Upload
ralph-walton
View
219
Download
0
Tags:
Embed Size (px)
Citation preview
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: 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