Gibbs Sampling from -Determinantal Point Processes Alireza Rezaei - - PowerPoint PPT Presentation

Gibbs Sampling from 𝑙-Determinantal Point Processes

Based on joint work with Shayan Oveis Gharan

Alireza Rezaei

University of Washington

Point Process: A distribution on subsets of 𝑂 = {1,2, … , 𝑂}. Determinantal Point Process: There is a PSD kernel 𝑀 ∈ ℝ𝑂×𝑂 such that ∀𝑇 ⊆ 𝑂 : ℙ 𝑇 ∝ det 𝑀𝑇

𝑴𝑻

Point Process: A distribution on subsets of 𝑂 = {1,2, … , 𝑂}. Determinantal Point Process: There is a PSD kernel 𝑀 ∈ ℝ𝑂×𝑂 such that ∀𝑇 ⊆ 𝑂 : ℙ 𝑇 ∝ det 𝑀𝑇 𝒍-DPP: Conditioning of a DPP on picking subsets of size 𝑙 if 𝑇 = 𝑙: ℙ 𝑇 ∝ det 𝑀𝑇

• therwise : ℙ 𝑇 = 0

Focus of the talk: Sampling from 𝑙- DPPs

𝑴𝑻

Point Process: A distribution on subsets of 𝑂 = {1,2, … , 𝑂}. Determinantal Point Process: There is a PSD kernel 𝑀 ∈ ℝ𝑂×𝑂 such that DPPs are Very popular probabilistic models in machine learning to capture diversity. ∀𝑇 ⊆ 𝑂 : ℙ 𝑇 ∝ det 𝑀𝑇 𝒍-DPP: Conditioning of a DPP on picking subsets of size 𝑙 if 𝑇 = 𝑙: ℙ 𝑇 ∝ det 𝑀𝑇

• therwise : ℙ 𝑇 = 0

Focus of the talk: Sampling from 𝑙- DPPs

Applications [Kulesza-Taskar’11, Dang’05, Nenkova-Vanderwende-McKeown’06, Mirzasoleiman-Jegelka-Krause’17]

— Image search, document and video summarization, tweet timeline generation, pose estimation, feature selection

𝑴𝑻

Continuous Domain

Input: PSD operator 𝑀: 𝒟 × 𝒟 → ℝ and 𝑙 select a subset 𝑇 ⊂ 𝒟 with 𝑙 points from a distribution with PDF function 𝑞(𝑇) ∝ det 𝑀(𝑦, 𝑧) 𝑦,𝑧∈𝑇

Continuous Domain

Input: PSD operator 𝑀: 𝒟 × 𝒟 → ℝ and 𝑙 select a subset 𝑇 ⊂ 𝒟 with 𝑙 points from a distribution with PDF function 𝑞(𝑇) ∝ det 𝑀(𝑦, 𝑧) 𝑦,𝑧∈𝑇

Applications.

— Hyper-parameter tuning [Dodge-Jamieson-Smith’17] — Learning mixture of Gaussians[Affandi-Fox-Taskar’13]

• Ex. Gaussian : 𝑀 𝑦, 𝑧 = exp −

𝑦−𝑧 Σ−1 𝑦−𝑧 2

Random Scan Gibbs Sampler for 𝐿-DPP

• 1. Stay at the current state 𝑇 = {𝑦1, … 𝑦𝑙} with prob

1 2.

• 2. Choose 𝑦𝑗 ∈ 𝑇 u.a.r
• 3. Choose 𝑧 ∉ 𝑇 from the conditional dist 𝜌 . 𝑇 − 𝑦𝑗 is

chosen) Continuous: PDF 𝑧 ∝ 𝜌 𝑦1, … 𝑦𝑗−1, 𝑧, 𝑦𝑗+1, … , 𝑦𝑙 )

S ∈ [𝑂] 𝑙

y 𝑦𝑗

Main Result

Given a 𝑙-DPP 𝜌, an “approximate” sample from 𝜌 can be generated by running the Gibbs sampler for 𝝊 = ෩ 𝑷 𝒍𝟓 ⋅ 𝐦𝐩𝐡 (𝐰𝐛𝐬𝝆

𝒒𝝂 𝒒𝝆 ) steps where 𝜈 is the starting dist.

Main Result

Given a 𝑙-DPP 𝜌, an “approximate” sample from 𝜌 can be generated by running the Gibbs sampler for 𝝊 = ෩ 𝑷 𝒍𝟓 ⋅ 𝐦𝐩𝐡 (𝐰𝐛𝐬𝝆

𝒒𝝂 𝒒𝝆 ) steps where 𝜈 is the starting dist.

Discrete: A simple greedy initialization gives 𝜐 = 𝑃 𝑙5log 𝑙 . Total running time is 𝑃 𝑂 . poly 𝑙 .

• Does not improve upon the previous MCMC methods. [Anari-Oveis Gharan-R’16]
• Mixing time is independent of 𝑂, so the running time in distributed settings is sublinear.

Main Result

Given a 𝑙-DPP 𝜌, an “approximate” sample from 𝜌 can be generated by running the Gibbs sampler for 𝝊 = ෩ 𝑷 𝒍𝟓 ⋅ 𝐦𝐩𝐡 (𝐰𝐛𝐬𝝆

𝒒𝝂 𝒒𝝆 ) steps where 𝜈 is the starting dist.

Discrete: A simple greedy initialization gives 𝜐 = 𝑃 𝑙5log 𝑙 . Total running time is 𝑃 𝑂 . poly 𝑙 .

• Does not improve upon the previous MCMC methods. [Anari-Oveis Gharan-R’16]
• Mixing time is independent of 𝑂, so the running time in distributed settings is sublinear.

Continuous: Given access to conditional oracles, 𝜈 can be found so 𝜐 = 𝑃(𝑙5log 𝑙).

• First algorithm with a theoretical guarantee for sampling from continuous 𝑙-DPP.

Being able to run the chain.

Main Result

Given a 𝑙-DPP 𝜌, an “approximate” sample from 𝜌 can be generated by running the Gibbs sampler for 𝝊 = ෩ 𝑷 𝒍𝟓 ⋅ 𝐦𝐩𝐡 (𝐰𝐛𝐬𝝆

𝒒𝝂 𝒒𝝆 ) steps where 𝜈 is the starting dist.

Discrete: A simple greedy initialization gives 𝜐 = 𝑃 𝑙5log 𝑙 . Total running time is 𝑃 𝑂 . poly 𝑙 .

• Does not improve upon the previous MCMC methods. [Anari-Oveis Gharan-R’16]
• Mixing time is independent of 𝑂, so the running time in distributed settings is sublinear.

Continuous: Given access to conditional oracles, 𝜈 can be found so 𝜐 = 𝑃(𝑙5log 𝑙).

• First algorithm with a theoretical guarantee for sampling from continuous 𝑙-DPP.
• Using a rejection sampler as the conditional oracles for Gaussian kernels 𝑀 𝑦, 𝑧 = exp(−

𝑦−𝑧 2 𝜏2

) defined a unit sphere in ℝ𝑒, the total running time is

• If 𝑙 =poly(d): poly(𝑒, 𝜏)
• If 𝑙 ≤ 𝑓𝑒1−𝜀 and 𝜏 = 𝑃 1 : poly 𝑒 ⋅ 𝑙𝑃(1

𝜀)

Being able to run the chain.