Phase Transitions in Semidefinite Relaxations Andrea Montanari - PowerPoint PPT Presentation

Spectral relaxation bad in the sparse regime! d ❂ 1 ✿ 5 d ❂ 15 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 p γ p γ eigenvalues/ eigenvalues/ Theorem ( Krivelevich, Sudakov 2003 + Vu 2005 ) With high probability, ✭ ♣ if d ✢ ✭ log n ✮ 4 ❀ 2 ✭ 1 ✰ o ✭ 1 ✮✮ ✕ max ✭ A cen ❂ d ✮ ❂ ♣ C log n ❂ ✭ log log n ✮✭ 1 ✰ o ✭ 1 ✮✮ if d ❂ O ✭ 1 ✮ ✿ Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 28 / 75

Example: d ❂ 20, ✕ ❂ 1 ✿ 2, n ❂ 10 4 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 0 2000 4000 6000 8000 10000 v 1 ✭ A cen ✮ Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 29 / 75

Example: d ❂ 3, ✕ ❂ 1 ✿ 2, n ❂ 10 4 0.10 0.05 0.00 0.05 0.10 0 2000 4000 6000 8000 10000 v 1 ✭ A cen ✮ Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 30 / 75

Why should SDP work better? ❤ A cen ❀ X ✐ ❀ maximize X ✷ R n ✂ n ❀ X ✗ 0 ❀ subject to X ii ❂ 1 ✿ Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 31 / 75

Recall the ultimate limit G ✭ n ❀ d ❀ ✕ ✮ graph distribution with parameters d ❂ a ✰ b a � b ❃ 1 ❀ ✕ ❂ ♣ 2 2 ✭ a ✰ b ✮ Theorem ( Mossel, Neeman, Sly, 2012) If ✕ ❁ 1 , then ✌ ✌ ✌ G ✭ n ❀ d ❀ 0 ✮ � G ✭ n ❀ d ❀ ✕ ✮ ✌ lim sup TV ❁ 1 ✿ n ✦✶ If ✕ ❃ 1 , then ✌ ✌ ✌ G ✭ n ❀ d ❀ 0 ✮ � G ✭ n ❀ d ❀ ✕ ✮ ✌ lim TV ❂ 1 ✿ n ✦✶ Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 32 / 75

SDP has nearly optimal threshold Theorem ( Montanari, Sen 2015) Assume G ✘ G ✭ n ❀ d ❀ ✕ ✮ . If ✕ ✔ 1 , then, with high probability, 1 ♣ SDP ✭ A cen G ✮ ❂ 2 ✰ o d ✭ 1 ✮ ✿ n d If ✕ ❃ 1 , then there exists ✁✭ ✕ ✮ ❃ 0 such that, with high probability, 1 ♣ SDP ✭ A cen G ✮ ❂ 2 ✰ ✁✭ ✕ ✮ ✰ o d ✭ 1 ✮ ✿ n d Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 33 / 75

Consequence ♣ ✭ if SDP ✭ A cen 1 G ✮ ✕ ✭ 2 ✰ ✍ ✮ n d , T SDP ✭ G ✮ ❂ 0 otherwise. Corollary ( Montanari, Sen 2015) Assume ✕ ✕ 1 ✰ ✧ . Then there exists d 0 ✭ ✧ ✮ and ✍ ✭ ✧ ✮ such that the SDP-based test succeeds with high probability, provided d ✕ d 0 ✭ ✧ ✮ . Namely ✄ ❂ 0 ✿ ✂ P 0 ✭ T SDP ✭ G ✮ ❂ 1 ✮ ✰ P 1 ✭ T SDP ✭ G ✮ ❂ 0 ✮ lim n ✦✶ Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 34 / 75

Earlier/related work Optimal spectral tests ◮ Massoulie 2013 ◮ Mossel, Neeman, Sly, 2013 ◮ Bordenave, Lelarge, Massoulie, 2015 SDP, d ❂ ✂✭ log n ✮ ◮ Abbe, Bandeira, Hall 2014 ◮ Hajek, Wu, Xu 2015 SDP, detection ◮ Guédon, Vershynin, 2015 (requires ✕ ✕ 10 4 , very different proof) Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 35 / 75

How does SDP work ‘in practice’? Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 36 / 75

Thresholds ◮ ✕ opt c ✭ d ✮ ✑ Threshold for optimal test ◮ ✕ SDP ✭ d ✮ ✑ Threshold for SDP-based test c Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 37 / 75

