Neural Networks for Network. Anoop Ghanwani. Duke University

Transcription

1 Neural Networks for Network Optimization Anoop Ghanwani M.S. Thesis Department of Electrical Engineering Duke University Durham, NC 7708 April 11, 1995 Anoop Ghanwani, MS Thesis, Duke University, #1

2 Overview Introduction The Random Neural Network (RNN) Model The Vertex Covering Problem (VCP) { Simulated Annealing { The Hopeld Network { The RNN Approach { Experimental Results Multipoint Routing in ATM Networks (Joint work with E. Gelenbe and V. Srinivasan) { The Steiner Problem in Networks (SPN) { The Minimum Spanning Tree Heuristic (MSTH) { The Average Distance Heuristic (ADH) { The RNN Approach { Experimental Results Anoop Ghanwani, MS Thesis, Duke University, #

3 Introduction NP-complete problems have no known polynomial-time algorithm (Cormen, Leiserson, Rivest). In the worst case, nding a solution is equivalent to enumerating all solutions. Solution search space for a problem grows exponentially with problem-size. Problems of moderately large size are intractable for even the most powerful computers (Table from Rosen). Problem Size Bit Operations Used n log n n n n n! ;9 sec 10 ;8 sec 10 ;7 sec 10 ;6 sec 3 10 ;3 sec ;9 sec 10 ;7 sec 10 ;5 sec yr * ;8 sec 10 ;6 sec 10 ;3 sec * * :3 10 ;8 sec 10 ;5 sec 10 ;1 sec * * Nevertheless, these problems nd applications in computer science and operations research. Need to nd heuristics that deliver near-optimal solutions. Anoop Ghanwani, MS Thesis, Duke University, #3

4 Combinatorial Optimization by Neural Networks Optimizing neural networks: Hopeld Network { The Traveling Salesman Problem (TSP) (Hopeld, Tank). { Poor performance for TSP (Wilson, Pawley). { Theoretical investigations: Convergence to invalid solutions (Aiyer, Niranjan, Fallside). Local minima, convergence regions and spurious states (Abe). The Random Neural Network { Dynamical RNN for the TSP (Gelenbe, Koubi, Pekergin). { Vertex covering problem (Gelenbe, Batty). Anoop Ghanwani, MS Thesis, Duke University, #4

5 The Random Neural Network Model Introduced by E. Gelenbe. Applied to various problems: { Image texture generation (Atalay, Gelenbe, Yalabik). { Combinatorial optimization (Gelenbe, Koubi, Pekergin and Gelenbe, Batty). { Image compression (Gelenbe and Sungur). Anoop Ghanwani, MS Thesis, Duke University, #5

6 The RNN (continued) Pulse-based neural network. Neurons are modeled as counters. The value of the counter is the neuron's potential. The state of the network is represented by a vector of neuron potentials. Positive and negative signals circulate in the network. An excitatory signal increases the potential of a neuron by 1. An inhibitory signal decreases the potential of a neuron by 1. A Neuron is said to be excited if its potential is positive. Anoop Ghanwani, MS Thesis, Duke University, #6

7 The RNN (continued) q q 1 r ( p + 1 1i + p 1i ) r ( p + _ + p ) i i _ Λ i _ + i i i1 i1 _ + i i i i q r ( p + p ) q r ( p + p ) q i q j _ + j ji ji r ( p + p ) λ i + _ i i ik ik q r ( p + p ) Λ i λi q r p q r p j j + ji _ j j ji q i arrival rate of exogenous excitatory signals arrival rate of exogenous inhibitory signals arrival rate of excitatory signals from neuron j arrival rate of inhibitory signals from neuron j probability that the neuron is excited The quantity of interest is the steady state probability that a neuron is excited: q i = i + P n j=1 q j r j p + ji r i + i + P n j=1 q j r j p ; ji Anoop Ghanwani, MS Thesis, Duke University, #7

8 RNN and the Connectionist Model Connectionist Model: One weight per connection. { Positive weights are excitatory. { Negative weights are inhibitory. Information is carried by the \amplitude" of the signals. Learning algorithm may not support recurrent structures as in the \backpropagation" network. Random Neural Network Model: A pair of weights is used to represent each connection separate weights for excitatory and inhibitory connections. Information is carried by the \frequency" of the pulses. Full support for recurrent network structures. Anoop Ghanwani, MS Thesis, Duke University, #8

