1. Synaptic DynamicsII : Supervised Learning
The Backpropagation Algorithm and Spport Vector Machines
JingLIU

2. History of BP Algorithm
Rumelhart [1986]popularized the BP algorithm in the Parallel Distributed Processing edited volume in the late 1980’s.
BP algorithm overcame the limitations of the perceptron algorithm, limitations that Minsky and Papert[1969] had carefully enumerated.
BP’s popularity begot waves of criticism. BP algorithm often failed to converge, and at best converged to local error minima.
The wave of criticism challenged BP’s historical priority.
The wave of criticism challenge whether the BP learning was new. The algorithm not offer a new kind of learning.

3. Multilayer feedforward NNs

4. Feedforward Sigmoidal Representation Theorems
Feedforward sigmoidal architectures can in principle represent any Borel-measurable function to any desired accuracy—if the network contains enough “hidden” neurons between the input and output neuronal fields.
So the MLP can solve the problems of nonlinear separable problems and function approximate.

5. Feedforward Sigmoidal Representation Theorems
We can explain the theorem in the following two aspects:
To improve the NNs’ classification ability, we must use multilayer networks, at least one hidden layers.
One the other hand, when feedforward –sigmoidal neural networks finely approximate complicated functions, the networks may suffer a like “explosion” of hidden neurons and interconnecting synapses.

6. How can a MLP learn?
We know only the random sample (xi,Yi) and the vector error Yi-N(Xi) Xi
Yi-N(Xi)

7. BP Algorithm The Principle of BP algorithm
Working signals are propagate forward to the output neurons.
Error signals are propagated backward to the input field.

8. BP Algorithm
The BP Algorithm for the learning process of Multilayer networks
The error signal of the jth output is:
The instantaneous summed squared error of the output is:
The objective function of learning process is:

9. BP Algorithm In the case of learning sample by sample:
The gradient of E(n) at nth iteration is:

10. BP Algorithm The modification quantity of weight wji is: where
For j is an output neuron:
For j is a hidden neuron:

11. BP Algorithm

12. Calculate output of each layer
initialize
Calculate output of each layer
Calculate the error of the
output layer
Calculate for all neurons
Modify the weights of
Output layer
Modify the weights of
Hidden layer

13. Some improvements for BP algorithm
The problems of BP algorithm
It convergence slowly.
There are local minima in the objective function.
Some methods to improvement BP algorithm, such as: conjugate gradient descent, using high order derivative and adding momentum term, etc.

14. Adding momentum term for BP Algorithm
Here we modify weights of neurons with:

15. Support Vector Machines
SVM was presented for the binary classification problems based on the principle of Structural Risk Minimization.

16. Support Vector Machines
To get the largest margin, we deduce the optimal classification function:
Using inner product to replace the dot product:

17. Support Vector Machines

18. Support Vector Machines

19. Support Vector Machines
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