What we know ◮ ✕ opt c ✭ d ✮ ❂ 1 [Mossel, Neeman, Sly, 2013] ◮ ✕ SDP ✭ d ✮ ❂ 1 ✰ o d ✭ 1 ✮ [Montanari, Sen, 2015] c How big is the o d ✭ 1 ✮ gap? Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 38 / 75

Simulations: d ❂ 5, N sample ❂ 500 (with Javanmard and Ricci) 1.0 GOE theory n =2 , 000 n =4 , 000 0.8 n =8 , 000 n =16 , 000 0.6 Overlap 0.4 0.2 0.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 q ( a − b ) / 2( a + b ) x SDP ✷ ❢ ✰ 1 ❀ � 1 ❣ n SDP estimator ˆ ✟☞ ☞✠ ✿ ☞ x ✮ ❂ 1 ☞ ❤ ˆ x SDP ✭ G ✮ ❀ x 0 ✐ Overlap n ✭ ❜ n E Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 39 / 75

Simulations: d ❂ 5, N sample ❂ 500 1.0 GOE theory n =2 , 000 n =4 , 000 0.8 n =8 , 000 n =16 , 000 0.6 Overlap 0.4 0.2 0.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 q ( a − b ) / 2( a + b ) ✕ SDP ✭ d ❂ 5 ✮ ✙ 1 ✿ c Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 40 / 75

Simulations: d ❂ 10, N sample ❂ 500 1.0 GOE theory n =2 , 000 n =4 , 000 0.8 n =8 , 000 n =16 , 000 0.6 Overlap 0.4 0.2 0.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 q a − b/ 2( a + b ) ✕ SDP ✭ d ❂ 10 ✮ ✙ 1 ✿ c Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 41 / 75

Simulations: d ❂ 10, N sample ❂ 500 1.0 GOE theory n =2 , 000 n =4 , 000 0.8 n =8 , 000 n =16 , 000 0.6 Overlap 0.4 0.2 0.0 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 q a − b/ 2( a + b ) How to estimate ✕ SDP ✭ d ✮ from data? c Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 42 / 75

A technique from physics: Binder cumulant Q ✭ G ✮ ✑ 1 x SDP ✭ G ✮ ❀ x 0 ✐ ❀ n ❤ ˆ ✟ Q ✭ G ✮ 4 ✠ Bind ✭ n ❀ ✕❀ d ✮ ✑ E ✟ Q ✭ G ✮ 2 ✠ 2 E CLT heuristics ✭ 3 if ✕ ❁ ✕ SDP ✭ d ✮ , c n ✦✶ Bind ✭ n ❀ ✕❀ d ✮ ❂ lim 1 if ✕ ❃ ✕ SDP ✭ d ✮ . c Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 43 / 75

Simulations: d ❂ 5, N sample ✕ 10 5 ! 3 n=2000 n=4000 n=8000 2.5 Binder 2 1.5 1 0.7 0.8 0.9 1 1.1 1.2 λ Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 44 / 75

Zoom ( ✘ 2 years CPU time) 2.1 n=2000 n=4000 2 n=8000 n=16000 1.9 1.8 d = 5 Binder 1.7 1.6 1.5 1.4 1.3 1 1.005 1.01 1.015 1.02 1.025 1.03 λ Estimate ✕ SDP ✭ d ✮ by the crossing point c Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 45 / 75

✕ c ✭ d ✮ SDP 1.02 1.01 ✕ SDP ✭ d ✮ c 1.00 0.99 1 10 d ◮ Dots: Numerical estimates ◮ Line: Non-rigorous analytical approximation (using statistical physics) ◮ At most 2 ✪ sub-optimal! Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 46 / 75

One last question Is this approach robust to model miss-specifications? Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 47 / 75

An experiment ◮ Select S ✒ V uniformly at random. with ❥ S ❥ ❂ n ☛ . ◮ For each i ✷ S , connect all of its neighbors. Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 48 / 75

An experiment 1 SDP, α = 0 0.9 BH, α = 0 SDP, α = 0.025 0.8 BH, α = 0.025 SDP, α = 0.05 0.7 BH, α = 0.05 0.6 0.5 0.4 0.3 0.2 0.1 0 0.8 1 1.2 1.4 1.6 1.8 2 ◮ Solid line:SDP ◮ Dashed line: Spectral (Non-backtracking walk [Krzakala, Moore, Mossel, Neeman, Sly, Zdeborova, Zhang, 2013]) Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 49 / 75

