Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks

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1 Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks Peter Tiňo School of Computer Science University of Birmingham, UK Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p./25

2 Motivation (Smith, 995) Self-Organizing Neural Network (SONN) - general methodology for solving 0- assignment problems Wide variety of successful applications - assembly line sequencing, frequency assignment in mobile communications, etc. Softmax renormalization - incorporated into SONN in (Guerrero et al., 2002). Can boost performance dramatically. Softmax function contains a free parameter - temperature T. At critical temperatures - powerful intermittent search for high-quality solutions. No theory of critical temperatures. Trial-and error settings. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.2/25

3 SONN with softmax weight renormalization finite set of items i I = {,2,...,N} to be assigned to items j J = {,2,...,M} so that global cost Q(A) of assignment A : I J is minimized. V (i,j) - Partial cost of assigning i I to j J. The "strength" of assigning i to j - "weight" w i,j (0,). 0- assignment solution - produced by imposing A(i) = j iff j = argmax k I w i,k. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.3/25

4 SONN Algorithm Initialize weights w i,j randomly to small values (around 0.5). Choose an j c J. Calculate partial costs V (i,j c ), i I, of all possible assignments to j c. "Winner" item i(j c ) I - the one that minimizes V (i,j c ). Neighborhood B L (i(j c )) of size L of i(j c ) - L items (nodes) i i(j c ) that yield the smallest partial costs V (i,j c ). Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.4/25

5 SONN Algorithm - cont d Weights from nodes i B L (i(j c )) to j c get strengthened: w i,jc w i,jc + η(i)( w i,jc ), i B L (i(j c )), where η(i) is the neighborhood function η(i) = β exp and k(j c ) = argmax i I V (i,j c ). { V (i(j } c),j c ) V (i,j c ), V (k(j c ),j c ) V (i,j c ) Weights w i = (w i,,w i,2,...,w i,n ) for each input node i I are normalized using softmax w i,j exp(w i,j T ) N k= exp(w i,k T ). Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.5/25

6 SONN strengthen assignment weights 2 j M SoftMax normalization 2 j M w i,j i N V(,j)V(2,j) V(i,j) V(N,j) i N B (i(j)) 2 Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.6/25

7 Weight updates w 3,j S for fixed input j update the assignment weights w 2,j w,j w 2,j w 3,j w,j Softmax renormalization Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.7/25

8 SoftMax renormalization - temperature T w 3,j S for fixed input j update the assignment weights w 2,j w,j w 2,j w 3,j w,j Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.8/25

9 Autonomous dynamics - ISM w 3,j w 3,j S S w 2,j w 2,j w,j w,j High temperature regime Low temperature regime Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.9/25

10 Critical temperatures Renormalization step is crucial for intermittent search by SONN for globally optimal assignment solutions. Symmetry breaking bifurcation of equilibria of the renormalization procedure - temperatures T at which optimal (both in terms of quality and quantity of found solutions) intermittent search takes place. Formulate analytical approximations to the critical temperatures. Good correspondence with experimentally found values. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.0/25

11 Iterative SoftMax (ISM) (N )-dimensional simplex in R N : S N = {w = (w,w 2,...,w N ) T R N w i 0, N w i = }. i= Given a temperature T > 0, the softmax maps R N into S N : where w F(w;T) = (F (w;t),f 2 (w;t),...,f N (w;t)) T, ISM on S N : F i (w;t) = exp( w i T ) N i =,2,...,N. k= exp(w k T ), w(t + ) = F(w(t);T). Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p./25

12 Equilibria of ISM Maximum entropy point w = (N,N,...,N ) is a fixed point of ISM for any T. Except for the maximum entropy fixed point all the other fixed points must have exactly 2 distinct coordinate values. Write the larger of the two fixed-point coordinates as γ (w;t) = αn, α (,N). The richest structure of equilibria is found in the center of S N (around w). The least number of fixed points exist around the vertices, i.e. in the "corner" areas of S N. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.2/25

