NOMAD: A Distributed Framework for Latent Variable Models Inderjit - PowerPoint PPT Presentation

NOMAD: A Distributed Framework for Latent Variable Models Inderjit S. Dhillon Department of Computer Science University of Texas at Austin Joint work with H.-F. Yu, C.-J. Hsieh, H. Yun, and S.V.N. Vishwanathan NIPS 2014 Workshop: Distributed Machine Learning and Matrix Computations Inderjit Dhillon (UT Austin.) Dec 12, 2014 1 / 40

Outline Challenges Matrix Completion Stochastic Gradient Method Existing Distributed Approaches Our Solution: NOMAD-MF Latent Dirichlet Allocation (LDA) Gibbs Sampling Existing Distributed Solutions: AdLDA, Yahoo LDA Our Solution: F+NOMAD-LDA Inderjit Dhillon (UT Austin.) Dec 12, 2014 2 / 40

Large-scale Latent Variable Modeling Latent Variable Models: very useful in many applications Latent models for recommender systems (e.g., MF) Topic models for document corpus (e.g., LDA) Fast growth of data Almost 2 . 5 × 10 18 bytes of data added each day 90% of the world’s data today was generated in the past two year Inderjit Dhillon (UT Austin.) Dec 12, 2014 3 / 40

Challenges Algorithmic as well as hardware level Many effective algorithms involve fine-grain iterative computation ⇒ hard to parallelize Many current parallel approaches bulk synchronization ⇒ wasted CPU power when communicating complicated locking mechanism ⇒ hard to scale to many machines asynchronous computation using parameter server ⇒ not serializable, danger of stale parameters Proposed NOMAD Framework access graph analysis to exploit parallelism asynchronous computation, non-blocking communication, and lock-free serializable (or almost serializable) successful applications: MF and LDA Inderjit Dhillon (UT Austin.) Dec 12, 2014 4 / 40

Matrix Factorization: Recommender Systems Inderjit Dhillon (UT Austin.) Dec 12, 2014 5 / 40

Recommender Systems Inderjit Dhillon (UT Austin.) Dec 12, 2014 6 / 40

Matrix Factorization Approach A ≈ WH T Inderjit Dhillon (UT Austin.) Dec 12, 2014 6 / 40

Matrix Factorization Approach � i h j ) 2 + λ � � � W � 2 F + � H � 2 ( A ij − w T min , F W ∈R m × k ( i , j ) ∈ Ω H ∈R n × k Ω = { ( i , j ) | A ij is observed } Regularized terms to avoid over-fitting A transform maps users/items to latent feature space R k the i th user ⇒ i th row of W , w T i , the j th item ⇒ j th column of H T , h j . w T i h j : measures the interaction. Inderjit Dhillon (UT Austin.) Dec 12, 2014 7 / 40

SGM: Stochastic Gradient Method SGM update: pick ( i , j ) ∈ Ω � � R ij ← A ij − w T h 1 h 2 h 3 i h j , w i ← w i − η ( λ | Ω i | w i − R ij h j ) ,     w T A 11 A 12 A 13 h j ← h j − η ( λ 1 Ω j | h j − R ij w i ) ,     | ¯     w T A 21 A 22 A 23     2     Ω i : observed ratings of i -th row.     w T A 31 A 32 A 33 ¯ 3 Ω j : observed ratings of j -th column. An iteration : | Ω | updates Time per update: O ( k ) Time per iteration: O ( | Ω | k ), better than O ( | Ω | k 2 ) for ALS Inderjit Dhillon (UT Austin.) Dec 12, 2014 8 / 40

Parallel Stochastic Gradient Descent for MF Challenge: direct parallel updates ⇒ memory conflicts. Multi-core parallelization Hogwild [Niu 2011] Jellyfish [Recht et al, 2011] FPSGD** [Zhuang et al, 2013] Multi-machine parallelization: DSGD [Gemulla et al, 2011] DSGD ++ [Teflioudi et al, 2013] Inderjit Dhillon (UT Austin.) Dec 12, 2014 9 / 40

DSGD/JellyFish [Gemulla et al, 2011; Recht et al, 2011] x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Synchronize and communicate x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Synchronize and communicate Inderjit Dhillon (UT Austin.) Dec 12, 2014 10 / 40