A simple robustness result Lemma ( Montanari, Sen, 2015) If ❢ G is obtained from the hidden partition model by flipping at most n ✧ edges, then ☞ ☞ ☞ ⑦ ☞ ✔ ✍ ✭ ✧ ✮ ✿ ✕ SDP ✭ d ✮ � ✕ SDP ✭ d ✮ c c Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 50 / 75

Proof ideas Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 51 / 75

What we want to prove Theorem ( Montanari, Sen 2015) Assume G ✘ G ✭ n ❀ d ❀ ✕ ✮ . If ✕ ✔ 1 , then, with high probability, 1 ♣ SDP ✭ A cen G ✮ ❂ 2 ✰ o d ✭ 1 ✮ ✿ n d If ✕ ❃ 1 , then there exists ✁✭ ✕ ✮ ❃ 0 such that, with high probability, 1 ♣ SDP ✭ A cen G ✮ ❂ 2 ✰ ✁✭ ✕ ✮ ✰ o d ✭ 1 ✮ ✿ n d Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 52 / 75

Strategy I. Prove equivalence to Gaussian model II. Analyze Gaussian model Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 53 / 75

Gaussian model: x 0 ✷ ❢ ✰ 1 ❀ � 1 ❣ n B ✭ ✕ ✮ ✑ ✕ n x 0 x 0 T ✰ W ✿ W ✘ GOE ✭ n ✮ : ◮ ✭ W ij ✮ i ❁ j ✘ iid N ✭ 0 ❀ 1 ❂ n ✮ ◮ W ❂ W T ◮ A lot is known about spectral properties of B Need to characterize the SDP value with Gaussian data Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 55 / 75

Notation 1 s ✭ ✕ ✮ ✑ lim sup n SDP ✭ B ✭ ✕ ✮✮ ❀ n ✦✶ 1 s ✭ ✕ ✮ ✑ lim inf n SDP ✭ B ✭ ✕ ✮✮ ✿ n ✦✶ ✟ ❤ B ❀ X ✐ ✿ X ✗ 0 ❀ X ii ❂ 1 ✽ i ✠ SDP ✭ B ✮ ✑ max Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 56 / 75

Phase transition at ✕ ❂ 1! Theorem ( Montanari, Sen, 2015) The following holds almost surely ✕ ✷ ❬ 0 ❀ 1 ❪ ✮ s ✭ ✕ ✮ ❂ s ✭ ✕ ✮ ❂ 2 ❀ ✕ ✷ ✭ 1 ❀ ✶ ✮ ✮ s ✭ ✕ ✮ ❃ 2 (strictly) ✿ For explicit probability bounds, see the paper Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 57 / 75

Proof of Gaussian phase transition Simple facts: ◮ ✭ 2 if ✕ ✷ ❬ 0 ❀ 1 ❪ , s ✭ ✕ ✮ ✔ lim n ✦✶ ✛ max ✭ B ✭ ✕ ✮✮ ❂ ✕ ✰ ✕ � 1 if ✕ ✷ ✭ 1 ❀ ✶ ✮ . [Baik, Ben Arous, Peche, 2005] ◮ 1 s ✭ ✕ ✮ ✔ lim n ❤ 1 ❀ B ✭ ✕ ✮ 1 ✐ ❂ ✕ n ✦✶ ◮ s ✭ ✕ ✮ , s ✭ ✕ ✮ are non-random, non-decreasing Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 58 / 75

Summarizing 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 λ ◮ Red: Upper bound ◮ Blue: Lower bound Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 59 / 75

Proof of Gaussian phase transition Part 1: Prove that s ✭ ✕ ❂ 0 ✮ ✕ 2 Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 60 / 75

Hence 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 λ ◮ Red: Upper bound ◮ Blue: Lower bound ◮ Purple: Non-trivial lower bound Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 61 / 75

Proof of Gaussian phase transition Part 1: Prove that s ✭ ✕ ❂ 0 ✮ ✕ 2 Part 2: Prove that s ✭ ✕ ❂ 1 ✰ ✧ ✮ ❃ 2 Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 62 / 75

Hence 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 λ ◮ Red: Upper bound ◮ Blue: Lower bound ◮ Purple: Non-trivial lower bound Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 63 / 75

Proof of Gaussian phase transition Part 1: Prove that s ✭ ✕ ❂ 0 ✮ ✕ 2 Part 2: Prove that s ✭ ✕ ❂ 1 ✰ ✧ ✮ ❃ 2 Technique: Construct feasible X , such that ❤ A ❀ X ✐ ✕ ✿ ✿ ✿ . Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 64 / 75

