Xavier initialization can also be adapted for sampling weights from a uniform distribution instead of a Gaussian one. A uniform distribution $$U(-a, a)$$ has a variance of $$\frac{a^2}{3}$$. By setting this equal to the Xavier variance condition $$\sigma^2 = \frac{2}{n_	extrm{in} + n_	extrm{out}}$$ and solving for $$a$$, we obtain $$a = \sqrt{\frac{6}{n_	extrm{in} + n_	extrm{out}}}$$. Therefore, the uniform version of Xavier initialization samples weights according to the distribution $$U\left(-\sqrt{\frac{6}{n_	extrm{in} + n_	extrm{out}}}, \sqrt{\frac{6}{n_	extrm{in} + n_	extrm{out}}}ight)$$.

Xavier Initialization with a Uniform Distribution

Xavier initialization, named after its creators Glorot and Bengio (2010), is a standard technique designed to mitigate vanishing and exploding gradients by carefully setting the initial weights of a neural network layer. To balance the variance during both forward and backward propagation, it typically samples weights from a Gaussian distribution with a mean of $$0$$ and a variance of $$\sigma^2 = \frac{2}{n_	extrm{in} + n_	extrm{out}}$$, where $$n_	extrm{in}$$ and $$n_	extrm{out}$$ represent the number of inputs and outputs of the layer respectively. While the underlying assumption of linear activations is often violated in practice, this initialization method has proven highly effective.

San Diego State University

Claude

Initial weights are applied to all the neurons.
It is necessary to set initial weights for the first forward pass. Two basic options are to set weights to zero or to randomize them. However, this can result in a vanishing or exploding gradient, which will make it difficult to train the model. To mitigate this problem, you can use a heuristic (a formula tied to the number of neuron layers) to determine the weights. A common heuristic used for the Tanh activation is called Xavier initialization.

On top of that if the architecture being used has already been trained by someone else (such as ImageNet) the node values can be initialized to those values. This is known as transfer learning and it can be very useful to substantially increase training speed and model accuracy.

Deep Learning Weight Initialization

When initializing network weights, we face a dilemma: to keep variance fixed during forward propagation, we need $$n_	extrm{in} \sigma^2 = 1$$, but for backpropagation, we need $$n_	extrm{out} \sigma^2 = 1$$. It is generally impossible to satisfy both conditions simultaneously unless the number of inputs equals the number of outputs. As a practical compromise, we try to satisfy the average of the two conditions: $$\frac{1}{2} (n_	extrm{in} + n_	extrm{out}) \sigma^2 = 1$$. This simplifies to the target weight standard deviation of $$\sigma = \sqrt{\frac{2}{n_	extrm{in} + n_	extrm{out}}}$$, which forms the mathematical condition for Xavier initialization.

Xavier Initialization Condition

Dive into Deep Learning

For a single neuron example: you want to train a deep network without the weights exploding or vanishing.

Example of Weight Initialization 

In a neural network with many time steps or layers, a gradient at the early layer is the product of all the terms from the later layers, which leads to an inherently unstable situation. Especially when the value of gradient has become so small, it no longer updates properly or is vanished eventually. Exploding gradient can be considered as the opposite of vanishing process. The updated weights using gradient descent become so large that they cause the whole network to become unstable, which leads to numerical overflow. 

Vanishing/exploding gradient

Symmetry breaking refers to a requirement of initializing machine learning models like neural networks.

When a neural network model has weights, all initialized to the same value, it can be difficult or impossible for the weights to differ as the model is trained. This is known as the “symmetry” problem.

Initializing the model to small random values breaks the symmetry and allows different weights to learn independently of each other.

Symmetry Breaking in Deep Learning


To prevent the gradients of the network’s activations from vanishing or exploding, we will stick to the following rules of thumb:

- The mean of the activations should be zero.
- The variance of the activations should stay the same across every layer.

Under these two assumptions, the backpropagated gradient signal should not be multiplied by values too small or too large in any layer.

Ensuring zero-mean and maintaining the value of the variance of the input of every layer guarantees no exploding/vanishing signal

How to Initialization Weights to Prevent Vanishing/Exploding Gradients

The longest part of the development of a deep learning model is that of the training stage. However, if the problem that is being tackled has already been studied and worked on by someone else and the model has been released to the public including the trained weights then a technique to substantially increase the training process is to utilize transfer learning.

Transfer Learning in Deep Learning

In multi-task learning, the goal is to try to have one neural network classify/predict multiple outputs at the same time, with each of the tasks helping in the execution of the other tasks, i.e., learning from multiple tasks. This contrasts with transfer learning where the tasks are done sequentially, with the previous task helping in the execution of the next.

Multi-task Learning in Deep Learning

When analyzing the scale distribution of an output $$o_i$$ for a fully connected layer without nonlinearities, the output is computed as $$o_i = \sum_{j=1}^{n_	extrm{in}} w_{ij} x_j$$. Assuming the inputs $$x_j$$ and weights $$w_{ij}$$ are drawn independently with a mean of $$0$$ and variances of $$\gamma^2$$ and $$\sigma^2$$ respectively, the expected value $$E[o_i]$$ is $$0$$. We can compute the variance as $$	extrm{Var}[o_i] = E[o_i^2] - (E[o_i])^2 = \sum_{j=1}^{n_	extrm{in}} E[w^2_{ij} x^2_j] - 0 = \sum_{j=1}^{n_	extrm{in}} E[w^2_{ij}] E[x^2_j] = n_	extrm{in} \sigma^2 \gamma^2$$. Note that the distribution does not have to be Gaussian, but the mean and variance must exist. To keep this variance fixed during forward propagation and prevent it from changing across layers, the initialization must satisfy the condition $$n_	extrm{in} \sigma^2 = 1$$.

Variance of Layer Output in Forward Propagation

When building a neural network, if a specific parameter initialization method is not explicitly defined by the user, the deep learning framework will apply a default random initialization method. This default approach, such as drawing weight values from a standard normal distribution, is often sufficient and works well in practice for models with moderate problem sizes.

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