Recurrent Neural Networks

A typical RNN has a short-term memory. It accomplishes this by using loops in the network of nodes. Here's a simple diagram:

    --(input)-->(node with loop)--(output)-->
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The loop allows information to be passed from one step in the network to the next. A delay parameter determines how many steps in the past will influence future operations.

An RNN receives an input vector and outputs a vector . The RNN also maintains hidden vectors . Each element in the input sequence is processed one at a time.

Here is how it works:

1. For each input , update the hidden state based on the previous hidden state and current input . The function f can be any function, such as Tanh or ReLU.

h_i = f(U*x_i + W*h_{i-1})
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2. Calculate output based on updated hidden state.

y_i = softmax(V*h_i)
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In these equations:

• is the weight matrix for inputs ,
• is the weight matrix for hidden states ,
• is usually a square matrix that connects the hidden layers to themselves,

The Softmax function ensures that we get probabilities as outputs.

Training an RNN involves finding parameters (, , and ) that minimize error on our training data. This is done using backpropagation through time (BPTT), which is essentially applying backpropagation on an unfolded version of the recurrent network.

Despite their flexibility and power, RNNs have a major drawback: they are difficult to train effectively. This is due to the so-called vanishing gradient problem, which makes it hard for an RNN to learn and tune its parameters when the sequences are long.

The vanishing gradient problem refers to the situation where gradients computed during the training phase tend to vanish as they are propagated back in time. This means that learning can be slow or even stop completely, and this can prevent RNNs from effectively learning long-range dependencies in the data.

To overcome this limitation, more advanced types of RNNs such as Long Short-Term Memory and Gated Recurrent Units (GRU) have been developed. These models include mechanisms that allow them to better propagate gradients and learn from data where relevant events have large gaps.

RNNs have been successfully applied in various fields including:

1. Natural Language Processing (NLP): RNNs are used for tasks like language modeling, machine translation, speech recognition etc.
2. Time series prediction: They can be used for predicting future values of stock market, weather etc.
3. Music Composition: They can generate music by predicting next notes based on previous ones.
4. Image Captioning: They can generate captions for images by processing sequences of words and visual information together.