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

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

3/17 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.

4/17 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 ).

5/17 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.

6/17 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.

This Talk A 7/17

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

The Problem 9/17

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

11/17 SDP relaxations

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

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

Tracking the properties 14/17

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

Trouble 16/17

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!

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