Exploding/vanshing gradients is a common problem in reccurrent neural networks. Consider a network where an input is multiplied by a matrix $\bold{W}$ $t$ times. Let $\bold{W}^t$ have eigendecomposition $\bold{V}\text{diag}(\bold{\lambda})^t\bold{V}^{-1}$.

We can see that if $\lambda>1$, our result will approach $\infty$ as $t$ gets large. This is an exploding gradient. Also, if $\lambda <1$, the result will approach $0$ as $t$ gets large. This is a vanishing gradient.

University of Connecticut

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

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

- Identity RNN with ReLU activation (solving vanishing gradient problem only)
- Gradient clipping
- Skip connections
- LSTM
- GRU

Solutions for vanishing/exploding gradient

A webpage explaining exploding gradients

https://machinelearningmastery.com/exploding-gradients-in-neural-networks/

A Gentle Introduction to Exploding Gradients in Neural Networks

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.

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