9 The Vertex Covering Problem v v 1 v 7 v 5 v 3 v 6 v 4 Given a graph G =(V E), nd a cover of minimum size. Acover of C of G is one that satises the following: { C V { For each edge(v i v j ) E, either v i C or v j C or both. The problem is NP-complete. { Proved by reducing the maximum clique problem to the VCP (Cormen, Leiserson, Rivest). Enumeration takes time proportional to O( jv j ). Anoop Ghanwani, MS Thesis, Duke University, #9

10 A Greedy Algorithm The following is a well-known greedy algorithm: 1. V 0 V E 0 E.. Select v V 0 having largest number of edges in E 0 adjacent to it. 3. C C S v. 4. V 0 V 0 ; v. 5. E 0 E 0 ;fall edges adjacent tovg. 6. If E 0 6=, gotostep. 7. The vertex cover is C. For a graph with maximum degree k, the worst-case error ratio is: P k i=1 1 i (Johnson). Anoop Ghanwani, MS Thesis, Duke University, #10

11 Simulated Annealing for the VCP Algorithm used is a modication of one used for the TSP by Teukolsky, Vetterling, Flannery. 1. Start with a high value of temperature.. Select a random set of vertices as the solution. 3. Repeat Steps 4 and 5 k times. 4. Do either of the following: Include in the cover a vertex not already in it. Remove a vertex from the cover. 5. Accept the new solution with probability P = exp( Cost old ; Cost new Temperature ). 6. If no solutions were accepted, terminate. 7. Reduce the temperature and go to Step 3. Anoop Ghanwani, MS Thesis, Duke University, #11

12 An Example of Simulated Annealing for Finding the Vertex Cover of a Graph With 00 Vertices x Energy Temperature Anoop Ghanwani, MS Thesis, Duke University, #1

13 The Hopeld Neural Network Introduced by John Hopeld. Neurons are modeled using ampliers with a high gain. Synaptic weights are modeled using resistors. { Excitatory connections using the non-inverting output. { Inhibitory connections using the inverting output. Anoop Ghanwani, MS Thesis, Duke University, #13

14 Hopeld Network Energy Function V T 1 1i V T i V i V T i V T i i1 i V T j ji I i V T i ij V i Tij output value of neuron i synaptic weight between neurons i and j Input to a neuron u i = P N j=1 V j T ij + I i. Output of a neuron V i = 1(1 + tanh u i u 0 ). State transitions minimize the following energy function: E = ; 1 NX NX i=1 j=1 V i V j T ij ; N X i=1 V i I i : Anoop Ghanwani, MS Thesis, Duke University, #14

15 Mapping Approach for the Hopeld Network (Ramanujam, Sadayappan) A single neuron V i is used to represent eachvertex in the graph. V i = 1 means the vertex is in the cover, and vice-versa. The following cost function is associated with the solution: 8 < NX E = C 1 : NX i=1 j=1 a ij ; NX NX i=1 j=1 V i a ij + NX NX i=1 j=1 V i V j a ij 9 = + C { A = [a ij ] is the adjacency matrix of the graph. { N is the number of vertices in the graph. { C 1 and C are the Lagrange multipliers. { Thersttermisgoestozerowhenthecover is valid. { The second term emphasizes minimality. { T ij = ;C 1 a ij and I i = C 1 P Ni=1 a ij ; C. NX i=1 V i Anoop Ghanwani, MS Thesis, Duke University, #15

16 Performance of the Hopeld Network Network always nds valid solutions. Performance was improved by nding a set of 100 solutions and selecting the best among them. { The order in which neurons are updated is randomized (similar to the Boltzmann machine). { Unlike the Boltzmann machine, the outputs are computed deterministically. Network performs poorly on large graphs. Anoop Ghanwani, MS Thesis, Duke University, #16

17 Mapping Approach for the RNN (Gelenbe, Batty) Two neurons N i and n i are used to represent each vertex in the graph. The vertex is in the cover if N i =1andn i =0. Provides for lateral inhibition. { N i and n i inhibit one another. { n i excites all neighbors of i. Anoop Ghanwani, MS Thesis, Duke University, #17

