How the Experts Algorithm Can Help Solve LPs Online Marco Molinaro TU Delft Anupam Gupta Carnegie...
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Transcript of How the Experts Algorithm Can Help Solve LPs Online Marco Molinaro TU Delft Anupam Gupta Carnegie...
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