then if we have two input features of the same scale but different distributions, why would this hinder the learning process. My current belief is that it isn't so much that normal distributions are important per se, but the individual features that are input into a neural network should have the same distributions for the fastest convergence and standard normal distributions are mathematically the easiest to work with.

I also don't understand what the impact of putting a batch normalization layer before and activation layer is. Wouldn't the BN layer normalize everything and then the activation layer denormalize every by putting all the values into its own distribution.

My questions are: -Why are normal distributions so important for neural networks(I understand that same scale input features is important for better convergence but why distribution) -Why do we place BN before activations, wouldn't the activations just change the distributions anyways?

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You seem to have mixed up two similar names.

  1. The normal (Gaussian) distribution, which is a “bell curve”
  2. A normalized distribution that has been centered and scaled by subtracting the mean and the dividing by the standard deviation

These are not the same.

Few regression techniques make distribution assumptions about the features, and neural networks are among those that kind of do not. Gaussian distributions of features are not important for neural networks. If you plot the distributions of pixels in MNIST images, I suspect there would be a fair amount of non-normality despite deep learning models being near-perfect on MNIST (though I have not done this plotting, at least not in a long time). In fact, neural networks can handle categorical features, which are guaranteed not to be normal.

Also, you do not theoretically need to do feature normalization in deep learning. Where this becomes useful has to do with the fact that computers need to calculate network optima in reasonable amounts of time, and feature scaling helps with the numerical optimization that must be performed, since neural networks do not have the kind of closed-form mathematical solutions for the optima like an ordinary least squares linear regression has.

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  • $\begingroup$ Okay thank you for you reply. I see that I have mixed these two term up. So when people reference Internal Covariate Shift inside the hidden layers of a neural network, they are simply referencing the fact that the activation values of any layer can change from a normalized distribution during training(which can hence hinder the calculations of the network) and this has nothing to do with Gaussian distributions? $\endgroup$ Jun 9, 2023 at 18:19
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Normal distributions, also known as Gaussian distributions, are essential in deep learning for several reasons:

Central Limit Theorem: The Central Limit Theorem states that the sum of a large number of independent and identically distributed random variables will tend to have a distribution that approaches a normal distribution. This theorem underlies the foundation of many statistical techniques and is crucial in deep learning because neural networks often involve the aggregation of numerous parameters and activations. The normal distribution allows for reliable statistical analysis and inference.

Initialization of Network Weights: In deep learning, network weights are typically initialized randomly to break the symmetry among neurons. A common practice is to initialize the weights with values drawn from a normal distribution, often with zero mean and small variance. This initialization strategy facilitates faster convergence during training and helps avoid the problem of dead neurons.

Activation Functions: Activation functions play a crucial role in neural networks by introducing non-linearities. Many popular activation functions, such as the sigmoid, tanh, and Gaussian error linear units (GELUs), are designed to operate well within the range of the normal distribution. They tend to map inputs to outputs that are distributed around the mean of the normal distribution, promoting stable and efficient learning.

Noise Modeling: In some deep learning applications, it is beneficial to model and incorporate noise into the learning process. Gaussian noise, which follows a normal distribution, is commonly used to model various types of noise, such as measurement noise, adversarial perturbations, or stochasticity in generative models. Normal distributions provide a convenient mathematical framework to model and analyze such noise.

Probabilistic Models: Deep learning has expanded beyond deterministic models and embraces probabilistic models that capture uncertainty. Probabilistic models often use normal distributions as the building blocks to model uncertainties, such as Gaussian mixture models (GMMs), variational autoencoders (VAEs), or Gaussian processes (GPs). The mathematical tractability of normal distributions simplifies the training and inference procedures in these probabilistic models.

Loss Functions: In many deep learning tasks, the choice of an appropriate loss function is critical. When dealing with regression problems, mean squared error (MSE) loss is commonly used, assuming that the errors follow a normal distribution. Similarly, in generative models, the maximum likelihood estimation (MLE) objective assumes that the generated samples follow a target distribution, often a normal distribution.

Overall, the use of normal distributions in deep learning enables effective initialization, efficient optimization, noise modeling, uncertainty quantification, and probabilistic modeling, making them fundamental to many aspects of the field.

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