( ) T. Reading. Lecture 22. Definition of Covariance. Imprinting Multiple Patterns. Characteristics of Hopfield Memory

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1 Part 3: Autonomous Agents /8/07 Reading Lecture 22 Flake, ch. 20 ( Genetics and Evolution ) /8/07 /8/07 2 Imprinting Multiple Patterns Let x, x 2,, x p be patterns to be imprinted Define the sum-of-outer-products matrix: W ij = n p " k= x i k x j k T W = n x k x k p " k= /8/07 3 Definition of Covariance Consider samples (x, y ), (x 2, y 2 ),, (x N, y N ) Let x = x k and y = y k Covariance of x and y values : C xy = ( x k " x )( y k " y ) = x k y k " x y k " x k y + x # y = x k y k " x y k " x k y + x # y = x k y k " x # y " x # y + x # y C xy = x k y k " x # y /8/07 4 Weights & the Covariance Matrix Sample pattern vectors: x, x 2,, x p Covariance of i th and j th components: C ij = x k k i x j " x i # x j If "i : x i = 0 (± equally likely in all positions) : C ij = x i k x j k " W = p n C " p = p x i k y j k k= 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 /8/07 5 /8/07 6

2 Part 3: Autonomous Agents /8/07 Stability of Imprinted Memories Suppose the state is one of the imprinted patterns x m [ " T k ] x m Then: h = Wx m = n x k x k = n " x k ( x k ) T x m k = x m n ( x m ) T x m + n " x k ( x k ) T x m = x m + n "( x k x m )x k /8/07 7 Interpretation of Inner Products x k x m = n if they are identical highly correlated x k x m = n if they are complementary highly correlated (reversed) x k x m = 0 if they are orthogonal largely uncorrelated x k x m measures the crosstalk between patterns k and m /8/07 8 Cosines and Inner products u" v = u v cos# uv If u is bipolar, then u 2 = u" u = n " uv u v Conditions for Stability Stability of entire pattern : x m = sgn' x m + n & ( x k cos" km * ) Hence, u" v = n n cos# uv = ncos# uv /8/07 9 Stability of a single bit : x m i = sgn' x m i + n & ( cos" km * ) /8/07 0 Sufficient Conditions for Instability (Case ) Suppose x i m = ". Then unstable if : (") + n cos# km > 0 km n cos" km > Sufficient Conditions for Instability (Case 2) Suppose x i m = +. Then unstable if : ( +) + n cos" km < 0 n cos" km < # km /8/07 /8/07 2 2

3 Part 3: Autonomous Agents /8/07 Sufficient Conditions for Stability cos" km n The crosstalk with the sought pattern must be sufficiently small Capacity of Hopfield Memory Depends on the patterns imprinted If orthogonal, p max = n but every state is stable trivial basins So p max < n Let load parameter α = p / n /8/07 3 /8/07 4 equations Single Bit Stability Analysis For simplicity, suppose x k are random Then x k x m are sums of n random ± binomial distribution Gaussian in range n,, +n with mean µ = 0 and variance σ 2 = n Probability sum > t: ) " erf # t &, 2 + (. * 2n '- [See Review of Gaussian (Normal) Distributions on course website] /8/07 5 Approximation of Probability Let crosstalk C i m = n x i k x k " x m We want Pr C i m > Note : nc i m = { } = Pr{ nc m i > n} p n "" k= j= x i k x j k x j m A sum of n(p ") # np random ±s Variance " 2 = np /8/07 6 Probability of Bit Instability { } = " erf n + 2 Pr nc i m > n = 2 ) # &, (. * + 2np '-. [ " erf( n 2p) ] Tabulated Probability of Single-Bit Instability P error α /8/07 (fig. from Hertz & al. Intr. Theory Neur. Comp.) 7 /8/07 (table from Hertz & al. Intr. Theory Neur. Comp.) 8 3

