Loading in 2 Seconds...

A note about gradient descent: Consider the function f(x)=(x-x 0 ) 2 Its derivative is:

Loading in 2 Seconds...

- 102 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' A note about gradient descent: Consider the function f(x)=(x-x 0 ) 2 Its derivative is:' - tad-harper

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

A note about gradient descent:

Consider the function f(x)=(x-x0)2

Its derivative is:

By gradient descent .

+ -

x0

Solving the differential equation:

or in the general form:

What is the solution of this type of equation:

Try:

THE PERCEPTRON:

(Classification)

Threshold unit:

where is the output for input pattern ,

are the synaptic weights and is the desired output

AND

w1 w2 w3 w4 w5

Perceptron learning rule:

Convergence proof:

Hertz, Krough, Palmer (HKP)

- did you receive the email?

Assignment 3a:

program in matlab a preceptron

with a perceptron learning rule

and solve the OR, AND and XOR problems. (Due before Feb 27)

w1 w2 w3 w4 w5

Show Demo

Linear single layer network:

( approximation, curve fitting)

*

or

Linear unit:

where is the output for input pattern ,

are the synaptic weights and is the desired output

Minimize mean square error:

w1 w2 w3 w4 w5

Linear single layer network:

( approximation, curve fitting)

Linear unit:

where is the output for input pattern ,

are the synaptic weights and is the desired output

Minimize mean square error:

w1 w2 w3 w4 w5

The best solution is obtained when E is minimal.

For linear neurons there is an exact solution for this called the pseudo-inverse (see HKP).

Looking for a solution by gradient descent:

-gradient

E

w

Chain rule

Sigmoidal neurons:

for

example:

Which types of problems can a sigmoidal networks solve?

Assignment 3b – Implement a one layer linear and sigmoidal network, fit a 1D a linear, a sigmoid and a quadratic function, for both networks.

Multi layer networks:

Output layer

- Can solve non linearly separable classification problems.
- Can approximate any arbitrary function, given ‘enough’ units in the hidden layer.

Hidden layer

Input layer

Gradient descent/ Back Propagation, the solution to the credit assignment problem:

Where:

{

From hidden layer to output weights:

and

For input to hidden layer:

Assignment 3c: Program a 2 layer network in matlab, solve the XOR problem. Fit the curve: x(x-1) between 0 and 1, how many hidden units did you need?

Formal neural networks can accomplish many tasks, for example:

- Perform complex classification
- Learn arbitrary functions
- Account for associative memory
- Some applications: Robotics, Character recognition, Speech recognition,
- Medical diagnostics.
- This is not Neuroscience, but is motivated loosely by neuroscience and carries important information for neuroscience as well.
- For example: Memory, learning and some aspects of development are assumed to be based on synaptic plasticity.

What did we learn today?

Is BackProp biologically realistic?

Download Presentation

Connecting to Server..