1 / 23

Stochastic Network Optimization with Non-Convex Utilities and Costs

Stochastic Network Optimization with Non-Convex Utilities and Costs. a 1 (t). Utility. a 2 (t). a K (t ). Attribute x. Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely Information Theory and Applications Workshop (ITA), Feb. 2010.

orsen
Download Presentation

Stochastic Network Optimization with Non-Convex Utilities and Costs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Stochastic Network Optimization with Non-Convex Utilities and Costs a1(t) Utility a2(t) aK(t) Attribute x Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely Information Theory and Applications Workshop (ITA), Feb. 2010 *Sponsored in part by the DARPA IT-MANET Program, NSF Career CCF-0747525, ARL

  2. ak(t) bk(t) • Problem Description: • K Queue Network --- (Q1(t), …, QK(t)) • Slotted time, t in {0, 1, 2, … } • w(t) = “Random Network Event” (e.g., arrivals, channels, etc.) • a(t) = “Control Action” (e.g., power allocation, routing, etc.) • Decision: • Observe w(t) every slot. Choose a(t) in Aw(t). • Affects, arrivals, service, and “Network Attributes”: • ak(t) = ak(w(t),a(t)) = arrivals to queue k on slot t • bk(t) = bk(w(t),a(t)) = service to queue k on slot t • xm(t) = xm(w(t),a(t)) = Network Attribute mon slot t (these are general functions, possibly non-convex, discontinuous)

  3. What are “Network Attributes” ? • x(t) = (x1(t), …, xM(t)) • Traditional: • Packet Admissions / Throughput • Power Expenditures • Packet Drops • Emerging Attributes for Network Science: • Quality of Information (QoI) Metrics • Distortions • Profit • Real-Valued Meta-Data

  4. Define Time Averages: • x = ( x1 , …, xM) • Goal: • Minimize : f( x ) • Subject to: 1) gn( x ) ≤ 0 for n in {1, …, N} • 2) x in X • 3) All queues Qk(t) stable • Where: • X is an abstract convex set • gn(x) are convex functions • f(x) is a possibly non-convex function!

  5. Example Problem 1: Maximizing non-concave thruput-utility x = ( x1 , …, xM) = time avg “thruput” attribute vector f(x) = Non-Concave Utility = f1(x1) + f2(x2) + … + fM(xM) Utility fm(x) Attribute x Utility is only large when thruput exceeds a threshold. Global Optimality can be as hard as combinatorial bin-packing.

  6. Example Problem 2: Risk-Aware Networking (Variance Minimization) Let p(t) = “Network Profit” on slot t. Define Attributes: x1(t) = p(t) x2(t) = p(t)2 Then: Var(p) = E{p2} – E{p}2 = x2 – ( x1 )2 Minimizing variance minimizes a non-convex function of a time-average! Non-Convex!

  7. Prior Work on Non-Stochastic (static) Non-Convex • Network Optimization: • Lee, Mazumdar, Shroff, TON 2005 • Chiang 2008 Utility fm(x) Attribute x

  8. Prior Work on Stochastic, Convex Network Optimization: • Dual-Based: • Neely 2003, 2005, Georgiadis, Neely, Tassiulas F&T 2006 • Explicit optimality, performance, convergence analysis via a • “drift-plus-penalty” alg: [O(1/V), O(V)] Performance-Delay tradeoff • Eryilmaz, Srikant 2005 (“fluid model,” infinite backlog) • Primal-Dual-Based: • Agrawal, Subramanian 2002 (no queues, infinite backlog) • Kushner, Whiting 2002 (no queues, infinite backlog) • Stolyar 2005, 2006 (with queues, but “fluid model”): • Proves optimality over a “fluid network.” • Conjectures that the actual network utility approaches • optimal when a parameter is scaled.

