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Distributed and Secure Computation of Convex Programs over a Network of Connected Processors

f 2 (x, p 2 ) b 2. 2. k. f 1 (x, p 1 ) b 1. f k (x, p k ) b k. 1. 3. Distributed and Secure Computation of Convex Programs over a Network of Connected Processors. Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely. Context:.

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Distributed and Secure Computation of Convex Programs over a Network of Connected Processors

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  1. f2(x, p2) b2 2 k f1(x, p1) b1 fk(x, pk) bk 1 3 Distributed and Secure Computation of Convex Programs over a Network of Connected Processors Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely

  2. Context: Parallel Processing and Distributed Sub-Gradient Algorithms: -Tsitsiklis, Bertsekas, Athens [1986] -Ferris, Mangasarian [1991] -Bertsekas, Tseng [1995] -Miller, Stout [1996] Sorting and Averaging over Graphs: -Nassimi, Sahni [1979] -Bordim, Nakano, Shen [2002] -Kempe, Dobra, Gehrke [2003] -Singh, Prasanna, Rolim [2003] Distributed Computation of Eigenvectors over Graphs: -Kempe, McSherry [2004] Distributed Computation of Linear Programs for Networks: -Bartal, Byers, Raz [2004]

  3. Problem A: A General Convex Program 2 K 1 3

  4. 2 K 1 3 Assign each set of constraints and utility term to a different processor…

  5. 2 K 1 3 Assign each set of constraints and utility term to a different processor…

  6. 2 K 1 3 Assign each set of constraints and utility term to a different processor…

  7. 2 K 1 3 Assign each set of constraints and utility term to a different processor…

  8. 2 K 1 3 Assign each set of constraints and utility term to a different processor… How to ensure all public variable constraints?

  9. 2 K 1 3 Idea: Define different variables at each node k. How to ensure all public variable constraints?

  10. 2 K 1 3 Idea: Define different variables at each node k.

  11. 2 K 1 3 Idea: Define different variables at each node k.

  12. 2 K 1 3 Idea: Define different variables at each node k.

  13. shortest path tree K 2 1 3 Problem B:

  14. Assume there is a point and a positive value e such that: Interior Point Assumption: e e

  15. Assume there is a point and a positive value e such that: Interior Point Assumption: emax emax

  16. Iterative Algorithm: We develop a distributed procedure where each node performs “update” computations every timeslot t = {0, 1, 2, …, t}.

  17. Algorithm motivated by Queueing Theory: Update Equation:

  18. Likewise, for constraints: (the d[t] vector is needed because there is no interior point associated with the above constraints)

  19. A measure of the parent-child inequalities -- Define:

  20. Initialize all queue backlogs to zero for t=0 Each node k transmits to its parent node. Each node k computes as solutions to: Each node k passes to its children Each node k updates according to the queueing eqs. The Distributed Algorithm: fix a parameter V > 0 On iteration t (where t=0, 1, 2, …) do:

  21. Algorithm Security:

  22. Analysis via Lyapunov Drift. Define Lyapunov function:

  23. f2(x, p2) b2 2 k f1(x, p1) b1 fk(x, pk) bk 1 3 Conclusions: Computation of General Convex Programs over Graphs Analysis via Lyapunov Drift / Queueing Theory Solution is given by an average, improved every slot (differs from classical subgradient methods, which often require solutions for each slot to be evaluated and compared). No initial seed point is necessary. Enables Distributed Computation and Maintains Privacy/Security

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