Can we synthesize learning? Sérgio Hortas Rodrigues IST, Aprendizagem Simbólica e Sub-Simbólica, Jun 2009

Transcription

1 Hierarchical Neural Netw rks Can we synthesize learning? Sérgio Hortas Rodrigues IST, Aprendizagem Simbólica e Sub-Simbólica, Jun 2009

2 Topics Brain review Artificial Neurons Basic Neural Networks Back Propagation Algorithm Hierachical l Neural Networks Neocognitron Convolutional

3 Artificial Neuronal Networks The Brain

4 Brain: Information-processor Longitudinal cut Left hemisfere Two hemisferes

5 Encephalization quotient (EQ) (Font: Fundamentals of Human Neuropsychology Kolb and Whishaw)

6 Hominids brain size evolution (Font: Fundamentals of Human Neuropsychology Kolb and Whishaw)

7 The neuron hypothesis ~ 1800: The first detailed descriptions of nerve cells were undertaken by Camillo Golgi and Santiago Ramón y Cajal Cajal showed that nervous tissue is not one continuous web but a network of discrete cells Cajal established that individual neurons are the elementary signaling elements of the nervous system

8 Processing information Structural: Neurons, regions of neurons and connections between them Physiological: The way the brain processes information as chemical and physical reactions and transmission of substances Cognitive: The way humans think

9 Structural level: Some numbers 11 Aproximated 100 billion (10 11 )neurons Aprox bilion sinapses (connections between neurons) Each neuron with an average of connections to other neurons At least of neuron types All have the same architecture (Font: Principles of Neural Science, Kandel et al.)

10 The biological neuron

11 Types of neurons (Font: Fundamentals of Human Neuropsychology Kolb and Whishaw)

12 Structural level: More numbers CNS Neuron: Soma[ 50μm ]: Storehouse of genetic info. Dentrites: Receive the input signals Axon: Conduct signals to other neurons Ø [0.2μm to 20μm ] long [0.1mm to 3m] ilepsy 2004 Reaseach Review) (Photo: The National Eociety for Ep Signals: Action potencials travels 1 to 100 m/s without distortion along the axon. amplitude: 100mV, duration: 1ms 80 μm (Font: Principles of Neural Science, Kandel et al.)

13 Sinapses Types: Excitatory & Inhibitory (Font: Fundamentals of Human Neuropsychology Kolb and Whishaw)

14 Signals and Neurotransmiters (Font: Fundamentals of Human Neuropsychology Kolb and Whishaw)

15 Artificial Neuronal Networks Fundamentals

16 Some History 1943 McCulloch & Pitts simplified neuron model 1969 Minsky & Papert shows the weekness of the perceptron models. Many researchers eft the field. 70s - Kohonen, Grossberg, Anderson and Fukushima continued their efforts. Early 80s New theoretical et results (e.g. (eg error back- propagation). The interest re-emerge. Present Research active ANN seen as a better Present Research active. ANN seen as a better alternative model in some fields (e.g. pattern recognition)

17 A framework for PDP A set of processing units ('neurons (neurons,''cells'); A state of activation y k for every unit, which equivalent to the outputof the unit; Connections between the units. Generally each connection is defined by a weight w jk which determines the effect which the signal of unit j has on unit k; A propagation rule which determines the A propagation rule, which determines the effective input s k of a unit from its external inputs;

18 A framework for PDP An activation function k, which determines the new level of activation based on the efective input s k(t) and the current activation y k (t) (i.e., the update); An external input (aka bias, offset) θ k for each unit; A method for information gathering (the learning rule); An environment within which the system must operate, providing input signals and, if necessary, error signals.

19 Artificial neuron model Conections between units: Positive w jk is considered excitation Negative w jk is considered inhibition jk

20 Activation and output rules The effect of the total input on the activation of the neuron is: y ( t 1) ( y ( t ), s ()) t k k k k yk( t1) k( sk( t)) k wjk( t) yj( t) k( t) j Generally, k is a nondecreasing function.

21 Deterministic Threshold functions Most used threshold function types: Examples: 1, sk 0 sgn( sk ) 1, s k 0 1, sk 1 Sat( sk) sk, 1sk 1 1, s k 1 y 1 ( s ) 1 e k k k s k y ( s ) tanh( s ) k k k k

22 Hidden Layer y Role Font: Artificial Neural Networks: A Tuturial-Jain, Mao and Moiuddin -1996

23 Stochatic Threshold functions Using for example a stochastic Boltzmann function, our neuron can fire with apropability p: 1 py ( k 1) k( sk) s 1 e s k T k k k T T is a parameter comparable with the (synthetic) temperature of the system. The network reachs a thermal equilibrium with the relative probability of two global states α and β following the Boltzmann distribution: P ( ) T e P P α is the probability of being in the α th global state and ε α is the energy of that state

24 Network topologies Feed-forward: Data flow from input to output units is strictly feed-forward. Can extend over multiple (layers of) units, but no feedback connections are present. Recurrent networks: Contain feedback connections. Dynamical properties of the network are important. In some cases, the network will evolve to a stable state in which these activations do not change anymore. In others the change of the activation values of the output neurons are significant, such that the dynamical behaviour constitutes t the output t of the network.