  9. Summary: • 1) Optimizing a time average of a non-convex function is Easy! • (can find global optimum Georgiadis, Neely, Tassiulas F&T 2006). • 2) Optimizing a non-convex function of a time average is Hard! • (CAN WE FIND A LOCAL OPTIMUM??) • Drift-Plus-Penalty with “Pure-Dual” Algorithm: • Works great for convex problems • Robust to changes, has explicit performance, convergence bounds • BUT: For non-convex problems, it would find global optimum of • the time average of f(x), which is not necessarily • even a local optimum of f( x ). • Drift-Plus-Penalty with “Primal-Dual” Component: • OUR NEW RESULT: Works well for non-convex! • Can find a local optimum of f( x )!

  10. Solving the Problem via a Transformation: Original Problem: Min: f( x ) Subject to: 1) gn( x ) ≤ 0 , n in {1,…,N} 2) x in X 3) All Queues Stable Transformed Problem: Min: f( x ) Subject to: 1) gn( g ) ≤ 0 , n in {1,…,N} 2) gm = xm , for all m 3) g(t) in X , for all t 4) All Queues Stable Auxiliary Variables: g(t) =(g1(t), …, gM(t)). These act as a proxy for x(t) = (x1(t), …, xM(t)). Constraints in the new problem are time averages of functions, not functions of time averages! And the problems are equivalent!

  11. Solving the Problem via a Transformation: Next Step: Lyapunov Optimization: Transformed Problem: Min: f( x ) Subject to: 1) gn( g ) ≤ 0 , n in {1,…,N} 2) gm = xm , for all m 3) g(t) in X , for all t 4) All Queues Stable • Define Virtual Queue for each inequality and equality constraint • Q(t) = vector of virtual and actual queues. • Use Quadratic Lyapunov function, Drift = D(t) • Use Min Drift-Plus-Penalty… Auxiliary Variables: g(t) =(g1(t), …, gM(t)). These act as a proxy for x(t) = (x1(t), …, xM(t)). Constraints in the new problem are time averages of functions, not functions of time averages! And the problems are equivalent!

  12. Use a “Primal” Derivative in Drift-Plus-Penalty: • Every slot t, observe w(t) and current queues Q(t). • Choose a(t) in Aw(t), a(t) in X to minimize… ∂ f( x(t) ) xm(w(t),a(t)) D(t) + V ∂ xm m where x(t) = (x1(t), …., xM(t)) = Empirical Running Time Avg. up to time t (starting from time 0) • Doesn’t need knowledge of traffic or channel statistics! • Can “approx” minimize to within constant C of infimum. Note: “Pure Dual” Algorithm Minimizes D(t) + Vf(g(t)), does not need running time average, is more robust to varying parameters and provides stronger guarantees, but only works for convex f() functions!

  13. ∂ f( x(t) ) xm(w(t),a(t)) D(t) + V ∂ xm m Theorem: Assuming the constraints are feasible, then for any parameter choice V ≥ 0, we have: All required constraints are satisfied. All queues strongly stable with: E{Delay} ≤ O(V) Assuming the attribute vector converges with prob. 1, then Time Average Attribute vector is a “Near-Local-Min”: ∂ f( x ) (xm - xm) ≥ -(B +C)/V * ∂ xm m where x* = (x1*, …, xM*) is any other feasible time average vector

  14. Proof Sketch: Very Simple Proof! Because we take actions to minimize the drift-plus-penalty every slot (given current queue states) to within C, we have: ∂ f( x(t) ) xm(w(t),a(t)) D(t) + V ∂ xm m ∂ f( x(t) ) xm(w(t), a*(t)) D*(t) + V ≤ C + ∂ xm m where D*(t) and a*(t) are the drift and decision under any other (possibly randomized) decision choices! But for any feasible time average vector x*, there are choices that make the drift zero (plus a constant B that is Independent of queue state)…so….

  15. Proof Sketch: Very Simple Proof! Because we take actions to minimize the drift-plus-penalty every slot (given current queue states) to within C, we have: ∂ f( x(t) ) xm(w(t),a(t)) D(t) + V ∂ xm m xm * B ∂ f( x(t) ) xm(w(t), a*(t)) D*(t) + V ≤ C + ∂ xm m where D*(t) and a*(t) are the drift and decision under any other (possibly randomized) decision choices! But for any feasible time average vector x*, there are choices that make the drift zero (plus a constant B that is Independent of queue state)…so….