18 Setting the RNN Parameters (Gelenbe, Batty) Ni ni Ni = 0 ni = 0 p + N i n j = 0 = D i, the degree of vertex i = i, a constant p ; N i n j = 1, for i = j p + n i N j = 1=D i,ifj is a neighbor of i p ; n i N j = 0 r Ni = i, a constant r ni = D i q ni = q Ni i D i. i D i + i q Ni. = D i i + 1 i P jn i ( i + D i ) i. j D j + q Nj j. Anoop Ghanwani, MS Thesis, Duke University, #18

19 Algorithm to Solve the VCP Using the RNN 1. Initialize the model for all vertices having at least one edge adjacent to it.. Compute the q i values of all the neurons in the network. This implies iteration until the network reaches a steady state. 3. Include vertex i 0, with the largest value of q Ni,in the cover. 4. Remove vertex i 0 and all its adjacent edges from the graph. 5. If there are no edges left in the graph, terminate. 6. Go to step 1. Anoop Ghanwani, MS Thesis, Duke University, #19

20 Experimental Results 1.1 Ratio to optimal or "best" solution Greedy SA Hopfield RNN No. of vertices in the graph, V % of optimal or "best" solutions Greedy SA Hopfield RNN No. of vertices in the graph, V Anoop Ghanwani, MS Thesis, Duke University, #0

21 Comparison of Execution Time Greedy SA Hopfield RNN Execution time (sec) No. of vertices in the graph, V Anoop Ghanwani, MS Thesis, Duke University, #1

22 Conclusions Hopeld network performs well only for small graphs. Simulated annealing performs the best among all methods but is computationally very expensive. RNN performs better than the greedy algorithm for graphs of all sizes. Anoop Ghanwani, MS Thesis, Duke University, #

23 Multipoint Routing in ATM Networks Future networks must support point to multipoint communication. Source sends data to a subset of nodes in the network. Broadcast wastes resources. Could establish point-to-point connections. { Too many copies could cause an overload at the source. To minimize redundancy of data in the network, switches must be multicast capable. Routing is done by nding a minimum cost tree that spans the sender and all the intended receivers. Equivalent graph theoretic solution nd a Steiner tree. Anoop Ghanwani, MS Thesis, Duke University, #3

24 Related Work Bharath-kumar and Jae. { Minimization of network cost and destination cost. { Show that the multipoint routing problem is NP-complete. { Discuss several heuristics MSTH is the best. Waxman. { Experimental comparison of MSTH and ADH. { Proposes the dynamic multipoint problem. Jiang. { Broadband stream multicast. Kompella, Polyzos and Pasquale. { Modify the MSTH to minimize cost while maintaining a bounded end-to-end delay. Anoop Ghanwani, MS Thesis, Duke University, #4

25 The Steiner Problem in Networks Proposed by Hakimi. a b a Steiner points b d e f 1 1 c d e f 1 1 c original graph optimal Steiner tree Given: Aweighted, undirected graph G =(V E c), c : E! R. A set of destination vertices D V. To nd: A tree T of minimum cost such that there is a path between every pair of vertices in D. Anoop Ghanwani, MS Thesis, Duke University, #5

26 SPN (Continued) The problem is NP-complete (Karp). When jdj =, reduces to the shortest path problem. When jdj = jv j, reduces to the minimum spanning tree problem. Exact Algorithms: { Spanning tree enumeration algorithm (Hakimi) takes time proportional to jv j;jdj. { Dynamic programming algorithm (Dreyfus and Wagner) takes time proportional to jdj. Anoop Ghanwani, MS Thesis, Duke University, #6

27 The Minimum Spanning Tree Heuristic (Kou, Markowsky, Berman) 1. Construct a complete graph G 0 = (D E 0 ) where COST G 0(u v) is the length of the shortest path from u to v in G.. Construct a minimum spanning tree T 0 for G Construct a subgraph G 00 of G with all the vertices of T Construct a minimum spanning tree T 00 for G Remove successively any pendants which are non-destination vertices in T 00 to form a solution, T MSTH. Points of interest: Time-complexity iso(jdjjv j ). Has a worst-case error bound which tends to. Anoop Ghanwani, MS Thesis, Duke University, #7

28 An Example Demonstrating the MSTH a b a b d e f 1 1 c 3 3 d c original graph distance graph a b a b d c d c min span tree of distance graph MSTH solution Anoop Ghanwani, MS Thesis, Duke University, #8

