1. University of Southern California
Distributed and Secure Computation of Convex Programs over a Network of Connected Processors
f2(x, p2) b2
f1(x, p1) b1
fk(x, pk) bk 1 2 3 k
Michael J. Neely
University of Southern California

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 Program2 K 1 3

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

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

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

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

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

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

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

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

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

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

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

15. Interior Point Assumption:
Assume there is a point and a positive
value e such that:
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. The Distributed Algorithm: fix a parameter V > 0
Initialize all queue backlogs
to zero for t=0
On iteration t (where t=0, 1, 2, …) do:
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.

21. Algorithm Security:

23. Analysis via Lyapunov Drift. Define Lyapunov function:

24. Conclusions: f2(x, p2) b2 fk(x, pk) bk 2 f1(x, p1) b1 k 1 3
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