Proposed Asynchronous Approach: NOMAD-MF [Yun et al, 2014] Inderjit Dhillon (UT Austin.) Dec 12, 2014 11 / 40

Motivation Most existing parallel approaches require Synchronization and/or E.g., ALS, DSGD/JellyFish, DSGD++, CCD++ Computing power is wasted: Interleaved computation and communication Curse of the last reducer Locking and/or E.g., parallel SGD, FPSGD** A standard way to avoid conflict and guarantee serializability Complicated remote locking slows down the computation Hard to implement efficient locking on a distributed system Computation using stale values E.g., Hogwild, Asynchronous SGD using parameter server Lack of serializability Q: Can we avoid both synchronization and locking but keep CPU from being idle and guarantee serializability ? Inderjit Dhillon (UT Austin.) Dec 12, 2014 12 / 40

Our answer: NOMAD A: Yes, NOMAD keeps CPU and network busy simultaneously Stochastic gradient update rule only a small set of variables involved Nomadic token passing widely used in telecommunication area avoids conflict without explicit remote locking Idea: “owner computes” NOMAD: multiple “active tokens” and nomadic passing Features: fully asynchronous computation lock-free implementation non-blocking communication serializable update sequence Inderjit Dhillon (UT Austin.) Dec 12, 2014 13 / 40

Access Graph for Stochastic Gradient Access graph G = ( V , E ): V = { w i } ∪ { h j } E = { e ij : ( i , j ) ∈ Ω } Connection to SG: each e ij corresponds to a SG update only access to w i and h j w i Parallelism: h j edges without common node can be updated in parallel identify “matching” in the graph Nomadic Token Passing: users mechanism s.t. active edges always items form a “matching” serializability guaranteed Inderjit Dhillon (UT Austin.) Dec 12, 2014 14 / 40

More Details Nomadic Tokens for { h j } : n tokens ( j , h j ): O ( k ) space Worker: x x p workers x x x x a computing unit + a concurrent x x x x x x token queue x x x x x x x x x a block of W : O ( mk / p ) x x x x x x a block row of A : O ( | Ω | / p ) x x x Inderjit Dhillon (UT Austin.) Dec 12, 2014 15 / 40

Illustration of NOMAD communication x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Inderjit Dhillon (UT Austin.) Dec 12, 2014 16 / 40

Comparison on a Multi-core System On a 32-core processor with enough RAM. Comparison: NOMAD, FPSGD**, and CCD++. (100M ratings) (250M ratings) Netflix, machines=1, cores=30, λ = 0 . 05, k = 100 Yahoo!, machines=1, cores=30, λ = 1 . 00, k = 100 0 . 95 NOMAD NOMAD FPSGD** FPSGD** 26 CCD++ CCD++ 0 . 94 test RMSE test RMSE 0 . 93 24 0 . 92 22 0 . 91 0 100 200 300 400 0 100 200 300 400 seconds seconds Inderjit Dhillon (UT Austin.) Dec 12, 2014 17 / 40

Comparison on a Distributed System On a distributed system with 32 machines. Comparison: NOMAD, DSGD, DSGD++, and CCD++. (100M ratings) (250M ratings) Netflix, machines=32, cores=4, λ = 0 . 05, k = 100 Yahoo!, machines=32, cores=4, λ = 1 . 00, k = 100 1 NOMAD NOMAD DSGD DSGD 26 DSGD++ 0 . 98 DSGD++ CCD++ CCD++ test RMSE test RMSE 0 . 96 24 0 . 94 22 0 . 92 0 20 40 60 80 100 120 0 20 40 60 80 100 120 seconds seconds Inderjit Dhillon (UT Austin.) Dec 12, 2014 18 / 40

NOMAD: A Distributed Framework for Latent Variable Models Inderjit - PowerPoint PPT Presentation

NOMAD: A Distributed Framework for Latent Variable Models Inderjit S. Dhillon Department of Computer Science University of Texas at Austin Joint work with H.-F. Yu, C.-J. Hsieh, H. Yun, and S.V.N. Vishwanathan NIPS 2014 Workshop: Distributed

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