4 Part 3: Autonomous Agents /8/07 Spurious Attractors Mixture states: sums or differences of odd numbers of retrieval states number increases combinatorially with p shallower, smaller basins 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 Basins of Mixture States x k x k 3 x mix /8/07 9 x k 2 x i mix = sgn x i k + x i k 2 + x i k 3 /8/07 20 Fraction of Unstable Imprints (n = 00) Number of Stable Imprints (n = 00) /8/07 (fig from Bar-Yam) 2 /8/07 (fig from Bar-Yam) 22 Number of Imprints with Basins of Indicated Size (n = 00) Summary of Capacity Results Absolute limit: p max < α c n = 0.38 n If a small number of errors in each pattern permitted: p max n If all or most patterns must be recalled perfectly: p max n / log n Recall: all this analysis is based on random patterns Unrealistic, but sometimes can be arranged /8/07 (fig from Bar-Yam) 23 /8/

5 Part 3: Autonomous Agents /8/07 Trapping in Local Minimum Stochastic Neural Networks (in particular, the stochastic Hopfield network) /8/07 25 /8/07 26 Escape from Local Minimum Escape from Local Minimum /8/07 27 /8/07 28 Motivation Idea: with low probability, go against the local field move up the energy surface make the wrong microdecision Potential value for optimization: escape from local optima Potential value for associative memory: escape from spurious states because they have higher energy than imprinted states The Stochastic Neuron Deterministic neuron : s " i = sgn h i Pr { s " i = +} = # h i Pr { s " i = } = # h i Stochastic neuron : Pr { s " i = +} = #( h i ) Pr { s " i = } =# h i Logistic sigmoid : " h = + exp #2 h T σ(h) h /8/07 29 /8/

6 Part 3: Autonomous Agents /8/07 Properties of With Varying T = " h + e #2h T As h +, σ(h) As h, σ(h) 0 σ(0) = /2 T varying from 0.05 to (/T = β = 0,, 2,, 20) /8/07 3 /8/07 32 T = 0.5 T = 0.0 Slope at origin = / 2T /8/07 33 /8/07 34 T = 0. T = /8/07 35 /8/

7 Part 3: Autonomous Agents /8/07 T = 0 T = 00 /8/07 37 /8/07 38 Pseudo-Temperature Temperature = measure of thermal energy (heat) Thermal energy = vibrational energy of molecules A source of random motion Pseudo-temperature = a measure of nondirected (random) change Logistic sigmoid gives same equilibrium probabilities as Boltzmann-Gibbs distribution /8/07 39 Transition Probability Recall, change in energy "E = #"s k h k = 2s k h k Pr { s " k = ±s k = m} = #( ±h k ) = #( s k h k ) Pr{ s k " #s k } = + exp 2s k h k T = + exp E T /8/07 40 Stability Are stochastic Hopfield nets stable? Thermal noise prevents absolute stability But with symmetric weights: average values s i become time - invariant Does Thermal Noise Improve Memory Performance? Experiments by Bar-Yam (pp ): n = 00 p = 8 Random initial state To allow convergence, after 20 cycles set T = 0 How often does it converge to an imprinted pattern? /8/07 4 /8/

8 Part 3: Autonomous Agents /8/07 Probability of Random State Converging on Imprinted State (n=00, p=8) Probability of Random State Converging on Imprinted State (n=00, p=8) T = / β /8/07 (fig. from Bar-Yam) 43 /8/07 (fig. from Bar-Yam) 44 Analysis of Stochastic Hopfield Network Complete analysis by Daniel J. Amit & colleagues in mid-80s See D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks, Cambridge Univ. Press, 989. The analysis is beyond the scope of this course Phase Diagram (D) all states melt (A) imprinted = minima (C) spin-glass states (B) imprinted, but s.g. = min. /8/07 45 /8/07 (fig. from Domany & al. 99) 46 Conceptual Diagrams of Energy Landscape Phase Diagram Detail /8/07 (fig. from Hertz & al. Intr. Theory Neur. Comp.) 47 /8/07 (fig. from Domany & al. 99) 48 8