What is the role of the bias in neural networks

Question

I m aware of the gradient descent and the back-propagation algorithm  What I don t get is  when is using a bias important and how do you use it   For example  when mapping the AND function  when I use 2 inputs and 1 output  it does not give the correct weights  however  when I use 3 inputs  1 of which is a bias   it gives the correct weights

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Two different kinds of parameters can be adjusted during the training of an ANN, the weights and the value in the activation functions. This is impractical and it would be easier if only one of the parameters should be adjusted. To cope with this problem a bias neuron is invented. The bias neuron lies in one layer, is connected to all the neurons in the next layer, but none in the previous layer and it always emits 1. Since the bias neuron emits 1 the weights, connected to the bias neuron, are added directly to the combined sum of the other weights (equation 2.1), just like the t value in the activation functions.1

The reason it's impractical is because you're simultaneously adjusting the weight and the value, so any change to the weight can neutralize the change to the value that was useful for a previous data instance... adding a bias neuron without a changing value allows you to control the behavior of the layer.

Furthermore the bias allows you to use a single neural net to represent similar cases. Consider the AND boolean function represented by the following neural network:

_{(source: aihorizon.com)}

w0 corresponds to b.
w1 corresponds to x1.
w2 corresponds to x2.

A single perceptron can be used to represent many boolean functions.

For example, if we assume boolean values of 1 (true) and -1 (false), then one way to use a two-input perceptron to implement the AND function is to set the weights w0 = -3, and w1 = w2 = .5. This perceptron can be made to represent the OR function instead by altering the threshold to w0 = -.3. In fact, AND and OR can be viewed as special cases of m-of-n functions: that is, functions where at least m of the n inputs to the perceptron must be true. The OR function corresponds to m = 1 and the AND function to m = n. Any m-of-n function is easily represented using a perceptron by setting all input weights to the same value (e.g., 0.5) and then setting the threshold w0 accordingly.

Perceptrons can represent all of the primitive boolean functions AND, OR, NAND ( 1 AND), and NOR ( 1 OR). Machine Learning- Tom Mitchell)

The threshold is the bias and w0 is the weight associated with the bias/threshold neuron.

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The term bias is used to adjust the final output matrix as the y-intercept does  For instance  in the classic equation  y   mx   c  if c   0  then the line will always pass through 0  Adding the bias term provides more flexibility and better generalisation to our Neural Network model

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Other than mentioned answers  I would like to add some other points   Bias acts as our anchor  It s a way for us to have some kind of baseline where we don t go below that  In terms of a graph  think of like y mx b it s like a y-intercept of this function   output   input times the weight value and added a bias value and then apply an activation function

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The bias helps to get a better equation   Imagine the input and output like a function  y   ax   b and you need to put the right line between the input x  and output y  to minimise the global error between each point and the line   if you keep the equation like this y   ax   you will have one parameter for adaptation only   even if you find the best a minimising the global error it will be kind of far from the wanted value   You can say the bias makes the equation more flexible to adapt to the best values

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The bias is not an NN term  it s a generic algebra term to consider   Y   M X   C  straight line equation   Now if C Bias    0 then  the line will always pass through the origin  i e   0 0   and depends on only one parameter  i e  M  which is the slope so we have less things to play with   C  which is the bias takes any number and has the activity to shift the graph  and hence able to represent more complex situations   In a logistic regression  the expected value of the target is transformed by a link function to restrict its value to the unit interval  In this way  model predictions can be viewed as primary outcome probabilities as shown  Sigmoid function on Wikipedia  This is the final activation layer in the NN map that turns on and off the neuron  Here also bias has a role to play and it shifts the curve flexibly to help us map the model

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I think that biases are almost always helpful   In effect  a bias value allows you to shift the activation function to the left or right  which may be critical for successful learning   It might help to look at a simple example   Consider this 1-input  1-output network that has no bias     The output of the network is computed by multiplying the input  x  by the weight  w0  and passing the result through some kind of activation function  e g  a sigmoid function    Here is the function that this network computes  for various values of w0     Changing the weight w0 essentially changes the  steepness  of the sigmoid   That s useful  but what if you wanted the network to output 0 when x is 2   Just changing the steepness of the sigmoid won t really work -- you want to be able to shift the entire curve to the right   That s exactly what the bias allows you to do   If we add a bias to that network  like so        then the output of the network becomes sig w0 x   w1 1 0    Here is what the output of the network looks like for various values of w1     Having a weight of -5 for w1 shifts the curve to the right  which allows us to have a network that outputs 0 when x is 2

