Dynamic Optimization and Learning for Renewal Systems

Michael J. Neely, University of Southern CaliforniaAsilomar Conference on Signals, Systems, and Computers, Nov. 2010

PDF of paper at: http://ee.usc.edu/stochastic-nets/docs/renewal-systems-asilomar2010.pdfSponsored in part by the NSF Career CCF-0747525, ARL Network Science Collaborative Tech. Alliance

tT/R

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T/RNetwork

Coordinator

Task 1

Task 2

Task 3

T[0] T[1] T[2]

A General Renewal System

tT[0] T[1] T[2]

y[2]y[1]y[0]

•Renewal Frames r in {0, 1, 2, …}.•π[r] = Policy chosen on frame r.•P = Abstract policy space (π[r] in P for all r).•Policy π[r] affects frame size and penalty vector on frame r.

π[r]•y[r] = [y0(π[r]), y1(π[r]), …, yL(π[r])]

•T[r] = T(π[r]) = Frame Duration

A General Renewal System

tT[0] T[1] T[2]

y[2]y[1]y[0]

•Renewal Frames r in {0, 1, 2, …}.•π[r] = Policy chosen on frame r.•P = Abstract policy space (π[r] in P for all r).•Policy π[r] affects frame size and penalty vector on frame r. These are random functions of π[r] (distribution depends on π[r]):

π[r]•y[r] = [y0(π[r]), y1(π[r]), …, yL(π[r])]

•T[r] = T(π[r]) = Frame Duration

A General Renewal System

tT[0] T[1] T[2]

y[2]y[1]y[0]

•Renewal Frames r in {0, 1, 2, …}.•π[r] = Policy chosen on frame r.•P = Abstract policy space (π[r] in P for all r).•Policy π[r] affects frame size and penalty vector on frame r. These are random functions of π[r] (distribution depends on π[r]):

π[r]•y[r] = [1.2, 1.8, …, 0.4]

•T[r] = 8.1 = Frame Duration

A General Renewal System

tT[0] T[1] T[2]

y[2]y[1]y[0]

•Renewal Frames r in {0, 1, 2, …}.•π[r] = Policy chosen on frame r.•P = Abstract policy space (π[r] in P for all r).•Policy π[r] affects frame size and penalty vector on frame r. These are random functions of π[r] (distribution depends on π[r]):

π[r]•y[r] = [0.0, 3.8, …, -2.0]

•T[r] = 12.3 = Frame Duration

A General Renewal System

tT[0] T[1] T[2]

y[2]y[1]y[0]

•Renewal Frames r in {0, 1, 2, …}.•π[r] = Policy chosen on frame r.•P = Abstract policy space (π[r] in P for all r).•Policy π[r] affects frame size and penalty vector on frame r. These are random functions of π[r] (distribution depends on π[r]):

π[r]•y[r] = [1.7, 2.2, …, 0.9]

•T[r] = 5.6 = Frame Duration

Example 1: Opportunistic Scheduling

S[r] = (S1[r], S2[r], S3[r])

•All Frames = 1 Slot•S[r] = (S1[r], S2[r], S3[r]) = Channel States for Slot r•Policy p[r]: On frame r: First observe S[r], then choose a channel to serve (i.,e, {1, 2, 3}).•Example Objectives: thruput, energy, fairness, etc.

Example 2: Markov Decision Problems

•M(t) = Recurrent Markov Chain (continuous or discrete)•Renewals are defined as recurrences to state 1.•T[r] = random inter-renewal frame size (frame r).•y[r] = penalties incurred over frame r.•π[r] = policy that affects transition probs over frame r.

•Objective: Minimize time average of one penalty subj. to time average constraints on others.

1

2

3

4

Example 3: Task Processing over Networks

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Network Coordinator

•Infinite Sequence of Tasks.•E.g.: Query sensors and/or perform computations.•Renewal Frame r = Processing Time for Frame r.•Policy Types:• Low Level: {Specify Transmission Decisions over Net}• High Level: {Backpressure1, Backpressure2, Shortest Path}

•Example Objective: Maximize quality of information per unit time subject to per-node power constraints.

Task 1Task 2Task 3T/R

Quick Review of Renewal-Reward Theory(Pop Quiz Next Slide!)

Define the frame-average for y0[r]:

The time-average for y0[r] is then:

*If i.i.d. over frames, by LLN this is the same as E{y0}/E{T}.

Pop Quiz: (10 points)

•Let y0[r] = Energy Expended on frame r.•Time avg. power = (Total Energy Use)/(Total Time)•Suppose (for simplicity) behavior is i.i.d. over frames.