25 Taxonomy of networks Font: Artificial Neural Networks: A Tuturial-Jain, Mao and Moiuddin -1996

26 Hierarchical Networks Back Propagation Algorithm

27 Train a NN w/ Backpropagation p Consider the Neural Network:

28 Train a NN w/ Backpropagation p Each neuron:

29 Train a NN w/ Backpropagation p

30 Train a NN w/ Backpropagation p

31 Train a NN w/ Backpropagation p

32 Train a NN w/ Backpropagation p

33 Train a NN w/ Backpropagation p

34 Train a NN w/ Backpropagation p Propagation of signals through the output layer:

35 Train a NN w/ Backpropagation p Out signal y is compared with desired out value z. The difference is called error signal δ of the output layer neuron

36 Train a NN w/ Backpropagation p

37 Train a NN w/ Backpropagation p

38 Train a NN w/ Backpropagation p

39 Train a NN w/ Backpropagation p

40 Train a NN w/ Backpropagation p

41 Train a NN w/ Backpropagation p

42 Train a NN w/ Backpropagation p

43 Train a NN w/ Backpropagation p

44 Train a NN w/ Backpropagation p

45 Train a NN w/ Backpropagation p

46 Hierarchical Networks Neocognitron

47 Biological Inspiration Hubel and Wiesel discovered that the primary visual cortex cells are organized in a hierarchical structure This structure re has two types of cells: Simple cells (S-cells) Extract features Complex cells (C-Cells) Cells) Introduce Invariance The hierarchical relation between S-Cells and C-Cells repeats several times (three or four stages for the pvc) The hierarchical relation between the retina and the cortex is topologically organized

48 Simple Cells After having finished learning, each S-cell react selectively to patterns with a specific orientation and position Their input connections are variable and modified through learning Have excitatory and inhibitory regions (Font: Eye, Brain and Vision H. Hubel)

49 Simple Cells Response of an S-Cell to different stimulus: (Font: Eye, Brain and Vision H. Hubel)

50 Complex Cells Like the S-cells: Respond over a limited region of the visual field, despite this beeing greater Respond only tospecifically oriented lines Unike the S-cells: Their behavior cannot be explained by a neat subdivision of the receptive field into excitatory and inhibitory regions Allow the pattern (bar, slit or edge) to be shifted The response adapts, this is, the brust of impulses is brief when the p p p line becomes stationary. Some respond better to one direction of movement than to the diametrically opposite direction

51 Complex Cells Response of an C-Cell Cell to different stimulus: (Font: Eye, Brain and Vision H. Hubel)

52 Hierarchical Neural Network Generally ANNs in each layer every unit receive input from all units of the previous layer In HNNs, in each layer everyer unit receivee input from a localized subset of units of the previous layer making it s view local. Advantages: Units in each layer have to deal with simpler problem Thenetworkcanworkwithmuchlessunits The global view is constructed as we move towards the The global view is constructed as we move towards the output layer.

53 The Neocognitron A neural model mainly for visual patterns recognition that can perform unsupervised learning (Font: Neocognitron: a new algorithm for pattern recognition K.Fukushima and S. Miyake)

54 Self-Organizaition Only the S-Cells have their input connections modified through learning Inhibitory V-cells receives like S-cells fixed excitatory connections from the same group of C-cells V-cells always respond with the average strength of the excitatory connections (Font: Neocognitron: a new algorithm for pattern recognition K.Fukushima and S. Miyake)

55 Pattern Recognition (Font: Neocognitron: a new algorithm for pattern recognition K.Fukushima and S. Miyake)

56 Invariance property p

57 Output of an S-cell u S 1 as ( t,, x ) u0 ( t, ) S x AS (, t x) (, ) (, ) S SbS t x vs t x Note: the effect of lateral inhibition among S-cells is neglected in this equation u ( t, ) is theoutput of the cell and [ ] is a S nonlinear function define by: x x, x 0 0, x 0