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In neural networks    Each Neuron has a bias You can view bias as threshold   generally opposite values of threshold  Weighted sum from input layers   bias decides activation of neuron  Bias increases the flexibility of the model    In absence of bias  the neuron may not be activated by considering only the weighted sum from input layer  If the neuron is not activated  the information from this neuron is not passed through rest of neural network    The value of bias is learn-able      Effectively  bias       threshold  You can think of bias as how easy it is to get the neuron to output a 1     with a really big bias  it   s very easy for the neuron to output a 1  but if the bias is very negative  then it   s difficult   in summary   bias helps in controlling the value at which activation function will trigger   Follow this video for more details  Few more useful links   geeksforgeeks  towardsdatascience

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Modification of neuron WEIGHTS alone only serves to manipulate the shape curvature of your transfer function  and not its equilibrium zero crossing point   The introduction of bias neurons allows you to shift the transfer function curve horizontally  left right  along the input axis while leaving the shape curvature unaltered  This will allow the network to produce arbitrary outputs different from the defaults and hence you can customize shift the input-to-output mapping to suit your particular needs   See here for graphical explanation  http   www heatonresearch com wiki Bias

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Just to add my two cents   A simpler way to understand what the bias is  it is somehow similar to the constant b of a linear function  y   ax   b  It allows you to move the line up and down to fit the prediction with the data better  Without b the line always goes through the origin  0  0  and you may get a poorer fit

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Expanding on  zfy explanation     The equation for one input  one neuron  one output should look   y   a   x   b   1    and out   f y    where x is the value from the input node and 1 is the value of the bias node  y can be directly your output or be passed into a function  often a sigmoid function  Also note that the bias could be any constant  but to make everything simpler we always pick 1  and probably that s so common that  zfy did it without showing  amp  explaining it    Your network is trying to learn coefficients a and b to adapt to your data  So you can see why adding the element b   1 allows it to fit better to more data  now you can change both slope and intercept   If you have more than one input your equation will look like   y   a0   x0   a1   x1         aN   1   Note that the equation still describes a one neuron  one output network  if you have more neurons you just add one dimension to the coefficient matrix  to multiplex the inputs to all nodes and sum back each node contribution   That you can write in vectorized format as   A    a0  a1      aN    X    x0  x1       1  Y   A   XT   i e  putting coefficients in one array and  inputs   bias  in another you have your desired solution as the dot product of the two vectors  you need to transpose X for the shape to be correct  I wrote XT a  X transposed    So in the end you can also see your bias as is just one more input to represent the part of the output that is actually independent of your input

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If you re working with images  you might actually prefer to not use a bias at all  In theory  that way your network will be more independent of data magnitude  as in whether the picture is dark  or bright and vivid  And the net is going to learn to do it s job through studying relativity inside your data  Lots of modern neural networks utilize this  For other data having biases might be critical  It depends on what type of data you re dealing with  If your information is magnitude-invariant --- if inputting  1 0 0 1  should lead to the same result as if inputting  100 0 10   you might be better off without a bias

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A layer in a neural network without a bias is nothing more than the multiplication of an input vector with a matrix   The output vector might be passed through a sigmoid function for normalisation and for use in multi-layered ANN afterwards but that   s not important    This means that you   re using a linear function and thus an input of all zeros will always be mapped to an output of all zeros  This might be a reasonable solution for some systems but in general it is too restrictive   Using a bias  you   re effectively adding another dimension to your input space  which always takes the value one  so you   re avoiding an input vector of all zeros  You don   t lose any generality by this because your trained weight matrix needs not be surjective  so it still can map to all values previously possible   2d ANN   For a ANN mapping two dimensions to one dimension  as in reproducing the AND or the OR  or XOR  functions  you can think of a neuronal network as doing the following   On the 2d plane mark all positions of input vectors  So  for boolean values  you   d want to mark  -1 -1    1 1    -1 1    1 -1   What your ANN now does is drawing a straight line on the 2d plane  separating the positive output from the negative output values   Without bias  this straight line has to go through zero  whereas with bias  you   re free to put it anywhere  So  you   ll see that without bias you   re facing a problem with the AND function  since you can   t put both  1 -1  and  -1 1  to the negative side   They are not allowed to be on the line   The problem is equal for the OR function  With a bias  however  it   s easy to draw the line   Note that the XOR function in that situation can   t be solved even with bias

