def __init__(self, dim_list, eta = 0.1):
"""
Constructor for network.
Params:
dim_list: a list of the number of dimension for each layer.
eta: learning rate for each gradient descent step
"""
depth = len(dim_list)
self.depth = depth
self.dim_list = dim_list
self.eta = eta
# 1. Initiate each layer: output, partial_output and weight,
# although partial_output is useless for the input layer, similarly
# weight and bias are useless for the output layer.
#
# 2. Partial_weight is an internal variable and will not be stored in
# a layer.
#
self.layers = [ {'output':Vector.fromIterable(0 for i in xrange(dim_list[l])),
'partial_output':Vector.fromIterable(0 for i in xrange(dim_list[l])),
'weight':Matrix.fromRandom(dim_list[l + 1], dim_list[l]),
'bias':Vector.fromRandom(dim_list[l + 1])}
for l in xrange(depth - 1) ]
# output layer
self.layers.append({'output':Vector.fromList([0] * dim_list[depth - 1]),
'partial_output':Vector.fromList([0] * dim_list[depth - 1]),
'weight': None, 'bias': None})
def _backward(self, x, y):
# output layer
layer_id = self.depth - 1
output = self.layers[layer_id]['output']
self.layers[layer_id]['partial_output'].assign(Vector.fromIterable(
output[i] - y[i] for i in xrange(self.dim_list[layer_id])
))
loss = sum((output[i] - y[i]) ** 2 for i in xrange(self.dim_list[layer_id]))
# hidden layer and input layer
for layer_id in xrange(self.depth - 2, -1, -1):
weight = self.layers[layer_id]['weight']
bias = self.layers[layer_id]['bias']
partial_output = self.layers[layer_id]['partial_output']
output = self.layers[layer_id]['output']
last_output = self.layers[layer_id + 1]['output']
last_partial = self.layers[layer_id + 1]['partial_output']
"""
Partial output for every layer except the output one is:
\frac {\partial E} {\partial O_k^{(l)}} =
\sum_i (\frac {\partial E} {\partial O_i^{ (l+1) }}
* O_i^{ (l+1) } * (1 - O_i^{ (l+1) }) * w_{ik}^{ (l) } )
But the partial output of the first layer is unnecessary,
thus we don't compute it.
"""
if layer_id > 0:
self.layers[layer_id]['partial_output'].assign(Vector.fromIterable(
sum(last_partial[i] * last_output[i] * (1 - last_output[i])
* weight.item(i, k)
for i in xrange(self.dim_list[layer_id + 1]))
/ (self.dim_list[layer_id] + 1.0)
for k in xrange(self.dim_list[layer_id] )))
"""
Partial weight for every layer except the output one:
\frac {\partial E} {\partial w_{ji}^{(l)}} =
\frac {\partial E} {\partial O_j^{(l + 1)}}
* O_j^{(l + 1)} * (1 - O_j^{(l+1)}) * O_i^{(l)}
"""
partial_weight = Matrix.fromIterable(weight.row_num, weight.col_num, (
last_partial[row_id]
* last_output[row_id] * (1 - last_output[row_id])
* output[col_id]
/ (self.dim_list[layer_id] + 1.0)
for row_id in xrange(self.dim_list[layer_id + 1])
for col_id in xrange(self.dim_list[layer_id])
))
self.layers[layer_id]['weight'] -= self.eta * partial_weight
"""
Partial bias is almost exact as the partial weight,
but for every item in the bias vector the last item is 1
\frac {\partial E}{\partial b_j^{(l)}} =
\frac {\partial E}{\partial O_j^{(l + 1)}}
O_j^{(l + 1)} (1 - O_j^{(l+1)}) * 1
"""
partial_bias = Vector.fromIterable(
last_partial[row_id]
* last_output[row_id] * (1 - last_output[row_id]) * 1
/ (self.dim_list[layer_id] + 1.0)
for row_id in xrange(self.dim_list[layer_id + 1])
)
self.layers[layer_id]['bias'] -= self.eta * partial_bias
return loss