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RCO論文輪読会(2016/07/29) “Ensuring Rapid Mixing and Low Bias for Asynchronous Gibbs Sampling”(ICML 2016) Christopher De Sa, Kunle Olukotun, Christopher Ré ICTソリューション局アドテクノロジーサービス開発部 高柳慎一

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（C）Recruit Communications Co., Ltd. Abstract 1

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（C）Recruit Communications Co., Ltd. 1. Introduction 2

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（C）Recruit Communications Co., Ltd. Gibbs samplingとは 3

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（C）Recruit Communications Co., Ltd. ビッグデータ時代におけるGibbs samplingの弱点 4

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（C）Recruit Communications Co., Ltd. 今回の中心となるアルゴリズム：HOGWILD! 5

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（C）Recruit Communications Co., Ltd. この論文でやること 6

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（C）Recruit Communications Co., Ltd. Total influence α 7

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（C）Recruit Communications Co., Ltd. ここから先やること(詳細後述) 8

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（C）Recruit Communications Co., Ltd. 2. Related Work 9

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（C）Recruit Communications Co., Ltd. HOGWILD!-Gibbs sampling アルゴリズム 10

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（C）Recruit Communications Co., Ltd. HOGWILD!-Gibbs sampling アルゴリズム 11

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（C）Recruit Communications Co., Ltd. 3. Modeling Asynchronicity 12

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（C）Recruit Communications Co., Ltd. 条件 13

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（C）Recruit Communications Co., Ltd. 条件 14

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（C）Recruit Communications Co., Ltd. 4. The First Challenge: Bias 15

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（C）Recruit Communications Co., Ltd. Total Variance Distance(以下、TVと呼称) 16

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（C）Recruit Communications Co., Ltd. Gibbs samplingの場合 17

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（C）Recruit Communications Co., Ltd. 4.1. Bias Example 18

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（C）Recruit Communications Co., Ltd. 4.1. Bias Example 19

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（C）Recruit Communications Co., Ltd. 4.1. Bias Example 20

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（C）Recruit Communications Co., Ltd. 4.2. Bounding the Bias 21

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（C）Recruit Communications Co., Ltd. Sparse Variation Distance(以下、SVと呼称) 22

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（C）Recruit Communications Co., Ltd. Sparse Variation Distance 23

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（C）Recruit Communications Co., Ltd. Sparse Estimation Time 24

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（C）Recruit Communications Co., Ltd. Total Influence 25

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（C）Recruit Communications Co., Ltd. Claim1 (詳細な証明はArxivの論文見ないとダメ) 26

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（C）Recruit Communications Co., Ltd. Claim1を証明するための定理 27

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（C）Recruit Communications Co., Ltd. 定理1 28

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（C）Recruit Communications Co., Ltd. 定理2 29

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（C）Recruit Communications Co., Ltd. 5. The Second Challenge: Mixing Times 30

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（C）Recruit Communications Co., Ltd. 5.1. Mixing Time Example 31

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（C）Recruit Communications Co., Ltd. 5.1. Mixing Time Example 32

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（C）Recruit Communications Co., Ltd. 5.2. Bounding the Mixing Time 33 ※前述の例は Dobrushin’s条 件を満たしてい ない例なのでこ この定理とは無 矛盾

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（C）Recruit Communications Co., Ltd. 5.2. Bounding the Mixing Time 34

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（C）Recruit Communications Co., Ltd. 5.3. A Positive Example: Ising Model 35

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（C）Recruit Communications Co., Ltd. 5.4. Proof Outline 36

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（C）Recruit Communications Co., Ltd. 6. Experiments 37

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（C）Recruit Communications Co., Ltd. 6. Experiments 38

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（C）Recruit Communications Co., Ltd. 7. Conclusion 39