To minimize time average power, which one should we minimize?

(a) (b)

Pop Quiz: (10 points)

•Let y0[r] = Energy Expended on frame r.•Time avg. power = (Total Energy Use)/(Total Time)•Suppose (for simplicity) behavior is i.i.d. over frames.

To minimize time average power, which one should we minimize?

(a) (b)

Two General Problem Types:

1) Minimize time average subject to time average constraints:

2) Maximize concave function φ(x1, …, xL) of time average:

Solving the Problem (Type 1):

Define a “Virtual Queue” for each inequality constraint:

Zl[r] clT[r]yl[r]

Zl[r+1] = max[Zl[r] – clT[r] + yl[r], 0]

Lyapunov Function and “Drift-Plus-Penalty Ratio”:

Z2(t)

Z1(t)

L[r] = Z1[r]2 + Z2[r]2 + … + ZL[r]2

Δ(Z[r]) = E{L[r+1] – L[r] | Z[r]} = “Frame-Based Lyap. Drift”

•Scalar measure of queue sizes:

•Algorithm Technique: Every frame r, observe Z1[r], …, ZL[r]. Then choose a policy π[r] in P to minimize:

Δ(Z[r]) + VE{y0[r]|Z[r]}

E{T|Z[r]}“Drift-Plus-Penalty Ratio” =

The Algorithm Becomes:

•Observe Z[r] = (Z1[r], …, ZL[r]). Choose π[r] in P to solve:

•Then update virtual queues:

Δ(Z[r]) + VE{y0[r]|Z[r]}

E{T|Z[r]}

Zl[r+1] = max[Zl[r] – clT[r] + yl[r], 0]

Theorem: Assume the constraints are feasible. Then under this algorithm, we achieve:

Δ(Z[r]) + VE{y0[r]|Z[r]}

E{T|Z[r]}DPP Ratio:

(a)

(b)

For all frames r in {1, 2, 3, …}

Solving the Problem (Type 2):

We reduce it to a problem with the structure of Type 1 via:• Auxiliary Variables γ[r] = (γ1[r], …, γL[r]).• The following variation on Jensen’s Inequality:

For any concave function φ(x1, .., xL) and any (arbitrarily correlated) vector of random variables (x1, x2, …, xL, T), where T>0, we have:

E{Tφ(X1, …, XL)}

E{T}E{T(X1, …, XL)}

E{T}φ( )≤

The Algorithm (type 2) Becomes:

•On frame r, observe Z[r] = (Z1[r], …, ZL[r]).•(Auxiliary Variables) Choose γ1[r], …, γL[r] to max the below deterministic problem:

•(Policy Selection) Choose π[r] in P to minimize:

•Then update virtual queues:Zl[r+1] = max[Zl[r] – clT[r] + yl[r], 0], Gl[r+1] = max[Gl[r] + γl[r]T[r] - yl[r], 0]

Example Problem – Task Processing:

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Network Coordinator

Task 1Task 2Task 3

•Every Task reveals random task parameters η[r]: η[r] = [(qual1[r], T1[r]), (qual2[r], T2[r]), …, (qual5[r], T5[r])]•Choose π[r] = [which node to transmit, how much idle] in {1,2,3,4,5} X [0, Imax] •Transmissions incur power•We use a quality distribution that tends to be better for higher-numbered nodes.•Maximize quality/time subject to pav≤ 0.25 for all nodes.

Setup Transmit Idle I[r]Frame r

Minimizing the Drift-Plus-Penalty Ratio:

•Minimizing a pure expectation, rather than a ratio, is typically easier (see Bertsekas, Tsitsiklis Neuro-DP).

•Define:

•“Bisection Lemma”:

Learning via Sampling from the past:

•Suppose randomness characterized by: {η1, η2, ..., ηW} (past random samples)

•Want to compute (over unknown random distribution of η):

•Approximate this via W samples from the past:

Simulation:

Sample Size W

Qua

lity

of In

form

ation

/ U

nit T

ime

Drift-Plus-Penalty Ratio Alg. With Bisection

Alternative Alg. With Time Averaging

Concluding Sims (values for W=10):

Quick Advertisement: New Book: M. J. Neely, Stochastic Network Optimization with Application to Communication and Queueing Systems. Morgan & Claypool, 2010.

http://www.morganclaypool.com/doi/abs/10.2200/S00271ED1V01Y201006CNT007

• PDF also available from “Synthesis Lecture Series” (on digital library)• Lyapunov Optimization theory (including these renewal system problems)• Detailed Examples and Problem Set Questions.