# Neural Networks

 Revision as of 06:04, 26 February 2011 (view source)Ang (Talk | contribs)← Older edit Latest revision as of 19:38, 6 April 2013 (view source)Wikiroot (Talk | contribs) Line 8: Line 8: diagram to denote a single neuron: diagram to denote a single neuron: - [[Image:SingleNeuron.png|400px|center]] + [[Image:SingleNeuron.png|300px|center]] This "neuron" is a computational unit that takes as input $x_1, x_2, x_3$ (and a +1 intercept term), and This "neuron" is a computational unit that takes as input $x_1, x_2, x_3$ (and a +1 intercept term), and - outputs $h_{W,b}(x) = f(W^Tx) = f(\sum_{i=1}^3 W_{i}x_i +b)$, where $f : \Re \mapsto \Re$ is + outputs $\textstyle h_{W,b}(x) = f(W^Tx) = f(\sum_{i=1}^3 W_{i}x_i +b)$, where $f : \Re \mapsto \Re$ is called the '''activation function'''.  In these notes, we will choose called the '''activation function'''.  In these notes, we will choose $f(\cdot)$ to be the sigmoid function: $f(\cdot)$ to be the sigmoid function: Line 31: Line 31: - +
The $\tanh(z)$ function is a rescaled version of the sigmoid, and its output range is The $\tanh(z)$ function is a rescaled version of the sigmoid, and its output range is $[-1,1]$ instead of $[0,1]$. $[-1,1]$ instead of $[0,1]$. - Note that unlike CS221 and (parts of) CS229, we are not using the convention + Note that unlike some other venues (including the OpenClassroom videos, and parts of CS229), we are not using the convention here of $x_0=1$.  Instead, the intercept term is handled separately by the parameter $b$. here of $x_0=1$.  Instead, the intercept term is handled separately by the parameter $b$. Line 49: Line 50: - == Neural Network formulation == + == Neural Network model == - + 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 + "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: Line 59: Line 59: 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 +1'' are called {\bf bias units}, and correspond to the intercept term. + labeled "+1" are called '''bias units''', and correspond to the intercept term. - The leftmost layer of the network is called the {\bf input layer}, and the + The leftmost layer of the network is called the '''input layer''', and the - rightmost layer the {\bf output layer} (which, in this example, has only one + rightmost layer the '''output layer''' (which, in this example, has only one - node).  The middle layer of nodes is called the {\bf hidden layer}, because its + node).  The middle layer of nodes is called the '''hidden layer''', because its values are not observed in the training set.  We also say that our example 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 + neural network has 3 '''input units''' (not counting the bias unit), 3 - hidden units}, and 1 {\bf output unit}. + '''hidden units''', and 1 '''output unit'''. We will let $n_l$ We will let $n_l$ Line 79: Line 79: the value +1.  We also let $s_l$ denote the number of nodes in layer $l$ (not counting the bias unit). the value +1.  We also let $s_l$ denote the number of nodes in layer $l$ (not counting the bias unit). - We will write $a^{(l)}_i$ to denote the {\bf activation} (meaning output value) of + We will write $a^{(l)}_i$ to denote the '''activation''' (meaning output value) of unit $i$ in layer $l$.  For $l=1$, we also use $a^{(1)}_i = x_i$ to denote the $i$-th input. unit $i$ in layer $l$.  For $l=1$, we also use $a^{(1)}_i = x_i$ to denote the $i$-th input. Given a fixed setting of Given a fixed setting of Line 95: Line 95: In the sequel, we also let $z^{(l)}_i$ denote the total weighted sum of inputs to unit $i$ in layer $l$, In the sequel, we also let $z^{(l)}_i$ denote the total weighted sum of inputs to unit $i$ in layer $l$, - including the bias term (e.g., $z_i^{(2)} = \sum_{j=1}^n W^{(1)}_{ij} x_j + b^{(1)}_i$), so that + including the bias term (e.g., $\textstyle z_i^{(2)} = \sum_{j=1}^n W^{(1)}_{ij} x_j + b^{(1)}_i$), so that $a^{(l)}_i = f(z^{(l)}_i)$. $a^{(l)}_i = f(z^{(l)}_i)$. + + Note that this easily lends itself to a more compact notation.  Specifically, if we extend the + activation function $f(\cdot)$ + to apply to vectors in an element-wise fashion (i.e., + $f([z_1, z_2, z_3]) = [f(z_1), f(z_2), f(z_3)]$), then we can write + the equations above more + compactly as: + :\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} + We call this step '''forward propagation.'''  More generally, recalling that we also use $a^{(1)} = x$ to also denote the values from the input layer, + then given layer $l$'s activations $a^{(l)}$, we can compute layer $l+1$'s activations $a^{(l+1)}$ as: + :\begin{align} + z^{(l+1)} &= W^{(l)} a^{(l)} + b^{(l)} \\ + a^{(l+1)} &= f(z^{(l+1)}) + \end{align} + 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. + + + 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 $\textstyle n_l$-layered network + where layer $\textstyle 1$ is the input layer, layer $\textstyle n_l$ is the output layer, and each + layer $\textstyle l$ is densely connected to layer $\textstyle l+1$.  In this setting, to compute the + output of the network, we can successively compute all the activations in layer + $\textstyle L_2$, then layer $\textstyle L_3$, and so on, up to layer $\textstyle L_{n_l}$, 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. + + + Neural networks can also have multiple output units.  For example, here is a network + with two hidden layers layers $L_2$ and $L_3$ and two output units in layer $L_4$: + + [[Image:Network3322.png|500px|center]] + + To train this network, we would need training examples $(x^{(i)}, y^{(i)})$ + where $y^{(i)} \in \Re^2$.  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 $x$ might give the input features of a + patient, and the different outputs $y_i$'s might indicate presence or absence + of different diseases.) + + + {{Sparse_Autoencoder}} + + + {{Languages|神经网络|中文}}