Primal Dual Combinatorial Algorithms Qihui Zhu May 11, 2009 1
Outline • Packing and covering • Primal and dual problems • Online prediction using feedback • Primal-dual combinatorial algorithm • Applications and extensions 2
Packing Problem (0 -1 Knapsack) • n objects with weights Wi and prices pi • A knapsack with capacity c • Pack objects maximizing the total price without exceeding the capacity
Packing Problem (0 -1 Knapsack) • n objects with weights Wi and prices pi • A knapsack with capacity c • Pack objects maximizing the total price without exceeding the capacity Knapsack IP
Packing Problem (General) • n objects with weights Wij and prices pi • m constraints with capacity cj to fit • Pack objects maximizing the total price without exceeding the capacities Packing IP
Covering Problem • n objects to cover at least pi times • m sets with costs cj cover each covers Wji objects • Cover all objects with the minimal cost
Covering Problem • n objects to cover at least pi times • m sets with costs cj cover each covers Wji objects • Cover all objects with the minimal cost Covering IP
Fractional Packing Original form 8
Fractional Packing Original form Matrix form 9
Fractional Packing and Covering Are Dual Packing Covering 10
Fractional Packing and Covering Are Dual Packing Covering: best upper bound on packing Packing: best lower bound on covering 11
Decision Version (PST 95) Feasibility Given constraint set such that For packing: Binary search for optimization 12
Flipped Sides of the Same Coin Think of the game of twenty questions. . . “Yes” certificate “No” certificate 13
Key Idea #1: Need Primal-Dual Algorithm Primal: “Yes” certificate Dual: “No” certificate 14
How to Generate the Certificates? Randomly guessing. . . 15
How to Generate the Certificates? Primal and Dual need to communicate Randomly guessing. . . 16
From Dual to Primal Efficient combinatorial algorithm 17
From Dual to Primal Oracle Given dual estimate Let , find and constraint set such that Efficient combinatorial algorithm 18
From Dual to Primal Oracle Given dual estimate Let , find and constraint set such that For packing, essentially ignore all capacity constraints Reduce to sorting over ! Complexity: 19
From Primal to Dual Combination of hyperplanes 20
From Primal to Dual Tilt the hyperplanes using feedback , One step not enough! Getting complicated over iterations. . . , etc. Online Prediction Combination of hyperplanes 21
Online Prediction • Experts predicting some uncertain event Expert 22
Online Prediction • Experts predicting some uncertain event Expert • Gain some value from the world (adversarial) Value 23
Online Prediction • Experts predicting some uncertain event Expert • Gain some value from the world (adversarial) • Linearly combine by weights Weight Value 24
Online Prediction • Experts predicting some uncertain event Expert • Gain some value from the world (adversarial) Time • Linearly combine by weights • Long term value over time Weight Value 25
A Simplified Case • Only 2 experts • Value history Combined 1 Expert 2 1 0 ? ? ？ ? 0 Expert 1 0 1 0 1 ? ? ? Value 1 0 Average playoff Regret 26
Strategy I Take the best from history? 1 0 Combined 0 0 0 1 0 1 1 1 0 1 0 0. 1 Expert 2 0 0. 9 Expert 1 0. 9 0. 5 1 0 27
Strategy II Linearly weighted by the cumulative values? 1 0 Combined 0 1 0 0 1 1 0 Expert 2 1 1 Expert 1 0 0 1 1 0 0 0. 5 1 2/3 1/3 2/3 1 1 0 1 2/3 2/3 0 0 1/3 1/3 28
Strategy III Exponentially weighted by the cumulative values! 1 0 Combined 0 1 1 0 1 0 1 1 0 0 ε ε 1 1 0 Expert 2 1 1 Expert 1 0 0. 5 ε 1 -ε 1 -ε 0 29
Key Idea #2: Need Multiplicative Feedback Primal: “Yes” certificate Dual: “No” certificate 30
From Primal to Dual: Continued Multiplicative Weight Update (MW) Let be the current state at step and be the “feedback” Combination of hyperplanes 31
Regret Bound Theorem (LW 94) Suppose predictions of experts have value. Each time the predictions are combined by weights , where. Update weights using. After time 32
Regret Bound Theorem (LW 94) Suppose predictions of experts have value. Each time the predictions are combined by weights , where. Update weights using. After time Proof. Consider the potential function Show that it is dominated by the best expert. 33
Primal-Dual Algorithms Primal Dual Oracle Multiplicative Weight Update 34
Algorithm Primal-Dual Combinatorial Algorithm Initialize: , Repeat Set Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution 35
Analysis Primal-Dual Combinatorial Algorithm Initialize: Oracle: if failed, infeasible , Repeat Set Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution 36
Analysis Primal-Dual Combinatorial Algorithm Initialize: Oracle: if failed, infeasible , Repeat Width: Set Oracle If then Regret bound: Return infeasible, output Set Multiplicative Weight Update Until Feasible solution! Complexity depends on Return feasible solution 37
Applications • Fractional packing and covering • Multicommodity flow • Held-Karp Bound for TSP • Semidefinite programming (SDP) relaxation • General convex programming • Boosting • Matrix game 38
Applications • Fractional packing and covering • Multicommodity flow • Held-Karp Bound for TSP • Semidefinite programming (SDP) relaxation • General convex programming • Boosting • Matrix game 39
Multicommodity Flow • Objective: maximizing total flow while respecting capacities 40
Multicommodity Flow • Objective: maximizing total flow while respecting capacities • Packing paths • Update: route some flow along shortest path Congestion Multicommodity Flow 41
Multicommodity Flow Primal-Dual Combinatorial Algorithm Initialize: P : polytope of graph flows , Repeat Oracle : shortest path Set keep pushing flows through Oracle If then Augment length (GK 98) Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution 42
SDP Relaxation • Better bounds on Max Cut, Sparsest Cut, Max-2 Sat, etc. Max Cut (GW Sparsest Cut (ARV 04) 95) • Finding good geometric embedding by SDP + rounding • Interior point method not scale. Primal SDP Dual SDP 43
Primal SDP Primal-Dual Combinatorial Algorithm Initialize: , P : set of semidefinite matrices Repeat Set Oracle : compute eigenvectors! Oracle If then Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution 44
Dual SDP Primal-Dual Combinatorial Algorithm Initialize: , Repeat P & Oracle : same as packing Multiplicative Weight Update Set Use matrix exponential Oracle If then ! Return infeasible, output Set Multiplicative Weight Update Until Return feasible solution 45
Weighted Majority Algorithm/Boosting Primal-Dual Combinatorial Algorithm Initialize: , Repeat Events/Classification results: not controlled by us! Set Oracle If then Return infeasible, output Error on training examples Set Multiplicative Weight Update Until Return feasible solution 46
Summary Combinatorial subroutines 47
Summary Combinatorial subroutines Fast algorithms! 48
Conclusion • A computational paradigm – Applied to numerous problems – Cover at least convex problems • Fast approximation algorithms – Trade accuracy for time: O(1/ε) – Fast combinatorial algorithms for oracle (without solving LP!) • Flexibility for algorithm design – Multiplicative weight update: a principle way to use feedback – Free to separate and combine constraints • Width management is important – Most algorithms polynomial to width – Separate high width constraints: nested primal-dual – Decompose constraint set 49
Thank you! Questions …
Regret Bound Theorem (LW 94) Suppose predictions of experts have value. Each time the predictions are combined by weights , where. Update weights using. After time Proof. Consider the potential function Show that it is dominated by the best expert. 51
Update rule 52
Update rule Exp function Width 53
Update rule Exp function Width 54
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