58 Output of an S-cell as ( t,, x)( 0) is the strength of the excitatory variable connection from cell u( ) in the previous layer u ( x ) b ( t, x )( 0) is the strength of the inhibitory S A S S variable from the V -cell is the radius of the connectable area of an S-cell S (0 S 1) is the constant tereshold that controls the selectivity of the S-cell S

59 Output of an V-cell 2 S 0 v (, t x ) c ( x ) u (, t ) S x A S c S ( x ) is a slightly bell-shaped, but almost flat, two-dimensional function

60 Output of an S-cell The difference in characteristics between S- and C-cells is created, not by the difference in network architecture, but by the difference in learning rules by which the connections are modified Remember The input connections from C-cells fixed and invariable, so they don t learn u C 1 ac ( t,, x ) u0 ( t, ) C x AC (, t x) (, ) (, ) C CbC t x vc t x Note: the effect of lateral inhibition among C-cells is neglected in this equation

61 Thinning out of cells The output size of the Neocognitron is 1 x 1. Input layer has the same size of the patterns to recognize, which implies reducing progressively the dimensions of the layers towards the output t The reducing takes place in the transition from S-layers to C-layers.

62 Improvements and limitations Incremental learning without affecting the learning speed and without damaging severely old memories. ThemaindifferenceresidesinthenumberofV-cells.EachS-cell plane has one V-cell plane instead of one V-cell pane for each S- cell layer Rotating invariance is added by determining first the orientation of an input pattern through a preprocessing layer and the fed it to a conventional Neocognitron Shift invariance of the Neocognitron is low, by means of the subsampling at the C-cell planes. A trade off bettween discriminatory power and shift invariance must be made.

63 Application examples Handwritten Character Recognition Symmetry Axis Extraction Detecting a target object Speech Recognition ii

64 Hierarchical Networks Convolutional Neural Networks

65 Convolutional neural networks Inspired on the work by Hubel and Wiesel Same built-in shift and distortion invariance as the Neocognitron Architecture is Neocognitron like: Receptive fields Shared Weights Subsampling Weights of several units synchronized making representing the samefeature in different position of the layers

66 Convolutional neural networks Key difference with ihthe Neocognitron is learning: Neocognitron is trained with crafted algorithm CNNs are trained with the back-propagation algorithm The shared weights reduce the free parameters involved The shared weights reduce the free parameters involved in computation, thus reducing the local minima problems due to the algorithm used

67 Convolutional neural networks i ij j j h w x y i ( h ) i

68 CNN Example Generic Object Recognition ((LeCun et. al CVPR 04) Task: Classify objects into 1 of 5 categories using stereo images (Animal, Human, Plane, Truck, Car) 50 toys from 5 categories 10 instances per category 5 training, 5 test 972 stereo pairs for each object instance: 18 azimuths 9 elevations 6 illuminations

69 Convolutional neural networks 90,857 parameters, 3,901,162 connections Algorithm Stereo Input Error K-NN (K=1) 2x96x % SVM (Gaussian Kernel) 2x96x % 1% Convolutional Net 2x96x96 6.6%

70 Application examples Face Recognition Medical Image Robot Navigation

71 Bibliography R. Hecht-Nielsen, Neurocomputing. Addison-Wesley, A. M. Cardoso, Time Delay Neocognitron, Master Thesis, Instituto Superior Técnico, K. Fukushima, S. Miyake, Neocognitron: a new algorithm for pattern recognition tolerant of deformations and shifts in position, Pattern Recognition 15 (6), , K. Fukushima, Self-organization of shift-invariant invariant receptive fields, Neural Networks 12, , K. Fukushima, Neocognitron for handwritten digit recognition, Neurocomputing 51, , K. Fukushima, Neocognitron capable of incremental learning, Neural Networks 17, 37 46, H. Hubel, Eye, Brain and Vision, Scientific American Library, 240 p., B. Krose and P. van der Smagt, An Introduction to Neural networks, University of Amsterdam,8th ed., 135 p., 1998.

72 Bibliography B. Kolb and I. Wishaw, Fundamentals Of Human Neuropsychology, W.H.Freeman & Co Ld Ltd, 5th ed, 763p., E. Kandel, J. Schwartz, T. Jessell, Principles of Neural Sciences, McGraw-Hill, 4th ed, 1414 p., J. Otuyama, Convolutional Neural Networks, Jun 2009, J. Bouvrie, Notes on Convolutional Neural Networks, MIT, M. Bernacki, P. Włodarczyk, Principles of backpropagation algorithm, Jun A. Jain, J. Mao, Artificial i Neural Networks: a tutorial, Institute of Electrical l and Electronics Engineers, Michigan State University, p , 1996.

73 THANK YOU Questions?