III Recurrent Neural Networks 1102022 1 A The
- Slides: 86
III. Recurrent Neural Networks 1/10/2022 1
A. The Hopfield Network 1/10/2022 2
Typical Artificial Neuron connection weights inputs output threshold 1/10/2022 3
Typical Artificial Neuron linear combination activation function net input (local field) 1/10/2022 4
Equations Net input: New neural state: 1/10/2022 5
Hopfield Network • • • Symmetric weights: wij = wji No self-action: wii = 0 Zero threshold: q = 0 Bipolar states: si {– 1, +1} Discontinuous bipolar activation function: 1/10/2022 6
What to do about h = 0? • There are several options: § § s(0) = +1 s(0) = – 1 or +1 with equal probability hi = 0 no state change (si = si) • Not much difference, but be consistent • Last option is slightly preferable, since symmetric 1/10/2022 7
Positive Coupling • Positive sense (sign) • Large strength 1/10/2022 8
Negative Coupling • Negative sense (sign) • Large strength 1/10/2022 9
Weak Coupling • Either sense (sign) • Little strength 1/10/2022 10
State = – 1 & Local Field < 0 h<0 1/10/2022 11
State = – 1 & Local Field > 0 h>0 1/10/2022 12
State Reverses h>0 1/10/2022 13
State = +1 & Local Field > 0 h>0 1/10/2022 14
State = +1 & Local Field < 0 h<0 1/10/2022 15
State Reverses h<0 1/10/2022 16
Net. Logo Demonstration of Hopfield State Updating Run Hopfield-update. nlogo 1/10/2022 17
Hopfield Net as Soft Constraint Satisfaction System • States of neurons as yes/no decisions • Weights represent soft constraints between decisions – hard constraints must be respected – soft constraints have degrees of importance • Decisions change to better respect constraints • Is there an optimal set of decisions that best respects all constraints? 1/10/2022 18
Demonstration of Hopfield Net Dynamics I Run Hopfield-dynamics. nlogo 1/10/2022 19
Convergence • Does such a system converge to a stable state? • Under what conditions does it converge? • There is a sense in which each step relaxes the “tension” in the system • But could a relaxation of one neuron lead to greater tension in other places? 1/10/2022 20
Quantifying “Tension” • If wij > 0, then si and sj want to have the same sign (si sj = +1) • If wij < 0, then si and sj want to have opposite signs (si sj = – 1) • If wij = 0, their signs are independent • Strength of interaction varies with |wij| • Define disharmony (“tension”) Dij between neurons i and j: Dij = – si wij sj Dij < 0 they are happy Dij > 0 they are unhappy 1/10/2022 21
Total Energy of System The “energy” of the system is the total “tension” (disharmony) in it: 1/10/2022 22
Review of Some Vector Notation (column vectors) (inner product) (outer product) (quadratic form) 1/10/2022 23
Another View of Energy The energy measures the number of neurons whose states are in disharmony with their local fields (i. e. of opposite sign): 1/10/2022 24
Do State Changes Decrease Energy? • Suppose that neuron k changes state • Change of energy: 1/10/2022 25
Energy Does Not Increase • In each step in which a neuron is considered for update: E{s(t + 1)} – E{s(t)} 0 • Energy cannot increase • Energy decreases if any neuron changes • Must it stop? 1/10/2022 26
Proof of Convergence in Finite Time • There is a minimum possible energy: – The number of possible states s {– 1, +1}n is finite – Hence Emin = min {E(s) | s { 1}n} exists • Must show it is reached in a finite number of steps 1/10/2022 27
Steps are of a Certain Minimum Size 1/10/2022 28
Conclusion • If we do asynchronous updating, the Hopfield net must reach a stable, minimum energy state in a finite number of updates • This does not imply that it is a global minimum 1/10/2022 29
Lyapunov Functions • A way of showing the convergence of discreteor continuous-time dynamical systems • For discrete-time system: – need a Lyapunov function E (“energy” of the state) – E is bounded below (E{s} > Emin) – DE < (DE)max 0 (energy decreases a certain minimum amount each step) – then the system will converge in finite time • Problem: finding a suitable Lyapunov function 1/10/2022 30
Example Limit Cycle with Synchronous Updating w>0 1/10/2022 w>0 31
The Hopfield Energy Function is Even • A function f is odd if f (–x) = – f (x), for all x • A function f is even if f (–x) = f (x), for all x • Observe: 1/10/2022 32
Conceptual Picture of Descent on Energy Surface 1/10/2022 (fig. from Solé & Goodwin) 33
Energy Surface 1/10/2022 (fig. from Haykin Neur. Netw. ) 34
Energy Surface + Flow Lines 1/10/2022 (fig. from Haykin Neur. Netw. ) 35
Flow Lines Basins of Attraction 1/10/2022 (fig. from Haykin Neur. Netw. ) 36
Bipolar State Space 1/10/2022 37
Basins in Bipolar State Space energy decreasing paths 1/10/2022 38
Demonstration of Hopfield Net Dynamics II Run initialized Hopfield. nlogo 1/10/2022 39
Storing Memories as Attractors 1/10/2022 (fig. from Solé & Goodwin) 40
Example of Pattern Restoration 1/10/2022 (fig. from Arbib 1995) 41
Example of Pattern Restoration 1/10/2022 (fig. from Arbib 1995) 42
Example of Pattern Restoration 1/10/2022 (fig. from Arbib 1995) 43
Example of Pattern Restoration 1/10/2022 (fig. from Arbib 1995) 44
Example of Pattern Restoration 1/10/2022 (fig. from Arbib 1995) 45
