Stochastic Network Optimization with Non-Convex Utilities and Costs

Michael J. NeelyUniversity of Southern California

http://www-rcf.usc.edu/~mjneelyInformation Theory and Applications Workshop (ITA), Feb. 2010

*Sponsored in part by the DARPA IT-MANET Program, NSF Career CCF-0747525, ARL

a1(t)

a2(t)

aK(t)

Utilit

y

Attribute x

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 m on slot t

ak(t) bk(t)

(these are general functions, possibly non-convex, discontinuous)

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

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!

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)U

tility

f m(x

)

Attribute x

Utility is only large when thruput exceeds a threshold.Global Optimality can be as hard as combinatorial bin-packing.

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!

Prior Work on Non-Stochastic (static) Non-ConvexNetwork Optimization:

•Lee, Mazumdar, Shroff, TON 2005

•Chiang 2008U

tility

f m(x

)

Attribute x

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.

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 )!

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!

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!

•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…

Solving the Problem via a Transformation:

Next Step: Lyapunov Optimization:

Use a “Primal” Derivative in Drift-Plus-Penalty:

∂ f( x(t) )∂ xmm

xm(w(t), a(t))D(t) + V

•Every slot t, observe w(t) and current queues Q(t).•Choose a(t) in Aw(t), a(t) in X to minimize…

where x(t) = (x1(t), …., xM(t)) = Empirical Running Time Avg. up to time t (starting from time 0)

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!

•Doesn’t need knowledge of traffic or channel statistics!•Can “approx” minimize to within constant C of infimum.

Theorem: Assuming the constraints are feasible, thenfor any parameter choice V ≥ 0, we have:1. All required constraints are satisfied. 2. All queues strongly stable with: E{Delay} ≤ O(V)3. Assuming the attribute vector converges with prob. 1, then Time Average Attribute vector is a “Near-Local-Min”:

∂ f( x(t) )∂ xmm

xm(w(t), a(t))D(t) + V

∂ f( x )∂ xmm

(xm - xm)* ≥ -(B +C)/V

where x* = (x1*, …, xM*) is any other feasible time average vector

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) )∂ xmm

xm(w(t), a(t))D(t) + V

∂ f( x(t) )∂ xmm

xm(w(t), a*(t))D*(t) + V≤ C +

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 isIndependent of queue state)…so….

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) )∂ xmm

xm(w(t), a(t))D(t) + V

∂ f( x(t) )∂ xmm

xm(w(t), a*(t))D*(t) + V≤ C + xmB *

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 isIndependent of queue state)…so….

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) )∂ xmm

xm(w(t), a(t))D(t) + V

∂ f( x(t) )∂ xmm

xm≤ C + B + V *

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

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 )∂ xmm

(xm - xm)* ≥ 0

where x* = (x1*, …, xM*) is any other feasible time average vector

All Constraints are still satisfied with this Variable-Valgorithm. However, queues are only “mean rate stable” (input rate = output rate) and have infinite averagecongestion and delay!

Extension 2: A 3-phase algorithm in special case whenUtility 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: bSubject to: 1) x = b qn

2) desired constraints 3) All queues stable

Unknown “Attribute Region”

Phase 2: Solve (to a local min) thedeterministic problem: Max: f(x1,…,xM)S.t.: (x1,…, xM) in Conv{b1q1, …,bnqn}

optimal x*

Extension 2: A 3-phase algorithm in special case whenUtility function f(x) is entrywise non-decreasing: Phase 3: Solve the convex stochastic net opt problem via pure dual method: Maximize: bSubject to: 1) x = b x*

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!

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 tochanges, convergence time issues unclear

•Second approach uses 3-phases, the stochastic partsare purely convex, and we can use the pure-dual methodto provide stronger performance guarantees.

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.

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.

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.