神经网络之梯度下降法（5）
反向传播算法是多层神经网络的训练中举足轻重的算法，本文着重讲解方向传播算法的原理和推导过程。因此对于一些基本的神经网络的知识，本文不做介绍。在理解反向传播算法前，先要理解神经网络中的前馈神经网络算法。
前馈神经网络
如下图，是一个多层神经网络的简单示意图：
给定一个前馈神经网络，我们用下面的记号来描述这个网络：
：表示神经网络的层数；
：表示第层神经元的个数；
：表示层神经元的**函数；
：表示层到第层的权重矩阵；
：表示层到层的偏置；
：表示第层神经元的输入；
：表示第层神经元的输出；
前馈神经网络通过如下的公式进行信息传播：
反向传播算法
在了解前馈神经网络的结构之后，我们一前馈神经网络的信息传递过程为基础，从而推到反向传播算法。首先要明确一点，反向传播算法是为了更好更快的训练前馈神经网络，得到神经网络每一层的权重参数和偏置参数。
在推导反向传播的理论之前，首先看一幅能够直观的反映反向传播过程的图，这个图取材于Principles
of training multi-layer neural network using back propagation。
Principles of training multi-layer neural network using backpropagation
|
The project describes teaching process of multi-layer neural network employing backpropagation algorithm. To illustrate this process the three layer neural network with two inputs and one output,which is shown in the picture below, is
used: Each neuron is composed of two units. First unit adds products of weights coefficients and input signals. The second unit realise nonlinear function, called neuron activation function. Signal e is adder output signal, and y = f(e) is output signal of nonlinear element. Signal y is also output signal of neuron. To teach the neural network we need training data set. The training data set consists of input signals (x1 and x2 ) assigned with corresponding target (desired output) z. The network training is an iterative process. In each iteration weights coefficients of nodes are modified using new data from training data set. Modification is calculated using algorithm described below: Each teaching step starts with forcing both input signals from training set. After this stage we can determine output signals values for each neuron in each network layer. Pictures below illustrate how signal is propagating through the network, Symbols w(xm)n represent weights of connections between network input xm and neuron n in input layer. Symbols yn represents output signal of neuron n. Propagation of signals through the hidden layer. Symbols wmn represent weights of connections between output of neuron m and input of neuron n in the next layer. Propagation of signals through the output layer. In the next algorithm step the output signal of the network y is compared with the desired output value (the target), which is found in training data set. The difference is called error signal d of output layer neuron. It is impossible to compute error signal for internal neurons directly, because output values of these neurons are unknown. For many years the effective method for training multiplayer networks has been unknown. Only in the middle eighties the backpropagation algorithm has been worked out. The idea is to propagate error signal d (computed in single teaching step) back to all neurons, which output signals were input for discussed neuron. The weights' coefficients wmn used to propagate errors back are equal to this used during computing output value. Only the direction of data flow is changed (signals are propagated from output to inputs one after the other). This technique is used for all network layers. If propagated errors came from few neurons they are added. The illustration is below: When the error signal for each neuron is computed, the weights coefficients of each neuron input node may be modified. In formulas below df(e)/de represents derivative of neuron activation function (which weights are modified). Coefficient h affects network teaching speed. There are a few techniques to select this parameter. The first method is to start teaching process with large value of the parameter. While weights coefficients are being established the parameter is being decreased gradually. The second, more complicated, method starts teaching with small parameter value. During the teaching process the parameter is being increased when the teaching is advanced and then decreased again in the final stage. Starting teaching process with low parameter value enables to determine weights coefficients signs. References Ryszard Tadeusiewcz "Sieci neuronowe", Kraków 1992 |
||
|
mgr inż. Adam Gołda (2005) Katedra Elektroniki AGH |
||
|
Last modified: 06.09.2004 |
||
A Step by Step Backpropagation Example
Background
Backpropagation is a common method for training a neural network. There is no shortage of papers online that attempt to explain how backpropagation works, but few that include an example with actual numbers. This post is my attempt to explain how it works with a concrete example that folks can compare their own calculations to in order to ensure they understand backpropagation correctly.
If this kind of thing interests you, you should sign up for my newsletter where I post about AI-related projects that I’m working on.
Backpropagation in Python
You can play around with a Python script that I wrote that implements the backpropagation algorithm in this Github repo.
