How the Experts Algorithm Can Help Solve LPs Online

Marco MolinaroTU Delft

Anupam GuptaCarnegie Mellon University

Applications: (optimal) gen load-balancing, packing/covering LPs

Primal-dual algo for online random order problems

using

black-box online learning to compute duals

• machines• Job: matrix , each column is a processing option• Algorithm chooses in simplex: fractional choice of

processing• Load vector …• Goal: Minimize makespan

GENERALIZED LOAD-BALANCING

.20.8

0

.9

.3

.4

.8

.1.7.2

.4

.2

.1

.1

.6

.5

.4

.1

.3

.3

.4

.7

.2

+ 𝐴2𝑝2+ …

∞

𝑚0

• Collection of matrices, unknown, adversarial• Job: matrix sampled without replacement • Algorithm: chooses in simplex (random)…• Goal: minimize makespan

GENERALIZED LOAD-BALANCING

Random permutation model

𝑨𝟏𝒑𝟏 𝑨𝟐𝒑𝟐+ +…∞

Entries of in [0,1]

Offline optimum:

Want: -competitive ratio (with high probability)

In iid model: as long as [Devanur et al. 11]– Primal-dual, exponential updates of dual– Asked if techniques worked for random permutation model.

How to handle dependencies?

GENERALIZED LOAD-BALANCING

Alg≤ (1+𝜖 )OPT

• Primal-dual, using black-box online linear optimization for dual

• Abstracts exponential update of Devanur et al., explains why works

• Abstraction allow us handle dependencies in random permutation

GENERALIZED LOAD-BALANCING

Thm: In the random permutation model, we get with high prob -approximation as long as

ALGORITHM

• Idea: Capture in a linear wayObj func:

• Algorithm: at time t– (primal step) chooses to minimize – (dual step) compute trying to maximize

ALGORITHM

• Idea: Capture in a linear wayObj func:

• Compute both ’s and “right” in online way

How? and unknown Online linear optimization

• Setup:– First, algo chooses vector in – Then, adversary chooses vector in – Reward:

• Goal: maximize reward

• Algorithms with good regret bound [Arora et al. 12]

ONLINE LINEAR OPTIMIZATION

¿

• Algorithm: at time t– (primal step) chooses to minimize – (dual step) compute trying to maximize

ALGORITHM

• Idea: Capture in a linear wayObj func:

• Compute both ’s and “right” in online way

• Algorithm: at time t– (primal step) chooses to minimize – (dual step) compute trying to maximize via online linear

optimization

ALGORITHM

• Idea: Capture in a linear wayObj func:

• Compute both ’s and “right” in online way

ANALYSIS (1/3)• Algorithm: at time t

– (primal step) chooses to minimize – (dual step) compute via online lin optimization with adv. vectors

?

• Want: if

• Let be the optimal solution. Then

(dual) guarantee of online lin optimization

(primal) greedy wrt duals

• Show in expectation: E[

• Issue: correlation between and

ANALYSIS (2/3)

𝐄 [𝒘 𝑡 (𝑨𝑡 �̂�𝑡 ) ]≈𝐄 [𝒘 𝑡 ] .𝐄 [𝑨𝑡 �̂�𝑡]• Uses a maximal Bernstein inequality to take care of all time

steps

Lemma: (low dependence) With high probability, we have for all

𝑨𝑡 �̂�𝑡 𝒘 𝑡𝑨𝟏 ,…, 𝑨𝑡 −1

in iid

ANALYSIS (3/3)

• Now with high probability:

• Issue: terms are not independent

• Martingale concentration

ONLINE PACKING/COVERING LP

• Packing/covering: non-negative data• Number of columns, right-hand-side known upfront• Columns (+coef in objective) come one by one, in random

order• Goal: feasible solution, maximize total reward

• Packing-only is well-studied [DH 09, Feldman et al. 10, MR 12, Kesselheim et al. 14, Agrawal et al. 14]

• No general results for packing/covering

𝐴𝑥 𝑏≤𝑥∈[0,1]𝑛

max𝑐𝑥

≥

ONLINE PACKING/COVERING LP

Thm: We get -approximation as long as

• Optimal guarantee for packing (indep Kesselheim et al. 14, Devanur-Agrawal 15)

• First general result for packing/covering (but requires technical assump)

• Idea: reduce online LP to gen load-balancing

• Elements– Handle slightly negative loads in gen load balancing (well-

bounded instances)– Simple reduction to gen load balancing assuming knows

OPT– Estimate OPT: pick out very valuable items, sampling +

chernoff on rest

• Cannot “scale down” solution to get feasibility– Crucially used in Kesselheim et al. 14, Devanur-Agrawal 15…

ONLINE PACKING/COVERING LP

Solving random order problems using duals from black-box online linear optimization

Clean abstraction, allows to handle dependencies in random perm.

– Separates “optimization” and “probability” parts

Applications– Generalized load-balancing– (optimal) guarantees for packing/covering LPs

Open questions

1. Seems very flexible. Apply techniques to other problems?

2. More general, realistic models

3. Remove technical assumption in packing/covering, or prove LB (minimax?)

CONCLUSION

THANK YOU!