1. Neural Nets: Something you can use and something to think about Cris Koutsougeras What are Neural Nets What are they good for Pointers to some models and formulations Methods that you can use Fun and open issues

2. What are they good for System of interest (Black box) If I have a finite set of observations (samples) of input-output behavior, can I figure out what function the box performs? System identification Prediction-forecasting Controls, Autogenerate a simulator Dealing with cases where only samples are known about the function of interest

3. Why Neural Nets Target function is unknown except from samples Target is known but is very hard to describe in finite terms (e.g. closed form expression) Target function is non-deterministic Something to think about

4. General Structure of a NN … … … Inputs Outputs

5. Function Approximation Output is a composition of the various node functions. Output is parametric on the inputs; weights-thresholds are parameters Bottomline: function approximators

6. Nets perform functional compositions Net output=

7. Output is a complex function of the inputs. The complexity comes from the deep nesting of the typical neuron functions. … Inputs f f ffff Y x1x2xn Y=f(f(f(f(x1, x2, … xn), f(f(f (x1, x2, … xn), f(f(f((x1, x2, … xn), …..)))

8. Net’s function is an “elastic” curve By adjusting the weights (curve parameters) the curve fits the samples Adjusting the weights is the key issue

9. How does it work We have a Sample set S i ={(x 1,t 1 ), (x 2,t 2 ),…(x n,t n )} We have the net producing: y i =f(x i, W) We define a Quality measure Q(W) that involves f and the targets t i. We adjust W iteratively:  W=-a  w Q until Q is optimized. A convenient Q is usually the mean square error

10. How does it work Target Net function Something you can use

11. Nonlinear regression Quality function is the sum of individual errors. Minimizing the error is like stretching the curve to fit the samples. Problem: How do we know that we are done?

12. Nonlinear regression

14. Problems Not enough non-linearity to fit, or Overfitting Need for minimal nonlinearity that can accomplish fitting

15. Gradient Decent can get stuck Weight Space Total Error (Q)

16. Simulated Annealing Turn a into a time function; start with very large values and gradually reduce it Theorem: If a is reduced at a slow enough rate the probability of landing at the global minimum asymptotically tends to 1 Something you can use

17. Simulated Annealing By starting with lots of energy and reducing it slowly enough, the probe will eventually have enough energy to jump out of local minima but not out of the global. If it remains long enough in that energy range it will get trapped in the global minimum area.

18. Let’s have some fun What network structure do we need? Particularly how many nodes?

19. Let’s have some fun

20. … Inputs X ij W rj V rj YjYj Y j = F(  W rj V rj ) So:  W rj V rj ) = F -1 (Y j ) r linear equations

21. Our Framework

22. New Class of Training Algorithms We conclude that after proper training (by any method) all intermediate normalized vectors Y project at the same point in the direction of W. Thus all Y’s are aligned on a plane that is perpendicular to W New class of algorithms: –Find weights for hidden layer that align all Y’s on a plane –W for the output layer is the normal to that plane

23. One such Algorithm didi YiYi W Minimizewhich is parametric on all weights. Thus use as and perform a gradient descent: quality function : Something you can use

24. Open Questions: What is the minimum number of neurons needed? What is the minimum nontrivial rank that the system can assume? This determines the number of neurons in the intermediate layer.

25. Interesting Results The local activation functions must be nonlinear for hidden layer but not for the output layer. We thus arrive at the same result as Kolmogorov’s theorem The solvability of proves universal approximation with only one necessary hidden layer The minimum nontrivial rank of the matrix provides the number of hidden layer neurons necessary for proper fitting Problem: the matrix is parametric and we have no effective method for computing the lowest (non trivial) rank We came up with other characterizations based on Vapnik- Chervonenkis dimension and PAC learning However, the problem of a precise optimum number for the hidden layer is at large still open (Something to think about)

26. Clustering Models (pattern recognition/classification) Neuron functions represent “discriminant functions” that can be used to construct borders among classes.

27. Clustering Models (pattern recognition/classification) Neuron functions represent “discriminant functions” that can be used to construct borders among classes.

28. Linear neurons (thresholding) Output = 1 if F(w 1 x 1 + w 2 x 2 +.. w n x n ) > T 0 if F(w 1 x 1 + w 2 x 2 +.. w n x n ) < T F x1x1 x2x2 xnxn... W1W1 WnWn W2W2 T + + + - - - - - - - -

29. Radial Basis Output = 1 if (w 1 - x 1 ) 2 + (w 2 -x 2 ) 2 +.. (w n -x n ) 2 > R 2 F x1x1 x2x2 xnxn... W1W1 WnWn W2W2 R 0 if (w 1 - x 1 ) 2 + (w 2 -x 2 ) 2 +.. (w n -x n ) 2 < R 2 + + + - - - - - - - -