A NEARLY-TIGHT SUM-OF-SQUARES LOWER BOUND FOR PLANTED CLIQUE

Transcription

1 A NEARLY-TIGHT SUM-OF-SQUARES LOWER BOUND FOR PLANTED CLIQUE Boaz Barak Sam Hopkins Jon Kelner Pravesh Kothari Ankur Moitra Aaron Potechin on the market!

2 PLANTED CLIQUE ("DISTINGUISHING") Input: graph G Goal: determine (with high probability) whether G G (n, 1 2 ) G G (n, 1 2 ("random distribution"), or ) + k-clique ("planted distribution"). FOR WHAT k IS THIS POSSIBLE IN POLYNOMIAL TIME?

3 [J '92, K '95] BASICS k = O(1): information-theoretically-impossible max clique in BRUTE FORCE k > G (n, ) 2 log n requires n O(log n) 1 2 [GM '75, M '76, BE '78] time ("quasipolynomial") SPECTRAL (ADJACENCY MATRIX) k > n, polynomial time [AKS '98]

4 HYPOTHESIS: NO POLYNOMIAL-TIME ALGORITHM FOR k = n 0.49 "SPECTRAL IS BEST"

5 IF SPECTRAL IS BEST, HARDNESS RESULTS GALORE! Sparse PCA [BR '13] Compressed Sensing [KZ '14] Property Testing [AAK+ '07] Mathematical Finance [DBL '10] Cryptography [JP '00, ABW '10] Computational Biology [PS000, MSOI+ '02, JM '15] Best Nash Equilibrium [HK '11, ABC '13] (and P N P ) Problems which have: a distribution on inputs n O(log n) -time algorithms TO BEAT SPECTRAL SEEMS TO REQUIRE NEW ALGORITHMIC IDEAS

6 WHY BELIEVE SPECTRAL IS BEST? no algorithmic progress in 20 years? bad science reductions? (3SAT planted clique) distribution on inputs Rule out large classes of algorithms Markov-Chain Monte Carlo [Jerrum '93] Lovasz-Schrijver+ Convex Hierarchy [Feige-Krauthgamer '04] Statistical Algorithms [Feldman et al '12] All do not beat spectral

7 WHY BELIEVE SPECTRAL COULD BE BEATEN?

8 THE SUM-OF-SQUARES (META-) ALGORITHM

9 Generalization of linear programming, basic semide nite programming, spectral algorithms Optimal among all (poly-sized) SDPs for constraint satisfaction [LRS '15] Solves all known hard instances of unique games, max cut in polynomial time [BBHKSZ '12, OZ '13, DMN '12] Best known algorithm for many planted problems, beating corresponding spectral algorithms! (dictionary learning, planted CSPs, tensor PCA, tensor decomposition, ) [BKS '15ab, AOW '15, RRS '16, HSS '15, HSSS '16, GM '15, MSS '16, ]

10 QUESTION: DOES SPECTRAL-IS-BEST WITHSTAND THE SUM-OF- SQUARES ALGORITHM?

11 Theorem (informal): The Sum-of-Squares hierarchy requires n Ω(log n) time to distinguish planted from random when k = n Spectral-is-best withstands the Sum-of-Squares algorithm.

12 WHAT IS THE SUM-OF-SQUARES ALGORITHM? nasty optimization problem

13 WHAT IS THE SUM-OF-SQUARES ALGORITHM? max S a clique in G S

14 WHAT IS THE SUM-OF-SQUARES ALGORITHM? A hierarchy of increasing-strength semide nite programming (SDP) relaxations of an underlying (nonconvex) problem. Generalizing linear programming, basic SDP, spectral methods.

15 WHAT IS THE SUM-OF-SQUARES ALGORITHM? ( ) d = n : 0 2 n 2 n ( ) d = 2 : 0 n 2 n 2 d = 1 : ( ) 0 n n Regime of interest:. d < o(log n) { "degree d" add variables, constriants huge, exact SDP small SDP, weak relaxation

16 Convex Relaxations for Planted Clique If max X So SIZE(X) S d (G) n 0.49, output PLANTED else RANDOM.

17 max X So SIZE(X) Question: How big is S d (G) in the random case? (If, we can beat the spectral algorithm!) n

18

19 Goal: X = X(G) So S (G) d so that n E G G(n,1/2) SIZE(X(G)) n 0.49

20 (G) Goal: X = X(G) So S d so that E G G(n,1/2) SIZE(X(G)) n 0.49 Prior work [FK '04, folklore]: X(G) feasible for LP, basic SDP, spectral, Sherali-Adams, Lovasz-Schrijver+ with SIZE(X(G)) n 0.49 w.h.p. Related [MPW '15, DM '15, HKPRS '16]: X(G) So Same (G) S d SIZE(X(G)) E G G(n,1/2) n 1/poly(d) n X(G) cannot work for tight SoS bound [Kelner]

21 Original Goal: X = X(G) So S d E G G(n,1/2) n 0.49 SIZE(X(G)) (G) so that X(G) So (G) Dif cult to construct S d ( highdimensional positive-semide nite matrices) New (harder) Goal (pseudocalibration) : Construct X = X(G) So S (G) d which shares more properties of a planted clique than just SIZE(X(G)) n 0.49.

22 New (harder) Goal (pseudocalibration) : X = X(G) So so that S (G) d (X(G)) = ( ) E G G(n,1/2) T G EplantedT 1 G clique for some family { T : R} G R n of G-dependent linear functions ("tests"), including SIZE. T ( 1 ) = G clique 4 Example: number of -cliques containing a typical clique vertex.

23 So (G) thinks there is an -size clique in S d n Lemma: If T G linear functions whose coef cients are low-degree polynomials in entries of A and X(G) G satis es 1. { } = E G G(n,1/2) T (X(G)) = ( ) G EplantedT 1 G clique X(G) 2. E G G(n,1/2) 2 2 is "small" X(G) So then with high probability S d d = o(log n) and SIZE(X(G)) n (G) for Lemma: X(G) satisfying these conditions exists.

24 G G (n, ). 2

25 WRAPPING UP (Nearly) resolved "SoS versus planted clique" using new pseudocalibration approach. Future work/open problems: Pseudocalibration suggests SoS lower bound construction for other planted problems.

26 THANKS! QUESTIONS?