Neural Networks
From Ufldl
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Here are plots of the sigmoid and <math>\tanh</math> functions: | Here are plots of the sigmoid and <math>\tanh</math> functions: | ||
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| - | + | [[Image:Sigmoid_Function.png|400px|center|Sigmoid activation function.]] | |
| - | + | [[Image:Tanh_Function.png|400px|center|Tanh activation function.]] | |
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| - | + | The <math>\tanh(z)</math> function is a rescaled version of the sigmoid, and its output range is | |
| - | + | <math>[-1,1]</math> instead of <math>[0,1]</math>. | |
| - | + | ||
| - | + | Note that unlike CS221 and (parts of) CS229, we are not using the convention | |
| + | here of <math>x_0=1</math>. Instead, the intercept term is handled separately by the parameter <math>b</math>. | ||
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| + | Finally, one identity that'll be useful later: If <math>f(z) = 1/(1+\exp(-z))</math> is the sigmoid | ||
| + | function, then its derivative is given by <math>f'(z) = f(z) (1-f(z))</math>. | ||
| + | (If <math>f</math> is the tanh function, then its derivative is given by | ||
| + | <math>f'(z) = 1- (f(z))^2</math>.) You can derive this yourself using the definition of | ||
| + | the sigmoid (or tanh) function. | ||
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| + | == Neural Network formulation == | ||
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| + | A neural network is put together by hooking together many of our simple | ||
| + | ``neurons,'' so that the output of a neuron can be the input of another. For | ||
| + | example, here is a small neural network: | ||
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| + | [[Image:Network331.png|400px|center]] | ||
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| + | In this figure, we have used circles to also denote the inputs to the network. The circles | ||
| + | labeled ``+1'' are called {\bf bias units}, and correspond to the intercept term. | ||
| + | The leftmost layer of the network is called the {\bf input layer}, and the | ||
| + | rightmost layer the {\bf output layer} (which, in this example, has only one | ||
| + | node). The middle layer of nodes is called the {\bf hidden layer}, because its | ||
| + | values are not observed in the training set. We also say that our example | ||
| + | neural network has 3 {\bf input units} (not counting the bias unit), 3 {\bf | ||
| + | hidden units}, and 1 {\bf output unit}. | ||
| + | |||
| + | We will let <math>n_l</math> | ||
| + | denote the number of layers in our network; thus <math>n_l=3</math> in our example. We label layer <math>l</math> as | ||
| + | <math>L_l</math>, so layer <math>L_1</math> is the input layer, and layer <math>L_{n_l}</math> the output layer. | ||
| + | Our neural network has parameters <math>(W,b) = (W^{(1)}, b^{(1)}, W^{(2)}, b^{(2)})</math>, where | ||
| + | we write | ||
| + | <math>W^{(l)}_{ij}</math> to denote the parameter (or weight) associated with the connection | ||
| + | between unit <math>j</math> in layer <math>l</math>, and unit <math>i</math> in layer <math>l+1</math>. (Note the order of the indices.) | ||
| + | Also, <math>b^{(l)}_i</math> is the bias associated with unit <math>i</math> in layer <math>l+1</math>. | ||
| + | Thus, in our example, we have <math>W^{(1)} \in \Re^{3\times 3}</math>, and <math>W^{(2)} \in \Re^{1\times 3}</math>. | ||
| + | Note that bias units don't have inputs or connections going into them, since they always output | ||
| + | the value +1. We also let <math>s_l</math> denote the number of nodes in layer <math>l</math> (not counting the bias unit). | ||
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| + | We will write <math>a^{(l)}_i</math> to denote the {\bf activation} (meaning output value) of | ||
| + | unit <math>i</math> in layer <math>l</math>. For <math>l=1</math>, we also use <math>a^{(1)}_i = x_i</math> to denote the <math>i</math>-th input. | ||
| + | Given a fixed setting of | ||
| + | the parameters <math>W,b</math>, our neural | ||
| + | network defines a hypothesis <math>h_{W,b}(x)</math> that outputs a real number. Specifically, the | ||
| + | computation that this neural network represents is given by: | ||
| + | :<math> | ||
| + | \begin{align} | ||
| + | a_1^{(2)} &= f(W_{11}^{(1)}x_1 + W_{12}^{(1)} x_2 + W_{13}^{(1)} x_3 + b_1^{(1)}) \\ | ||
| + | a_2^{(2)} &= f(W_{21}^{(1)}x_1 + W_{22}^{(1)} x_2 + W_{23}^{(1)} x_3 + b_2^{(1)}) \\ | ||
| + | a_3^{(2)} &= f(W_{31}^{(1)}x_1 + W_{32}^{(1)} x_2 + W_{33}^{(1)} x_3 + b_3^{(1)}) \\ | ||
| + | h_{W,b}(x) &= a_1^{(3)} = f(W_{11}^{(2)}a_1^{(2)} + W_{12}^{(2)} a_2^{(2)} + W_{13}^{(2)} a_3^{(2)} + b_1^{(2)}) | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | In the sequel, we also let <math>z^{(l)}_i</math> denote the total weighted sum of inputs to unit <math>i</math> in layer <math>l</math>, | ||
| + | including the bias term (e.g., <math>z_i^{(2)} = \sum_{j=1}^n W^{(1)}_{ij} x_j + b^{(1)}_i</math>), so that | ||
| + | <math>a^{(l)}_i = f(z^{(l)}_i)</math>. | ||