  16. Proof Sketch: Very Simple Proof! Because we take actions to minimize the drift-plus-penalty every slot (given current queue states) to within C, we have: ∂ f( x(t) ) xm(w(t),a(t)) D(t) + V ∂ xm m ∂ f( x(t) ) * xm ≤ C + B + V ∂ xm m • The rest follows by (see [Georgiadis, Neely, Tassiulas, F&T 2006)]: • Iterated Expectations: E{E{X|Y}} = E{X} • Telescoping Sums: • [f(4) – f(3)] + [f(3) –f(2)] + [f(2) – f(1)] + [f(1) – f(0)] = f(4) – f(0) • Rearranging Terms and Taking Limits

  17. Extension 1: Using a “Variable V(t)” algorithm with increasing V(t): V(t) = (1+t)d (for 0 < d < 1) gives a true local min: ∂ f( x ) (xm - xm) ≥ 0 * ∂ xm m where x* = (x1*, …, xM*) is any other feasible time average vector All Constraints are still satisfied with this Variable-V algorithm. However, queues are only “mean rate stable” (input rate = output rate) and have infinite average congestion and delay!

  18. Extension 2: A 3-phase algorithm in special case when Utility function f(x) is entrywise non-decreasing: Phase 1: Pick Directions {q1, …, qN}. Solve the convex stochastic net opt problem via pure dual method: Maximize: b Subject to: 1) x = bqn 2) desired constraints 3) All queues stable Unknown “Attribute Region” Phase 2: Solve (to a local min) the deterministic problem: Max: f(x1,…,xM) S.t.: (x1,…, xM) in Conv{b1q1, …,bnqn} optimal x*

  19. Extension 2: A 3-phase algorithm in special case when Utility function f(x) is entrywise non-decreasing: Phase 3: Solve the convex stochastic net opt problem via pure dual method: Maximize: b Subject to: 1) x = bx* 2) desired constraints 3) All queues stable x* This involves 1 purely deterministic non-convex phase (any static solver can be used) and 2 purely convex stochastic network optimizations!

  20. Conclusions: • We have studied techniques for non-convex stochastic • network optimization. • “Primal-Dual” partial derivative info used with • Drift-Plus-Penalty metric for achieving local min. • Requires a running time average, not as robust to • changes, convergence time issues unclear • Second approach uses 3-phases, the stochastic parts • are purely convex, and we can use the pure-dual method • to provide stronger performance guarantees.

  21. Some Possible Questions: • 1) Why do we use auxiliary variables? • They allow treatment of the abstract set • constraint • They allow the constraints of the problem • to be transformed into constraints on time • averages of functions, rather than functions • of time averages. • This enables explicit bounds on convergence • times. • It also ensures the constraint satisfaction • is robust to system changes, even if the • non-convex utility optimization is not.

  22. Some Possible Questions: • 2) How is the first method different from • prior stochastic primal-dual methods? • We use auxiliary variables • We treat the convex inequality constraints via • a “pure-dual” (no derivatives) to get stronger • proof that all constraints are met, and to within • a known convergence time • We treat abstract set constraints • We treat the non-convex problem (the lack of • convergence time knowledge for the utility part • is due to the “primal” component, but this is • the price of treating non-convex problems!) • We treat joint queue stability and utility optimization, • with a proof that is even simpler than the fluid limit • proof given for the special case of convex problems in • Stolyar 05, 06.

  23. Some Possible Questions: • 3) Why do we consider the 3-phase algorithm? • Uses 2 pure convex stochastic problems (and • so the stochastic parts have stronger and • more explicit convergence time guarantees, do • not require derivatives to exist). • The 1 non-convex optimization is a pure • deterministic problem, from which we can use • any known deterministic solver (such as “brute • force,” or “Nelder-Mead,” or “Newton-type” • methods that do not necessarily restrict to small • step sizes.

More Related