Example of Pattern Completion 1/10/2022 (fig. from Arbib 1995) 46
Example of Pattern Completion 1/10/2022 (fig. from Arbib 1995) 47
Example of Pattern Completion 1/10/2022 (fig. from Arbib 1995) 48
Example of Pattern Completion 1/10/2022 (fig. from Arbib 1995) 49
Example of Pattern Completion 1/10/2022 (fig. from Arbib 1995) 50
Example of Association 1/10/2022 (fig. from Arbib 1995) 51
Example of Association 1/10/2022 (fig. from Arbib 1995) 52
Example of Association 1/10/2022 (fig. from Arbib 1995) 53
Example of Association 1/10/2022 (fig. from Arbib 1995) 54
Example of Association 1/10/2022 (fig. from Arbib 1995) 55
Applications of Hopfield Memory • • 1/10/2022 Pattern restoration Pattern completion Pattern generalization Pattern association 56
Hopfield Net for Optimization and for Associative Memory • For optimization: – we know the weights (couplings) – we want to know the minima (solutions) • For associative memory: – we know the minima (retrieval states) – we want to know the weights 1/10/2022 57
Hebb’s Rule “When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased. ” —Donald Hebb (The Organization of Behavior, 1949, p. 62) 1/10/2022 58
Example of Hebbian Learning: Pattern Imprinted 1/10/2022 59
Example of Hebbian Learning: Partial Pattern Reconstruction 1/10/2022 60
Mathematical Model of Hebbian Learning for One Pattern For simplicity, we will include self-coupling: 1/10/2022 61
A Single Imprinted Pattern is a Stable State • Suppose W = xx. T • Then h = Wx = xx. Tx = nx since • Hence, if initial state is s = x, then new state is s = sgn (n x) = x • May be other stable states (e. g. , –x) 1/10/2022 62
Questions • How big is the basin of attraction of the imprinted pattern? • How many patterns can be imprinted? • Are there unneeded spurious stable states? • These issues will be addressed in the context of multiple imprinted patterns 1/10/2022 63
Imprinting Multiple Patterns • Let x 1, x 2, …, xp be patterns to be imprinted • Define the sum-of-outer-products matrix: 1/10/2022 64
Definition of Covariance Consider samples (x 1, y 1), (x 2, y 2), …, (x. N, y. N) 1/10/2022 65
Weights & the Covariance Matrix Sample pattern vectors: x 1, x 2, …, xp Covariance of ith and jth components: 1/10/2022 66
Characteristics of Hopfield Memory • Distributed (“holographic”) – every pattern is stored in every location (weight) • Robust – correct retrieval in spite of noise or error in patterns – correct operation in spite of considerable weight damage or noise 1/10/2022 67
Demonstration of Hopfield Net Run Malasri Hopfield Demo 1/10/2022 68
Stability of Imprinted Memories • Suppose the state is one of the imprinted patterns xm • Then: 1/10/2022 69
Interpretation of Inner Products • xk xm = n if they are identical – highly correlated • xk xm = –n if they are complementary – highly correlated (reversed) • xk xm = 0 if they are orthogonal – largely uncorrelated • xk xm measures the crosstalk between patterns k and m 1/10/2022 70
Cosines and Inner products u v 1/10/2022 71
Conditions for Stability 1/10/2022 72
Sufficient Conditions for Instability (Case 1) 1/10/2022 73
Sufficient Conditions for Instability (Case 2) 1/10/2022 74
Sufficient Conditions for Stability The crosstalk with the sought pattern must be sufficiently small 1/10/2022 75
Capacity of Hopfield Memory • Depends on the patterns imprinted • If orthogonal, pmax = n – but every state is stable trivial basins • So pmax < n • Let load parameter a = p / n 1/10/2022 equations 76
Single Bit Stability Analysis • For simplicity, suppose xk are random • Then xk xm are sums of n random 1 § § binomial distribution ≈ Gaussian in range –n, …, +n with mean m = 0 and variance s 2 = n • Probability sum > t: [See “Review of Gaussian (Normal) Distributions” on course website] 1/10/2022 77
Approximation of Probability 1/10/2022 78
Probability of Bit Instability 1/10/2022(fig. from Hertz & al. Intr. Theory Neur. Comp. ) 79
Tabulated Probability of Single-Bit Instability a Perror 1/10/2022 0. 1% 0. 105 0. 36% 0. 138 1% 0. 185 5% 0. 37 10% 0. 61 (table from Hertz & al. Intr. Theory Neur. Comp. ) 80
Spurious Attractors • Mixture states: – – – sums or differences of odd numbers of retrieval states number increases combinatorially with p shallower, smaller basins of mixtures swamp basins of retrieval states overload useful as combinatorial generalizations? self-coupling generates spurious attractors • Spin-glass states: – not correlated with any finite number of imprinted patterns – occur beyond overload because weights effectively random 1/10/2022 81
Basins of Mixture States 1/10/2022 82
Fraction of Unstable Imprints (n = 100) 1/10/2022 (fig from Bar-Yam) 83
Number of Stable Imprints (n = 100) 1/10/2022 (fig from Bar-Yam) 84
Number of Imprints with Basins of Indicated Size (n = 100) 1/10/2022 (fig from Bar-Yam) 85
Summary of Capacity Results • Absolute limit: pmax < acn = 0. 138 n • If a small number of errors in each pattern permitted: pmax n • If all or most patterns must be recalled perfectly: pmax n / log n • Recall: all this analysis is based on random patterns • Unrealistic, but sometimes can be arranged 1/10/2022 III B 86
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