Neural Networks
From Ufldl
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diagram to denote a single neuron: | diagram to denote a single neuron: | ||
| - | [[Image:SingleNeuron.png| | + | [[Image:SingleNeuron.png|300px|center]] |
This "neuron" is a computational unit that takes as input <math>x_1, x_2, x_3</math> (and a +1 intercept term), and | This "neuron" is a computational unit that takes as input <math>x_1, x_2, x_3</math> (and a +1 intercept term), and | ||
| - | outputs <math>h_{W,b}(x) = f(W^Tx) = f(\sum_{i=1}^3 W_{i}x_i +b)</math>, where <math>f : \Re \mapsto \Re</math> is | + | outputs <math>\textstyle h_{W,b}(x) = f(W^Tx) = f(\sum_{i=1}^3 W_{i}x_i +b)</math>, where <math>f : \Re \mapsto \Re</math> is |
called the '''activation function'''. In these notes, we will choose | called the '''activation function'''. In these notes, we will choose | ||
<math>f(\cdot)</math> to be the sigmoid function: | <math>f(\cdot)</math> to be the sigmoid function: | ||
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| - | + | <div align=center> | |
| - | [[Image:Sigmoid_Function.png|400px| | + | [[Image:Sigmoid_Function.png|400px|top|Sigmoid activation function.]] |
| - | [[Image:Tanh_Function.png|400px| | + | [[Image:Tanh_Function.png|400px|top|Tanh activation function.]] |
| + | </div> | ||
The <math>\tanh(z)</math> function is a rescaled version of the sigmoid, and its output range is | 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>. | <math>[-1,1]</math> instead of <math>[0,1]</math>. | ||
| - | Note that unlike | + | Note that unlike some other venues (including the OpenClassroom videos, 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>. | here of <math>x_0=1</math>. Instead, the intercept term is handled separately by the parameter <math>b</math>. | ||
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| - | == Neural Network | + | == Neural Network model == |
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A neural network is put together by hooking together many of our simple | 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: | example, here is a small neural network: | ||
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In this figure, we have used circles to also denote the inputs to the network. The circles | In this figure, we have used circles to also denote the inputs to the network. The circles | ||
| - | labeled | + | labeled "+1" are called '''bias units''', and correspond to the intercept term. |
The leftmost layer of the network is called the '''input layer''', and the | The leftmost layer of the network is called the '''input layer''', and the | ||
rightmost layer the '''output layer''' (which, in this example, has only one | rightmost layer the '''output layer''' (which, in this example, has only one | ||
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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>, | 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 | + | including the bias term (e.g., <math>\textstyle 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>. | <math>a^{(l)}_i = f(z^{(l)}_i)</math>. | ||
| + | |||
| + | Note that this easily lends itself to a more compact notation. Specifically, if we extend the | ||
| + | activation function <math>f(\cdot)</math> | ||
| + | to apply to vectors in an element-wise fashion (i.e., | ||
| + | <math>f([z_1, z_2, z_3]) = [f(z_1), f(z_2), f(z_3)]</math>), then we can write | ||
| + | the equations above more | ||
| + | compactly as: | ||
| + | :<math>\begin{align} | ||
| + | z^{(2)} &= W^{(1)} x + b^{(1)} \\ | ||
| + | a^{(2)} &= f(z^{(2)}) \\ | ||
| + | z^{(3)} &= W^{(2)} a^{(2)} + b^{(2)} \\ | ||
| + | h_{W,b}(x) &= a^{(3)} = f(z^{(3)}) | ||
| + | \end{align}</math> | ||
| + | We call this step '''forward propagation.''' More generally, recalling that we also use <math>a^{(1)} = x</math> to also denote the values from the input layer, | ||
| + | then given layer <math>l</math>'s activations <math>a^{(l)}</math>, we can compute layer <math>l+1</math>'s activations <math>a^{(l+1)}</math> as: | ||
| + | :<math>\begin{align} | ||
| + | z^{(l+1)} &= W^{(l)} a^{(l)} + b^{(l)} \\ | ||
| + | a^{(l+1)} &= f(z^{(l+1)}) | ||
| + | \end{align}</math> | ||
| + | By organizing our parameters in matrices and using matrix-vector operations, we can take | ||
| + | advantage of fast linear algebra routines to quickly perform calculations in our network. | ||
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| + | We have so far focused on one example neural network, but one can also build neural | ||
| + | networks with other '''architectures''' (meaning patterns of connectivity between neurons), including ones with multiple hidden layers. | ||
| + | The most common choice is a <math>\textstyle n_l</math>-layered network | ||
| + | where layer <math>\textstyle 1</math> is the input layer, layer <math>\textstyle n_l</math> is the output layer, and each | ||
| + | layer <math>\textstyle l</math> is densely connected to layer <math>\textstyle l+1</math>. In this setting, to compute the | ||
| + | output of the network, we can successively compute all the activations in layer | ||
| + | <math>\textstyle L_2</math>, then layer <math>\textstyle L_3</math>, and so on, up to layer <math>\textstyle L_{n_l}</math>, using the equations above that describe the forward propagation step. This is one | ||
| + | example of a '''feedforward''' neural network, since the connectivity graph | ||
| + | does not have any directed loops or cycles. | ||
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| + | Neural networks can also have multiple output units. For example, here is a network | ||
| + | with two hidden layers layers <math>L_2</math> and <math>L_3</math> and two output units in layer <math>L_4</math>: | ||
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| + | [[Image:Network3322.png|500px|center]] | ||
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| + | To train this network, we would need training examples <math>(x^{(i)}, y^{(i)})</math> | ||
| + | where <math>y^{(i)} \in \Re^2</math>. This sort of network is useful if there're multiple | ||
| + | outputs that you're interested in predicting. (For example, in a medical | ||
| + | diagnosis application, the vector <math>x</math> might give the input features of a | ||
| + | patient, and the different outputs <math>y_i</math>'s might indicate presence or absence | ||
| + | of different diseases.) | ||
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| + | {{Sparse_Autoencoder}} | ||
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| + | {{Languages|神经网络|中文}} | ||
