What is universal approximation theorem? How can real-time recurrent learning be achieved.

The Universal Approximation Theorem is a fundamental theory in the field of machine learning and neural networks. It states that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact subsets of R^n, under mild assumptions on the activation function. The theorem is instrumental in the sense that it provides a guarantee that neural networks can represent a wide variety of interesting functions when given appropriate parameters.

However, it's important to note that while the theorem tells us that a neural network can represent such functions, it does not tell us how to train the network to achieve this. It also doesn't say anything about how many neurons are needed in the hidden layer to approximate a given function.

Real-time recurrent learning (RTRL) is a method for training recurrent neural networks. Here's a step-by-step explanation:

1. Initialize the weights in the network randomly.
2. For each time step in your input sequence, do the following:
   • Compute the output of the network for the current input.
   • Compute the error of the network, which is the difference between the network's output and the desired output.
   • Update the weights in the network to reduce the error. This is done by computing the gradient of the error with respect to the weights, and then adjusting the weights in the direction that reduces the error.
3. Repeat step 2 for many iterations, until the network's performance is satisfactory.

RTRL is an online learning algorithm, meaning it updates the weights continuously as each new input comes in, rather than batching inputs and updating weights once per batch. This makes it suitable for tasks where you need to process data in real-time. However, RTRL is computationally expensive because it requires computing and storing the gradients for all weights for each time step.