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This thread really helped me developing my own project  Here are some further illustrations showing the result of a simple 2-layer feed forward neural network with and without bias units on a two-variable regression problem  Weights are initialized randomly and standard ReLU activation is used  As the answers before me concluded  without the bias the ReLU-network is not able to deviate from zero at  0 0

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To think in simple way if you have y w1 x where y is your output and w1 is the weight imagine a condition where x 0 then y w1 x equals to 0 If you want to update your weight you have to compute how much change by delw target-y where target is your target output in this case  delw  will not change since y is computed as 0 So suppose if you can add some extra value it will help y w1 x w0 1 where bias 1 and weight can be adjusted to get a correct bias Consider the example below     In terms of line Slope-intercept is a specific form of linear equations    y mx b  check the image  image  here b is  0 2   if you want to increase it to  0 3  how will you do it by changing the value of b which will be your bias

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For all the ML books I studied  the W is always defined as the connectivity index between two neurons   which means the higher connectivity between two neurons   the stronger the signals will be transmitted from the firing neuron to the target neuron or Y  w   X as  a result to maintain the biological character of neurons   we need to keep the 1   W    -1   but in the real regression  the W will end up with  W    1 which contradict with how the Neurons are working  as a result I propose W  cos theta    while 1     cos  theta     and Y  a   X   W   X   b   while a   b   W   b   cos  theta    b is an integer

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In a couple of experiments in my masters thesis  e g  page 59   I found that the bias might be important for the first layer s   but especially at the fully connected layers at the end it seems not to play a big role   This might be highly dependent on the network architecture   dataset

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Bias determines how much angle your weight will rotate  In 2-dimensional chart  weight and bias can help us to find the decision boundary of outputs  Say we need to build a AND function  the input p -output t  pair should be   p  0 0   t 0   p  1 0   t 0   p  0 1   t 0   p  1 1   t 1    Now we need to find a decision boundary  the ideal boundary should be   See  W is perpendicular to our boundary  Thus  we say W decided the direction of boundary  However  it is hard to find correct W at first time  Mostly  we choose original W value randomly  Thus  the first boundary may be this   Now the boundary is pareller to y axis  We want to rotate boundary  how  By changing the W  So  we use the learning rule function  W  W P    W  W P  is equivalent to W  W bP  while b 1   Therefore  by changing the value of b bias   you can decide the angle between W  and W  That is  quot the learning rule of ANN quot   You could also read Neural Network Design by Martin T  Hagan   Howard B  Demuth   Mark H  Beale  chapter 4  quot Perceptron Learning Rule quot

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When you use ANNs  you rarely know about the internals of the systems you want to learn  Some things cannot be learned without a bias  E g   have a look at the following data   0  1    1  1    2  1   basically a function that maps any x to 1    If you have a one layered network  or a linear mapping   you cannot find a solution  However  if you have a bias it s trivial   In an ideal setting  a bias could also map all points to the mean of the target points and let the hidden neurons model the differences from that point

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In particular  Nate   s answer  zfy   s answer  and Pradi   s answer are great   In simpler terms  biases allow for more and more variations of weights to be learnt stored     side-note  sometimes given some threshold   Anyway  more variations mean that biases add richer representation of the input space to the model s learnt stored weights   Where better weights can enhance the neural net   s guessing power     For example  in learning models  the hypothesis guess is desirably bounded by y 0 or y 1 given some input  in maybe some classification task    i e some y 0 for some x  1 1  and some y 1 for some x  0 1    The condition on the hypothesis outcome is the threshold I talked about above  Note that my examples setup inputs X to be each x a double or 2 valued-vector  instead of Nate s single valued x inputs of some collection X    If we ignore the bias  many inputs may end up being represented by a lot of the same weights  i e  the learnt weights mostly occur close to the origin  0 0    The model would then be limited to poorer quantities of good weights  instead of the many many more good weights it could better learn with bias   Where poorly learnt weights lead to poorer guesses or a decrease in the neural net   s guessing power   So  it is optimal that the model learns both close to the origin  but also  in as many places as possible inside the threshold decision boundary  With the bias we can enable degrees of freedom close to the origin  but not limited to origin s immediate region

[machine-learning] What is the role of the bias in neural networks?

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