The proof for the general representation of gradient with respect to 'W' in recurrent neural networks involves understanding the backpropagation through time (BPTT) algorithm. Here's a step-by-step explanation:

1. Define the network: A recurrent neural network (RNN) is a type of artificial neural network where connections between nodes form a directed graph along a temporal sequence. This allows it to exhibit temporal dynamic behavior. The weight matrix 'W' is what we want to optimize.

2. Forward pass: During the forward pass, the RNN computes and stores all the hidden states h(t) and outputs y(t) for t = 1 to T, where T is the sequence length.

3. Compute the loss: The loss for a single input (x, y) pair is computed as L = Σ L(t), where L(t) is the loss at each time step, computed as the squared difference between the predicted output y(t) and the actual output.

4. Backward pass (BPTT): The gradient of the loss L with respect to the weights W is computed by applying the chain rule of differentiation. This involves computing the gradient of the loss at each time step with respect to the hidden state (dh(t)/dW), and then summing these gradients.

5. Gradient computation: The gradient dh(t)/dW can be computed as dh(t)/dh(t-1) * dh(t-1)/dW. The term dh(t)/dh(t-1) can be computed directly from the RNN equations, and dh(t-1)/dW is the gradient computed at the previous time step.

6. Update the weights: Once the gradient is computed, the weights can be updated using a gradient descent step: W = W - α * dL/dW, where α is the learning rate.

This is a high-level overview of the process. The actual computations involve dealing with matrices and vectors, and require a good understanding of calculus and linear algebra. The key point is that the gradient is computed by backpropagating the error through the network, and this involves summing the contributions from all time steps.

Question

The proof for the general representation of gradient with respect to 'W' in recurrent neural networks involves understanding the backpropagation through time (BPTT) algorithm. Here's a step-by-step explanation:

1. Define the network: A recurrent neural network (RNN) is a type of artificial neural network where connections between nodes form a directed graph along a temporal sequence. This allows it to exhibit temporal dynamic behavior. The weight matrix 'W' is what we want to optimize.

2. Forward pass: During the forward pass, the RNN computes and stores all the hidden states h(t) and outputs y(t) for t = 1 to T, where T is the sequence length.

3. Compute the loss: The loss for a single input (x, y) pair is computed as L = Σ L(t), where L(t) is the loss at each time step, computed as the squared difference between the predicted output y(t) and the actual output.

4. Backward pass (BPTT): The gradient of the loss L with respect to the weights W is computed by applying the chain rule of differentiation. This involves computing the gradient of the loss at each time step with respect to the hidden state (dh(t)/dW), and then summing these gradients.

5. Gradient computation: The gradient dh(t)/dW can be computed as dh(t)/dh(t-1) * dh(t-1)/dW. The term dh(t)/dh(t-1) can be computed directly from the RNN equations, and dh(t-1)/dW is the gradient computed at the previous time step.

6. Update the weights: Once the gradient is computed, the weights can be updated using a gradient descent step: W = W - α * dL/dW, where α is the learning rate.

This is a high-level overview of the process. The actual computations involve dealing with matrices and vectors, and require a good understanding of calculus and linear algebra. The key point is that the gradient is computed by backpropagating the error through the network, and this involves summing the contributions from all time steps.

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