Backpropagation Visualization
For an interactive visualization showing a neural network as it learns, check out my Neural Network visualization.
Additional Resources
If you find this tutorial useful and want to continue learning about neural networks and their applications, I highly recommend checking out Adrian Rosebrock’s excellent tutorial on Getting Started with Deep Learning and Python.
Overview
For this tutorial, we’re going to use a neural network with two inputs, two hidden neurons, two output neurons. Additionally, the hidden and output neurons will include a bias.
Here’s the basic structure:
In order to have some numbers to work with, here are the initial weights, the biases, and training inputs/outputs:
The goal of backpropagation is to optimize the weights so that the neural network can learn how to correctly map arbitrary inputs to outputs.
For the rest of this tutorial we’re going to work with a single training set: given inputs 0.05 and 0.10, we want the neural network to output 0.01 and 0.99.
The Forward Pass
To begin, lets see what the neural network currently predicts given the weights and biases above and inputs of 0.05 and 0.10. To do this we’ll feed those inputs forward though the network.
We figure out the total net input to each hidden layer neuron, squash the total net input using an activation function (here we use the logistic function), then repeat the process with the output layer neurons.
Here’s how we calculate the total net input for :
We then squash it using the logistic function to get the output of :
Carrying out the same process for we
get:
We repeat this process for the output layer neurons, using the output from the hidden layer neurons as inputs.
Here’s the output for :
And carrying out the same process for we
get:
Calculating the Total Error
We can now calculate the error for each output neuron using the squared error function and sum them to get the total error:
For example, the target output for is
0.01 but the neural network output 0.75136507, therefore its error is:
Repeating this process for (remembering
that the target is 0.99) we get:
The total error for the neural network is the sum of these errors:
The Backwards Pass
Our goal with backpropagation is to update each of the weights in the network so that they cause the actual output to be closer the target output, thereby minimizing the error for each output neuron and the network as a whole.
Output Layer
Consider . We want to know how much
a change in
affects the total error,
aka
.
By applying the chain rule we know that:
Visually, here’s what we’re doing:
We need to figure out each piece in this equation.
First, how much does the total error change with respect to the output?
Next, how much does the output of change
with respect to its total net input?
The partial derivative of the logistic function is the output multiplied by 1 minus the output:
Finally, how much does the total net input of change
with respect to
?
Putting it all together:
You’ll often see this calculation combined in the form of the delta rule:
Alternatively, we have and
which
can be written as
,
aka
(the
Greek letter delta) aka the node delta. We can use this to rewrite the calculation above:
Therefore:
Some sources extract the negative sign from so
it would be written as:
To decrease the error, we then subtract this value from the current weight (optionally multiplied by some learning rate, eta, which we’ll set to 0.5):
We can repeat this process to get the new weights ,
,
and
:
We perform the actual updates in the neural network after we have the new weights leading into the hidden layer neurons (ie, we use the original weights, not the updated weights, when we continue the backpropagation algorithm below).
Hidden Layer
Next, we’ll continue the backwards pass by calculating new values for ,
,
,
and
.
Big picture, here’s what we need to figure out:
Visually:
We’re going to use a similar process as we did for the output layer, but slightly different to account for the fact that the output of each hidden layer neuron contributes to the output (and therefore error) of multiple output neurons. We know that affects
both
and
therefore
the
needs
to take into consideration its effect on the both output neurons:
Starting with :
We can calculate using
values we calculated earlier:
And is
equal to
:
Plugging them in:
Following the same process for ,
we get:
Therefore:
Now that we have ,
we need to figure out
and
then
for
each weight:
We calculate the partial derivative of the total net input to with
respect to
the same as we did for
the output neuron:
Putting it all together:
You might also see this written as:
We can now update :
Repeating this for ,
,
and
Finally, we’ve updated all of our weights! When we fed forward the 0.05 and 0.1 inputs originally, the error on the network was 0.298371109. After this first round of backpropagation, the total error is now down to 0.291027924. It might not seem like much, but after repeating this process 10,000 times, for example, the error plummets to 0.000035085. At this point, when we feed forward 0.05 and 0.1, the two outputs neurons generate 0.015912196 (vs 0.01 target) and 0.984065734 (vs 0.99 target).
If you’ve made it this far and found any errors in any of the above or can think of any ways to make it clearer for future readers, don’t hesitate to drop me a note. Thanks!