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

A note about gradient descent: Consider the function f(x)=(x-x 0 ) 2 Its derivative is: By gradient descent. + -. x 0. Solving the differential equation:. or in the general form:. What is the solution of this type of equation:. Try:. THE PERCEPTRON: (Classification).

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

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  1. A note about gradient descent: Consider the function f(x)=(x-x0)2 Its derivative is: By gradient descent . + - x0

  2. Solving the differential equation: or in the general form: What is the solution of this type of equation: Try:

  3. 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

  4. 1 0 1 -1.5 1 1 AND Linearly seprable

  5. -0.5 1 1 OR 1 0 1 Linearly separable

  6. 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

  7. Summary – what can perceptrons do and how?

  8. 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

  9. 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

  10. 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

  11. and Since: Error: Therefore: Which types of problems can a linear network solve?

  12. 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.

  13. 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

  14. Note: is not a vector but a matrix

  15. Solving linearly inseparable problems XOR Hint: XOR = or and not and

  16. XOR -.5 1 0.5 .5 0 0.5 -0.5 1 -1 How do we learn a multi-layer network The credit assignment problem !

  17. Gradient descent/ Back Propagation, the solution to the credit assignment problem: Where: { From hidden layer to output weights:

  18. For input to hidden layer: { Where: and and

  19. 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?

  20. 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.

  21. What did we learn today? Is BackProp biologically realistic?

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