I did watch the course DeepLearning of Andrew Ng and he told that we should create parameter w small like:

parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[l - 1]) ** 0.001 

But in the last application assignment. They choose another way:

layers_dims = [12288, 20, 7, 5, 1]
def initialize_parameters_deep(layer_dims):
    parameters = {}
    L = len(layer_dims)

    for l in range(1, L):
        parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[l - 1]) / np.sqrt(layer_dims[l - 1])
        parameters['b' + str(l)] = np.zeros((layer_dims[l], 1))
        assert (parameters['W' + str(l)].shape == (layer_dims[l], layer_dims[l - 1]))
        assert (parameters['b' + str(l)].shape == (layer_dims[l], 1))
    return parameters

And the result of this way is very good but if I choose w like the old above, It's just have 34% correct!

So do you can explain ?

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    – nbro
    Apr 14 '20 at 13:19

Weights initialisation is strictly related to the vanishing/exploding gradient problem. For a complete explanation, please check this awesome page (also from deeplearning.ai). Here I'll summarise the main concepts:

  • initialising weight all to zero will cause all weights to have the same derivative value with respect to the loss function, hence the network would be incapable of learning anything.
  • initialising to zero the biases has no drawback since they are constants (no effect at all when computing the derivative anyway).

  • initialising weights with too high or too small values will lead to an exploding gradient (oscillating gradient values without convergence) or a vanishing gradient (small gradient values variation that converge before reaching the loss global minimum).

In order to avoid these problems, a method called Xavier initialisation has been proposed: the weights should be initialised in such a way that they will generate activations scores with a distribution that has:

  • Mean 0
  • Constant variance across layers (i.e. no vanishing/exploding)

The value "np.sqrt(layer_dims[l - 1])" pup up when imposing the second constrain. For a formal prove check the page I linked. To grasp the concept just focus on the fact that the variance of the weights of a layer depends on the amount of nodes of the previous layer. This mean that for layers preceded by layers with a big amount of hidden nodes, the weights will be initialised with a small variance, and this is ok cause we don't want a small bunch nodes to have stronger influence on the subsequent activations. But in layers which instead are preceded by layers with a small amount of nodes, it's ok to allow the weights to vary more.


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