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The problem of vanishing and exploding gradients is a common issue in the training of deep neural networks, especially in gradient-based optimization algorithms like stochastic gradient descent (SGD) or variants of it. This problem arises when the gradients of the loss function with respect to the model parameters become very small (vanishing) or very large (exploding) as they are backpropagated through the layers of the network during training.

Vanishing gradients occur when the gradients of the loss function with respect to the parameters of the model become extremely small as they are backpropagated through the network. This means that the weights of the early layers in the network are updated very little, and as a result, these layers may not learn meaningful representations.

The causes of vanishing gradients are:

1. Activation Functions: Certain activation functions, such as the sigmoid or hyperbolic tangent (tanh) functions, squash their input into a small range. When the network is deep, the repeated application of these functions can lead to very small gradients, causing the vanishing gradient problem.

2. Weight Initialization: Poor choices for weight initialization can exacerbate the vanishing gradient problem. If weights are initialized to very small values, the gradients during backpropagation can quickly become negligible.

3. Deep Networks: The deeper the network, the more likely it is to suffer from vanishing gradients. As gradients are backpropagated through numerous layers, their values can diminish exponentially.

The consequences of vanishing gradients are:

• Layers early in the network may not learn meaningful features.
• Training might be very slow, or the model may not converge at all.

The strategies for mitigating exploding gradients are:

1. Weight Initialization: Careful initialization of weights, such as using techniques like He initialization, can help alleviate the vanishing gradient problem.

2. Activation Functions: ReLU (Rectified Linear Unit) and its variants are popular choices for activation functions, as they do not suffer from the vanishing gradient problem to the same extent as sigmoid or tanh.

3. Batch Normalization: Normalizing the inputs to each layer can help mitigate both vanishing and exploding gradients.

4. Proper Learning Rate: Adjust the learning rate appropriately to avoid both vanishing and exploding gradients.

5. LSTM and GRU Networks: For sequences, especially in natural language processing, Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) networks are designed to mitigate the vanishing gradient problem in recurrent neural networks.

Given a neural network with the output,

          ------------------------------- [3715a]

where,

          x is the input to the network.

          W^[L] is a 2x2 matrix:

                 ------------------------------- [3715b]

Then, the output y^ ​ is the result of multiplying several weight matrices together.

Vanishing gradients occur when the gradients become very small during backpropagation, leading to negligible updates for the weights. In the given network, if the entries of the weight matrix W^[L] are such that they are less than 1, and if this pattern continues through the earlier layers, the gradients can diminish exponentially as they are backpropagated through the network.

For example, if a = 0.2 , b = 0, c = 0, d = 0.2 and L = 10, the repeated multiplication of these small values during backpropagation can result in vanishing gradients, particularly for the weights in the earlier layers. In this case, we have,

              ------------------------------- [3715c]

Since a =0.2 and d = 0.2 are both less than 1, each multiplication by the weight matrix W^[L] will scale down the values in the input. After L layers, this can lead to the vanishing gradient problem.

Assuming all the weight matrices W^[1], W^[2], … , W^[L] are identical with the given values a = 0.2, b = 0, c = 0, and d = 0.2, then the network simplifies, and the output y^ ​ can be expressed as follows:

              ------------------------------- [3715d]

Then,

              ------------------------------- [3715e]

This means that after passing through 10 layers, the input x is scaled down significantly, contributing to the vanishing gradient problem during backpropagation. The network may struggle to learn meaningful representations, especially in the earlier layers. The gradients with respect to the weights in those layers are likely to be very small, making the training process challenging. Mitigation strategies, such as weight initialization techniques, activation functions like ReLU, and normalization methods, become important to address the vanishing gradient problem and facilitate effective training in deep neural networks.

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