13 Number of ISM Equilibria Theorem: Fix α (,N) and write γ = αn. Let l min be the smallest natural number greater than (α )/γ. Then, for l {l min,l min +,...,N }, at temperature T e (γ ;N,l) = (α ) [ ( l ln α )], lγ there exist ( N l ) distinct fixed points of ISM, with (N l) coordinates having value γ and l coordinates equal to γ 2 = γ (N l). l Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.3/25

14 Number of ISM Equilibria (N = 0) # possible fixed points gamma Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.4/25

15 Stable ISM Equilibria Theorem: Consider a fixed point w S N of ISM with one of its coordinates equal to N γ <. Then, if T s (γ ) = { Ts,2 (γ ) = γ, if γ [N,/2) T s, (γ ) = 2 γ ( γ ), if γ [/2,). Then, if T > T s (γ ), the fixed point w is stable. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.5/25

16 Unstable ISM Equilibria Theorem: Consider a fixed point w S N of ISM with one of its coordinates equal to N γ <. Let N l be the number of coordinates of value γ. Then if T < T u (γ ;N,l) = γ (2 Nγ ) N l Nl + N Nl, w is not stable. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.6/25

17 stability and existence of ISM fixed points (N=0) stable equilibria potentially saddle type equilibria N = stable equilibria T T N = N =4 N =3 unstable equilibria gamma N=5 N=0 N=5 N=40 N= gamma N - number of coordinates with value γ. Bold line - T s (γ ) Solid normal lines - T u (γ ;N,l) Dashed lines - T e (γ ;N,l) Bifurcation temperature is decreasing with increasing N. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.7/25

18 Approximations to the critical temperature. Expand T e (γ ;N,N ) around γ 0 polynomial T (2) N (γ ). = 0.9 as a second-order 2. Solve T (2) N (γ ) = T s (γ ) for γ. 3. Plug the solution γ (2) back to T s, i.e. calculate T (2) (N) = T s (γ (2) ) A cheaper approximation T () (N) can be obtained using linear expansions. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.8/25

19 Example: N-Queen problem Originally posed in 9th century. A benchmark for constraint satisfaction and assignment problems. Place N queens onto an N N chessboard without attacking each other. J = {,2,...,N} and I = {,2,...,N} index the columns and rows, respectively, of the chessboard. Partial cost V (i,j) evaluates the diagonal and column contributions to the global cost Q of placing a queen on column j of row i. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.9/25

20 Experiments Increasing problem size N. Experimentally found the best regimes for intermittent search: optimal neighborhood size L optimal temperature T Compared with analytically predicted critical temperatures T () (N) and T (2) (N). Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.20/25

21 bifurcation temperatures T B (qdr. approx) T B (lin. approx) T B T * (N queens) 0.2 T max. entropy point stable 0.05 max. entropy point unstable N bold solid and dashed lines - T (2) (N) and T () (N), respectively circles - experimentally found bifurcation temperatures stars - best performing temperatures for intermittent search in the N-queens problems Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.2/25

22 Understanding intermittent search in SONN Equilibria that can be guaranteed to be stable - maximum entropy point w and "one-hot" solutions at the vertexes of the N dimensional simplex. Each "one-hot" solution corresponds to one particular assignment of SONN inputs to SONN outputs. Most powerful intermittent search - close to the critical temperature at which "one-hot" equilibria lose stability. "One-hot" solutions do not exhibit a strong attractive force in the ISM state space. SONN weight updates can easily jump from one assignment to another, occasionally being pulled by the neutralizing power of the stable maximum entropy equilibrium w. Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.22/25

23 Detailed bifurcation structure Bifurcation structure of ISM equilibria (N=0) N = T E (0,4) T E (0,3) T E (0,2) T E (0,) 0.8 gamma N =3 N =2 N = temperature T Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.23/25

24 Positions of ISM Equilibria Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.24/25

25 Stable/unstable manifolds w 3 w 3 w 2 w 2 w w Critical Temperatures for Intermittent Search in Self-Organizing Neural Networks p.25/25