annotate weights.py @ 516:2b0e10ac6929

misc
author Olivier Breuleux <breuleuo@iro.umontreal.ca>
date Mon, 03 Nov 2008 00:10:18 -0500
parents 4f3c66146f17
children
rev   line source
466
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
1 """
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
2 Routine to initialize weights.
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
3
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
4 @note: We assume that numpy.random.seed() has already been performed.
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
5 """
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
6
484
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
7 from math import pow, sqrt
466
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
8 import numpy.random
484
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
9
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
10 sqrt3 = sqrt(3.0)
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
11 def random_weights(nin, nout, scale_by=1./sqrt3, power=0.5):
466
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
12 """
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
13 Generate an initial weight matrix with nin inputs (rows) and nout
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
14 outputs (cols).
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
15 Each weight is chosen uniformly at random to be in range:
484
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
16 [-scale_by*sqrt(3)/pow(nin,power), +scale_by*sqrt(3)/pow(nin,power)]
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
17 @note: Play with scale_by, but reasonable values are <=1, maybe 1./sqrt3
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
18 power=0.5 is strongly recommanded (see below).
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
19
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
20 Suppose these weights w are used in dot products as follows:
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
21 output = w' input
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
22 If w ~ Uniform(-r,r) and Var[input_i]=1 and x_i's are independent, then
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
23 Var[w]=r2/3
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
24 Var[output] = Var[ sum_{i=1}^d w_i input_i] = d r2 / 3
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
25 To make sure that variance is not changed after the dot product,
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
26 we therefore want Var[output]=1 and r = sqrt(3)/sqrt(d). This choice
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
27 corresponds to the default values scale_by=sqrt(3) and power=0.5.
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
28 More generally we see that Var[output] = Var[input] * scale_by.
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
29
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
30 Now, if these are weights in a deep multi-layer neural network,
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
31 we would like the top layers to be initially more linear, so as to let
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
32 gradients flow back more easily (this is an explanation by Ronan Collobert).
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
33 To achieve this we want scale_by smaller than 1.
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
34 Ronan used scale_by=1/sqrt(3) (by mistake!) and got better results than scale_by=1
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
35 in the experiment of his ICML'2008 paper.
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
36 Note that if we have a multi-layer network, ignoring the effect of the tanh non-linearity,
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
37 the variance of the layer outputs would go down roughly by a factor 'scale_by' at each
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
38 layer (making the layers more linear as we go up towards the output).
466
23221eefb70e Added pylearn.sandbox.weights.random_weights
Joseph Turian <turian@iro.umontreal.ca>
parents:
diff changeset
39 """
484
3daabc7f94ff Added Yoshua's explanation
Joseph Turian <turian@gmail.com>
parents: 466
diff changeset
40 return (numpy.random.rand(nin, nout) * 2.0 - 1) * scale_by * sqrt3 / pow(nin,power)