Unsupervised Learning in Neural Networks Keith L. Downing The - PowerPoint PPT Presentation

Unsupervised Learning in Neural Networks Keith L. Downing The Norwegian University of Science and Technology (NTNU) Trondheim, Norway keithd@idi.ntnu.no January 18, 2014 Keith L. Downing Unsupervised Learning in Neural Networks

Unsupervised Learning No ”instructor” feedback Network tunes itself, often using local mechanisms (i.e., those involving a neuron and a few of its neighbors, but no global controller) to detect patterns that are prevalent in the inputs. Appears to be common in neocortex and hippocampus of brains. Hopfield Networks (a.k.a. Attractor Networks) and Kohonen (a.k.a. Self-Organizing Maps (SOMs)) are common implementations. Hebbian synaptic modifications commonly used. Keith L. Downing Unsupervised Learning in Neural Networks

Neural Coding Schemes Local Semi-Local or Sparsely Distributed Fully Distributed Grandma!! Given N neurons � N � = N patterns Local: 1 � N N ! � Sparsely Distributed: = ( N − k )! k ! patterns. N = 20, k= 3 → 1140 k � N N ! � Fully Distributed: = 2 !) 2 patterns. N = 20 → 184756 N ( N 2 *But, to store 184756 patterns with 20 nodes in a useful manner isn’t easy! Keith L. Downing Unsupervised Learning in Neural Networks

Hopfield Networks Use Distributed/Population Codes Storage efficiency - in theory, k neurons with m 1 differentiable states can store m k patterns. Each node represents a pattern component (e.g. pixel). 2 Each (bi-directional) arc represents the correlation 3 (observed in the full set of training patterns) between those components. Pattern completion - given part of a pattern, the network 4 can often fill in the rest. Content addressable memory - patterns are retrieved 5 using portions of the pattern (not memory addresses) as keys. Robustness - if a few neurons die, each pattern may be 6 slightly corrupted, but none is lost completely. Keith L. Downing Unsupervised Learning in Neural Networks

Auto-Associative Training Phase: Pattern Storage a a b b +1 -1 a w b c d Learning Intra-Pattern Correlations P w jk ← 1 ∑ c pk c pj P p = 1 where P = # of patterns to learn, c p k = value of kth component in the pth pattern, and w jk = weight between the jth and kth neuron. Keith L. Downing Unsupervised Learning in Neural Networks

Simple Example of Auto-Associative Learning 1. Patterns to Learn On (+1) p1 p2 p3 Off (-1) 1 2 1 2 1 2 3 4 3 4 3 4 Neutral (0) 2. Hebbian Weight Initialization 3. Build Network p1 p2 p3 Avg w12 +1 +1 -1 +1/3 1 2 +1/3 w13 +1 -1 -1 -1/3 +1/3 +1/3 w14 -1 +1 +1 +1/3 -1/3 -1/3 w23 +1 -1 +1 +1/3 w24 -1 +1 -1 -1/3 3 4 -1 w34 -1 -1 -1 -1 Keith L. Downing Unsupervised Learning in Neural Networks

Auto-Associative Testing Phase: Pattern Recall a b c d Oscillation + Convergence to an Attractor Pattern C ∑ c k ( t + 1 ) ← sign ( w kj c j ( t )+ I k ) j = 1 where c k ( t ) = binary activation value of the kth neuron at time t, and I k is the constant forcing input to neuron k from the initial pattern. Keith L. Downing Unsupervised Learning in Neural Networks

Simple Example of Auto-Associative Recall On (+1) Off (-1) 1 2 1 2 3 4 3 4 Neutral (0) 1 2 +1/3 1 +1/3 2 +1/3 +1/3 +1/3 +1/3 -1/3 -1/3 -1/3 -1/3 4 4 3 -1 3 -1 Enter partial pattern and run to quiescence Keith L. Downing Unsupervised Learning in Neural Networks

Hetero-Associative Training Phase a a b b +1 a w a b b w - 1 Learning Inter-Pattern Correlations P w jk ← 1 ∑ i pk o pj P p = 1 where i pk and o pj are the values of the kth and jth components of the input and output patterns, respectively, for the pth pair of patterns. Keith L. Downing Unsupervised Learning in Neural Networks

Hetero-Associative Testing Phase a a b b Oscillation + Convergence to a PAIR of attractor patterns C ∑ c k ( t + 1 ) ← sign ( w kj c j ( t )+ I k ) j = 1 *Same as in the auto-associative network, but alternate between updating neurons in the left and right networks. Keith L. Downing Unsupervised Learning in Neural Networks

Interference → Spurious Recall in Hopfield Networks a b c d Keith L. Downing Unsupervised Learning in Neural Networks