Part 1: ✕ ❂ 0 Limiting Spectral Density − 2 2 First idea: ◮ v 1 ❂ v 1 ✭ B ✮ ✑ principal eginvector of B ◮ Take X ❂ n v 1 v T 1 ◮ Wrong: X ii ✙ N ✭ 0 ❀ 1 ✮ 2 ✻ ❂ 1 Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 65 / 75

Part 1: ✕ ❂ 0 Limiting Spectral Density − 2 2 Good idea: ◮ Let U ❂ ❬ v 1 ❥ v 2 ❥ ✁ ✁ ✁ ❥ v n ✍ ❪ ✷ R n ✂ n ✍ , ✍ small. ◮ D ✑ Diag ✭ U U T ✮ ✷ R n ✂ n . Claim D ✙ ✍ I ( D ii ✘ n � 1 ✤ n ✍ ) ◮ Set X ❂ D � 1 ❂ 2 ✭ U U T ✮ D � 1 ❂ 2 Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 66 / 75

Part 2: ✕ ❂ 1 ✰ ✧ Limiting Spectral Density − 2 2 Construction: ◮ T ✭ x ✮ ✑ max ✭ min ✭ x ❀ ✰ 1 ✮ ❀ � 1 ✮ , ϕ ✷ R n ✬ i ✑ T ✭ ✧ ♣ n v 1 ❀ i ✮ ✿ ◮ U ❂ ❬ v 2 ❥ v 3 ❥ ✁ ✁ ✁ ❥ v n ✍ ✰ 1 ❪ ✷ R n ✂ n ✍ q ◮ D ✷ R n ✂ n diagonal D ii ✑ 1 � ✬ 2 i ❂ ❦ U e i ❦ 2 . ◮ X ✑ ϕϕ T ✰ DU U T D Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 67 / 75

A parenthesis Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 68 / 75

The Gaussian model is very interesting 1.0 Max Likelihood Bayes 0.8 SDP PCA SDP, n =200 0.6 SDP, n =400 MSE SDP, n =800 SDP, n =1600 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 λ Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 69 / 75

We want to prove 1 ♣ SDP ✭ A cen ✮ ✙ SDP ✭ B ✭ ✕ ✮✮ d Lindeberg method: Replace the entries one-by-one Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 71 / 75

A simple Lindeberg lemma ◮ X 1 ❀ X 2 ❀ ✿ ✿ ✿ X M iid ✽ ✏ ✑ 1 � d with probability d ❁ 1 n ❀ ♣ n d ✏ ✑ X i ❂ ♣ ✿ d 1 � d � with probability ✿ n n E ❢ X i ❣ ❂ 0, E ❢ X 2 i ❣ ❂ ✭ 1 ❂ n ✮ � ✭ d ❂ n 2 ✮ ◮ Z 1 ❀ Z 2 ❀ ✿ ✿ ✿ Z M ✘ i ✿ i ✿ d ✿ N ✭ 0 ❀ 1 ❂ n ✮ Lemma Assume M ❂ n ✭ n � 1 ✮ ❂ 2 , F ✷ C 3 ✭ R M ✮ , d ✔ n 2 ❂ 3 ❂ 10 . Then ☞ ☞ ✏✌ ✑ n ✌ ✌ ✌ ☞ ☞ ✌ ❅ 2 ✌ ✌ ❅ 3 ✌ ☞ E F ✭ X ✮ � E F ✭ Z ✮ ♣ max i F i F ☞ ✔ ✶ ❴ ✿ ✶ 3 d i ✷ ❬ M ❪ i F ✭ x ✮ ✑ ❅ ❵ F where ❅ ❵ i , and ❦ ❅ ❵ i F ❦ ✶ ✑ sup x ✷ R M ❥ ❅ ❵ i F ✭ x ✮ ❥ . ❅ x ❵ Problem: F ✭ ✁ ✮ ❂ SDP ✭ ✁ ✮ ✻✷ C 3 ✭ R M ✮ Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 72 / 75

Phase Transitions in Semidefinite Relaxations Andrea Montanari - PowerPoint PPT Presentation

Phase Transitions in Semidefinite Relaxations Andrea Montanari [with Adel Javanmard, Federico Ricci-Tersenghi, Subhabrata Sen] Stanford University December 7, 2015 Andrea Montanari (Stanford) SDP Phase Transitions December 7, 2015 1 / 75

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