29 The Average Distance Heuristic (Rayward-Smith) 1. Begin with a forest F of single node trees of the vertices in D.. Choose v V such thatf(v) is minimum where f(v) := min SF jsj>1 1 jsj ; 1 X T S 3. Let T 1 and T be two closest trees to v. d(v T): 4. Join T 1 and T by a shortest path through v. 5. If jf j > 1, go to Step, else terminate. Points of interest: Time-complexity iso(jv j 3 ). Finds a solution no worse than twice the optimal (Waxman). Anoop Ghanwani, MS Thesis, Duke University, #9

30 An Example Demonstrating the ADH a b a b d e f 1 1 c d c original graph start with 4 isolated trees a b a b d c d c join trees closest to a join trees closest to a a b f1 f f3 d join trees closest to b c (ADH solution) a b c d e f Anoop Ghanwani, MS Thesis, Duke University, #30

31 Mapping approach for the RNN The RNN is used as a tool for nding potential Steiner vertices not already in the solution. A single neuron n i is used to represent eachvertex in the graph. The network parameters are set as follows: { w + ij = A=Aij if A ij 6= 1. { w ; ij = 1 if A ij = 1. { A is the average edge cost in the graph. { All other network parameters are set to zero. { Compute r i = P j (w + ij + w ; ij). Anoop Ghanwani, MS Thesis, Duke University, #31

32 The Improved Heuristic 1. Set Y D and T 0 T.. Set up the Neural Network as discussed. (a) For every vertex y Y, set q y = 1. (b) Compute the output values q for all the other neurons. 3. Y Y S i,wherei = Y is the vertex corresponding to the neuron with highest output. 4. If i X, go to Step Create subgraph G 0 using the vertices in X S Y and the edges adjacent to those vertices. 6. Compute T 00, the minimum spanning tree for this subgraph. 7. Recursively remove all pendants from T 00 that are non-destination vertices. 8. If COST(T 00 ) < COST(T 0 ), T 0 T If Y 6= V,gotoStep. 10. Steiner tree produced by thernnist 0. Anoop Ghanwani, MS Thesis, Duke University, #3

33 Random Graph Model for Testing the Heuristics (Waxman) Place vertices at random integer coordinates on a jv jjv j grid. Edge cost is simply the Euclidean distance between the vertices. Retain an edge with the following probability: ;d(u v) P (u v) = exp L { L is the maximum length of an edge in the graph. { d(u v) is the cost of the edge (u v). A higher value of increases the proportion of larger edges. is used to vary the overall edge density. Anoop Ghanwani, MS Thesis, Duke University, #33

34 Results for 5 Node Networks 1.03 Ratio to optimal solution MSTH ADH RNN/MSTH RNN/ADH No. of destination vertices, D % of optimal solutions MSTH ADH RNN/MSTH RNN/ADH No. of destination vertices, D Anoop Ghanwani, MS Thesis, Duke University, #34

35 Results for 100 Node Networks MSTH ADH RNN/MSTH RNN/ADH Ratio to "best" solution No. of destination vertices, D % of "best" solutions MSTH ADH RNN/MSTH RNN/ADH No. of destination vertices, D Anoop Ghanwani, MS Thesis, Duke University, #35

36 An Example Where RNN Improves MSTH MSTH tree cost RNN-MSTH tree cost Anoop Ghanwani, MS Thesis, Duke University, #36

37 An Example Where RNN Improves ADH ADH tree cost RNN-ADH tree cost Anoop Ghanwani, MS Thesis, Duke University, #37

38 The Advantage of Using an Improved Heuristic A Simple Example Ability to support more connections leading to better utilization of resources. To minimize the average delay, we need to minimize: X F ij : (i j) C ij ; F ij a 3,1 c a c 3,1 5,1 (a) b 5,1 d (b) b d a 3,3 c 3,3 5,3 (c) b 5,3 d a c a c (d) b d (e) b d Anoop Ghanwani, MS Thesis, Duke University, #38

39 Conclusions RNN improves the quality of solutions found by MSTH and ADH. Quality of the solution depends largely on the solution found by the initial heuristic. Reduction in tree cost leads to better utilization of available resources. Anoop Ghanwani, MS Thesis, Duke University, #39