Recall = Search in an Energy Landscape Hopfield Energy ANN Activation State Hopfield Energy Metric C C C ∑ ∑ ∑ E = − a w jk c j c k − b I k c k k = 1 j = 1 k = 1 Keith L. Downing Unsupervised Learning in Neural Networks

Hopfield Search in the Brain mouth eyes beak Duck ears neck mouth eyes beak Rabbit ears neck Flip-Flop Figures Sharing of features between the two figures + Habituation of neurons ⇒ repeated switching between network attractors = scene interpretations. Keith L. Downing Unsupervised Learning in Neural Networks

Competitive Neural Networks Characteristics Single neurons in Intermediate and/or Output Layers function as detectors for multi-neuron patterns of activity in upstream layers. Detector neurons tend to inhibit one another. The weights on afferent (i.e. incoming) arcs to each detector serve as a prototype of the class of patterns that it detects. These weights are modified by local Hebbian mechanisms. Keith L. Downing Unsupervised Learning in Neural Networks

Competitive ANNs Input Nodes Input Output Classes Case 1 wi1 .5 2 .3 wi2 i .7 .1 3 wi4 j wj4 4 Class j Class i .5 .9 .9 .1 .2 .3 .9 .1 .1 .3 .7 .7 .8 .1 .3 .1 .8 .1 .3 .8 Keith L. Downing Unsupervised Learning in Neural Networks

Prototype Learning Weight Adjustment by the Winning Node w ij ← w ij + η ( P j − w ij ) (1) Throught multiple training epochs, this simple update produces input weight vectors = prototype input patterns - but only if all input weights (to each detector node, i) are normalized : n w 2 ∑ ij = 1 (2) j = 1 * See compendium for mathematical proof. Keith L. Downing Unsupervised Learning in Neural Networks

Competitive Networks in the Brain Lateral Random Inhibition P2 becomes Activation & a detector for of Learning pattern 1001 Principal Cells P1 P2 P1 P2 P3 P3 Random Lateral Activation Inhibition P3 becomes of & a detector for Principal Learning pattern 0101 Cells P1 P2 P1 P2 P3 P3 Keith L. Downing Unsupervised Learning in Neural Networks

Spatially Correlated Neural Populations Sensory Neurons Pre-Motor Neurons Inputs 1 a Pi 2 i b - vs - b* j 3 Pj c 4 If similar inputs map to neighboring neurons, and those in turn map to neighboring neurons, etc., then: Generalization occurs naturally Small errors in perception still lead to the correct action Neural wiring can be reduced. Keith L. Downing Unsupervised Learning in Neural Networks

Topological Maps in the Brain 10 kHz 20 kHz 1 kHz 4 kHz Auditory Cortex MGN Source Localization Inferior via Delay Lines Colliculus Superior Olive 1 kHz Ventral Cochlear 4 kHz Nucleus 10 kHz Spiral Ganglion 20 kHz Cochlea Cochlea (Inner Ear) Isomorphism between 2 Spaces Spaces: Sound Frequencies + A Layer of Neurons If points p and q are close (distant) in the sound frequency space, then the neurons that detect frequencies p and q, n p and n q , are also close neighbors (distant) in the neuron layer. Keith L. Downing Unsupervised Learning in Neural Networks

Self-Organizing Visual Topological Maps Visual Field Visual Neurons Keith L. Downing Unsupervised Learning in Neural Networks

Artificial Self-Organizing Maps (SOMs) Uncorrelated Correlated Euclidean Neighbor (closest weight vectors) Topological Neighbor (closest grid location) Competition + Cooperation Nodes compete for input patterns, but then share the win by allowing grid neighbors to also update their input weights to more closely match the input pattern. Keith L. Downing Unsupervised Learning in Neural Networks

There Goes the Neighborhood Neuron Space R = 1 R = 2 Self-Organizing Learning R = 3 Keith L. Downing Unsupervised Learning in Neural Networks

TSP Using SOM (.96, .34) (.37, .08) Neuron Ring (.57, .11) (.83, .66) (.25, .80) Neighborhood Spaces Euclidean: City Locations Neural: A ring of neurons ⇒ Each neuron has 2 neighbors. Keith L. Downing Unsupervised Learning in Neural Networks

Emerging Tours Before After Y Y X X TSP City Neuron Keith L. Downing Unsupervised Learning in Neural Networks

Hebbian Learning Rules General Hebbian: Basic Homosynaptic △ w i = λ u i v △ w i = λ ( v − θ v ) u i Basic Heterosynaptic BCM △ w i = λ v ( u i − θ i ) △ w i = λ u i v ( v − θ v ) Homosynaptic All active synapses are modified the same way, depending only on the strength of the postsynaptic activity. Heterosynaptic Active synapses can be modified differently, depending upon the strength of their presynaptic activity. Keith L. Downing Unsupervised Learning in Neural Networks