Feed-Forward mapping networks KAIST 바이오및뇌공학과 정재승

Transcription

1 Feed-Forward mapping networks KAIST 바이오및뇌공학과 정재승

2 How much energy do we need for brain functions?

3 Information processing: Trade-off between energy consumption and wiring cost

4 Trade-off between energy consumption (wiring cost) and maximal information processing

5 Optical character recognition (OCR) OCR is the process of optically scanning an image and interpreting this digital image so that the computer understands its meaning.

6 Two feed-forward process The perception of a letter, the physical sensing of an image of a letter. Process attaching meaning to such an image.

7 The perception of a letter Sensory feature vector: the number of feature values defines the dimensionality of the feature space.

8 Mapping functions: the recognition process to a vector function

9 Perceptron The perceptron is a type of artificial neural network invented in 1957 at the Cornell Aeronautical Laboratory by Frank Rosenblatt. It can be seen as the simplest kind of feedforward neural network: a linear classifier.

10 Population node as perceptron Linear perceptron

11 Output manifold of population node with two input channels

12 Boolean algebra Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the 1840s. These turn out to coincide with the set of all operations on the set {0,1} that take only finitely many arguments; there are 2 2n such operations when there are n arguments.

13 Boolean algebra It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation x replaced by the respective logical operations of conunction x y, disunction x y, and negation x. The Boolean operations are these and all other operations that can be built from these, such as x (y z).

14 Boolean functions: the threshold node

15 Look-up table, graphical representation, and single threshold population nodes

16 The history of perceptron A feed-forward neural network with two or more layers (i.e., a multilayer perceptron) had far greater processing power than perceptrons with one layer (i.e., a single layer perceptron). Single layer perceptrons are only capable of learning linearly separable patterns; in 1969 a famous book entitled Perceptrons by Marvin Minsky and Seymour Papert showed that it was not possible for these classes of network to learn an XOR function.

17 The history of perceptron Both Minsky and Papert already knew that multi-layer perceptrons were capable of producing an XOR Function. Three years later, Stephen Grossberg published a series of papers introducing networks capable of modelling differential, contrast-enhancing and XOR functions.

18 The history of perceptron Nevertheless the often-miscited Minsky/Papert text caused a significant decline in interest and funding of neural network research. It took ten more years until neural network research experienced a resurgence in the 1980s. This text was reprinted in 1987 as "Perceptrons - Expanded Edition" where some errors in the original text are shown and corrected.

19 The number of nonlinear separable functions grows rapidly with the dimension of the feature space and soon outgrows the number of linear separable functions

20 The weight matrix will be changed by small amounts in an attempt to find a better answer. Learning delta rule The delta rule is a gradient descent learning rule for updating the weights of the artificial neurons in a singlelayer perceptron.

21 Supervised learning Given examples Find perceptron such that R N 0,1 x 1 y 1 y Hw T x a a x 2 y 2 x 3 y 3

22 Example: handwritten digits Find a perceptron that detects two s.

23 Delta rule w y Hw T xx Learning from mistakes. delta : difference between desired and actual output. Also called perceptron learning rule

24 Two types of mistakes False positive Make w less like x. y 0, Hw T x1 False negative Make w more like x. w x y 1, Hw T x 0 w x The update is always proportional to x.

25 Gradient update Obective function w e w ew,x,y y Hw T x w T x Stochastic gradient descent on Ew ew,x,y E=0 means no mistakes.

26 If examples are nonseparable The delta rule does not converge. Obective function is not equal to the number of mistakes. No reason to believe that the delta rule minimizes the number of mistakes.

27 Contrast with Hebb rule w yx w y y x Hebb rule Perceptron learning rule Assume that the teacher can drive the perceptron to produce the desired output. What are the obective functions?

28 Is the delta rule biological? Actual output: anti-hebbian w Hw T xx Desired output: Hebbian w yx Contrastive

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30 Obective function Hebb rule distance from inputs Delta rule error in reproducing the output

31 Multi-layer Perceptrons

32 Multilayer Perceptrons: Architecture Input layer Output layer Hidden Layers

33 A solution for the XOR problem x 1 x 1 x 2 x 1 xor x x 2 x 1 x if v > 0 (v) = -1 if v 0 is the sign function.

34 NEURON MODEL Sigmoidal Function v ) ( 1 Increasing a (v ) 1 e av v induced field of neuron Most common form of activation function a threshold function Differentiable v v w i0,..., m i y i

35 Learning Algorithms Back-propagation algorithm Function signals Forward Step Error signals Backward Step It adusts the weights of the NN in order to minimize the average squared error.

36 Average Squared Error Error signal of output neuron at presentation of n-th training example: Total energy at time n: Average squared error: Measure of learning performance: e (n) E(n) E AV d N (n) - y C N n1 e 2 (n) (n) E(n) C: Set of neurons in output layer N: size of training set Goal: Adust weights of NN to minimize E AV

37 Notation e y v Error at output of neuron Output of neuron w i0,..., m i y i Induced local field of neuron

38 Weight Update Rule Update rule is based on the gradient descent meth take a step in the direction yielding the maximum decrease of E w i E - w i Step in direction opposite to the gradient w i With to neuron weight associated to the link from neuro

39 Definition of the Local Gradient of neuron v - E Local Gradient ) (v e We obtain because ) (v ' 1) ( e v y y e e v E E

40 Update Rule We obtain because w i y i E E v w i v w i E v v w i y i

41 Error e of output neuron Single layer Perceptron: output neuron e d - y Then ( d - y ) ' (v )

42 Multi-layer Perceptron

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