Download presentation

Presentation is loading. Please wait.

Published byApril Drakeford Modified over 2 years ago

1
1/17 Deterministic Discrepancy Minimization Nikhil Bansal (TU Eindhoven) Joel Spencer (NYU)

2
2/17 Combinatorial Discrepancy Universe: U= [1,…,n] Subsets: S 1,S 2,…,S m Problem: Color elements red/blue so each subset is colored as evenly as possible. CS: Computational Geometry, Comb. Optimization, Monte-Carlo simulation, Machine learning, Complexity, Pseudo-Randomness, … Math: Dynamical Systems, Combinatorics, Mathematical Finance, Number Theory, Ramsey Theory, Algebra, Measure Theory, … S1S1 S2S2 S3S3 S4S4

3
3/17 General Set System Universe: U= [1,…,n] Subsets: S 1,S 2,…,S m Find : [n] ! {-1,+1} to Minimize | (S)| 1 = max S | i 2 S (i) | For simplicity consider m=n henceforth.

4
4/17 Simple Algorithm Random: Color each element i independently as x(i) = +1 or -1 with probability ½ each. Thm: Discrepancy = O (n log n) 1/2 Pf: For each set, expect O(n 1/2 ) discrepancy Standard tail bounds: Pr[ | i 2 S x(i) | ¸ c n 1/2 ] ¼ e -c 2 Union bound + Choose c ¼ (log n) 1/2 Analysis tight: Random actually incurs (n log n) 1/2 ).

5
5/17 Better Colorings Exist! [Spencer 85]: (Six standard deviations suffice) Always exists coloring with discrepancy · 6n 1/2 Tight: Cannot beat 0.5 n 1/2 (Hadamard Matrix, “orthogonal” sets) Inherently non-constructive proof (pigeonhole principle on exponentially large universe) Challenge: Can we find it algorithmically ? (Certain algorithms do not work) Conjecture [Alon-Spencer]: May not be possible.

6
6/17 Algorithmic Results [Bansal 10]: Efficient (randomized) algorithm for Spencer’s result. Technique: SDPs (new rounding idea) Use several SDPs over time (guided by the non-constructive method). General: Geometric problems, Beck Fiala setting, k-permutation problem, pseudo-approximation of discrepancy, … Thm: Deterministic Algorithm for Spencer’s (and other) results.

7
This Talk A 7/17

8
Derandomizing Chernoff (Pessimistic estimators, exp. moments, hyp. cosine rule, …) 8/17

9
The Problem 9/17

10
10/17 Algorithm (at high level) Cube: {-1,1} n start finish Each dimension: A variable Each vertex: A rounding

11
11/17 SDP relaxations

12
12/17 Analysis (at high level) Cube: {-1,1} n Analysis: Progress: Few steps to reach a vertex (walk has high variance) Low Discrepancy: For each equation, discrepancy random walk has low variance start finish Each dimension: An Element Each vertex: A Coloring

13
Making it Deterministic Need to find an update that i)Makes Progress ii)Adds low discrepancy to equations. Recall, for Chernoff: Round one variable at a time (Progress) Whether -1 or +1, guided by the potential. (Low Discrepancy) 13/17

14
Tracking the properties 14/17

15
Our fix 15/17 origin x(t-1) x(t): New position

16
Trouble 16/17

17
17/17 Concluding Remarks Idea: Add new constraints to force a deterministic choice to exist. Works more generally for other discrepancy problems. Can potentially have other applications. Thank You!

18
Techniques 18/17 Entropy Method Spencer’s Result Bansal’s Result SDPs New “orthogonality” idea (based on entropy) + K-wise independence, pessimistic estimators, … This Result

Similar presentations

OK

Discrepancy and SDPs Nikhil Bansal (TU Eindhoven, Netherlands ) August 24, ISMP 2012, Berlin.

Discrepancy and SDPs Nikhil Bansal (TU Eindhoven, Netherlands ) August 24, ISMP 2012, Berlin.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on paintings and photographs related to colonial period years Ppt on superconductors applications Ppt on marie curie biography Ppt on static circuit breaker Ppt on tunnel diode circuits Ppt on blood groups in humans Ppt on as 14 amalgamation synonyms Ppt on indian constitution for class 8 Ppt on sustainable rural development Ppt on ministry of corporate affairs