If we initialize $W$s as a matrix of 0s, then $\frac{d\mathcal L}{dz}$ in each neuron from each layer would get the same value and no matter how long you train the network, gradient descent would get stuck at the same point. This is usually called the symmetry breaking problem.

What happens if we initialize the weights of a feed forward network to 0s?

Suppose you have built a neural network. You decide to initialize the weights and biases to be zero. Which of the following statements is true?

If all parameters of a hidden layer are initialized to a constant, such as $$\mathbf{W}^{(1)} = c$$, every hidden unit will receive the same inputs and parameters, producing identical activations during forward propagation. Consequently, during backpropagation, the gradients of the output with respect to the parameters $$\mathbf{W}^{(1)}$$ will all take the exact same value. Because gradient-based algorithms like minibatch stochastic gradient descent update the parameters using these uniform gradients, all elements of $$\mathbf{W}^{(1)}$$ will continue to have identical values after every iteration. The hidden layer will thus behave as if it has only a single unit, failing to realize the network's expressive power.

The Problem with Constant Initialization

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.

University of Michigan - Ann Arbor

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

https://machinelearning.wtf/terms/symmetry-breaking/#:~:text=Symmetry%20breaking%20refer%20to%20a,This%20is%20the%20%E2%80%9Csymmetry%E2%80%9D.

Symmetry breaking


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 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.

Default Random Initialization

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.

Learn Before

Related

Learn After