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Comparison of Random Walk based techniques for estimating network averages Jithin K. Sreedharan INRIA, France Arun Kadavankandy Konstantin Avrachenkov Vivek S. Borkar INRIA, France INRIA, France IIT Bombay, India CSoNet 2016, August 2

  1. Reinforcement Learning technique (contd.) Stochastic Approximation Seed set For each node 𝑗 in Algorithm Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

  2. Reinforcement Learning technique (contd.) Stochastic Approximation Seed set For each node 𝑗 in Function sum Algorithm inside a tour Cost function Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

  3. Reinforcement Learning technique (contd.) Stochastic Approximation Seed set For each node 𝑗 in Function sum Algorithm inside a tour Cost function ……. sample 1 sample 2 sample 𝑙 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

  4. Reinforcement Learning technique (contd.) Stochastic Approximation Seed set For each node 𝑗 in Function sum Algorithm inside a tour Cost function ……. sample 1 sample 2 sample 𝑙 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

  5. Reinforcement Learning technique (contd.) Stochastic Approximation Seed set For each node 𝑗 in Function sum Algorithm inside a tour Cost function Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

  6. Which Random Walk method to select ? Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

  7. Which Random Walk method to select ?  Mixing time Not a good criterion here due to burn-in period. Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

  8. Which Random Walk method to select ?  Mixing time Not a good criterion here due to burn-in period. X k : rejected sample : accepted sample ……. X 1 X 2 Burn-in period Approximate stationary regime Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

  9. Which Random Walk method to select ?  Mixing time Not a good criterion here due to burn-in period. X k : rejected sample : accepted sample ……. X 1 X 2 Burn-in period Approximate stationary regime Reinforcement Learning technique does not require burn-in period Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

  10. Which Random Walk method to select ?  Mixing time Not a good criterion here due to burn-in period.  Efficiency of the estimator: How many samples are needed to achieve certain accuracy Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

  11. Asymptotic Variance Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 11

  12. Asymptotic Variance Asymptotic variance of the estimator Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 11

  13. Asymptotic Variance Asymptotic variance of the estimator Also from Central Limit Theorem equivalent Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 11

  14. Asymptotic Variance (contd.) Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 12

  15. Asymptotic Variance (contd.)  For Metropolis-Hastings Sampling, where Fundamental matrix of Markov chain Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 12

  16. Asymptotic Variance (contd.)  For Metropolis-Hastings Sampling, where Fundamental matrix of Markov chain  For Respondent Driven Sampling, Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 12

  17. Asymptotic Variance (contd.)  For Metropolis-Hastings Sampling, where Fundamental matrix of Markov chain  For Respondent Driven Sampling,  For Reinforcement Learning based sampling, Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 12

  18. Numerical Studies Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

  19. Numerical Studies Normalized Root Mean Square Error (NRMSE) vs Budget B Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

  20. Numerical Studies Normalized Root Mean Square Error (NRMSE) vs Budget B Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

  21. Numerical Studies Normalized Root Mean Square Error (NRMSE) vs Budget B Why MSE ? Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

  22. Numerical Studies Normalized Root Mean Square Error (NRMSE) vs Budget B Why MSE ? Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

  23. Numerical Studies Normalized Root Mean Square Error (NRMSE) vs Budget B Why MSE ? Budget B: number of allowed samples Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

  24. Les Misérables network Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 14

  25. Les Misérables network Number of nodes: 77, number of edges: 254. Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 14

  26. Les Misérables network Number of nodes: 77, number of edges: 254. Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 14

  27. Les Misérables network Number of nodes: 77, number of edges: 254. Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 14

  28. Les Misérables network contd. Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 15

  29. Les Misérables network contd. Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 15

  30. Les Misérables network contd. Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 15

  31. Les Misérables network contd. Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

  32. Les Misérables network contd. Study of asymptotic variance Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

  33. Les Misérables network contd. Study of asymptotic variance Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

  34. Les Misérables network contd. Study of asymptotic variance Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

  35. Les Misérables network contd. Study of asymptotic variance Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

  36. Friendster network Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 17

  37. Friendster network Number of nodes ~ 65K number of edges ~ 1.25M Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 17

  38. Friendster network Number of nodes ~ 65K number of edges ~ 1.25M Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 17

  39. Friendster network Number of nodes ~ 65K number of edges ~ 1.25M Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 17

  40. Friendster network contd. Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

  41. Friendster network contd. Stability of sample paths: Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

  42. Friendster network contd. Stability of sample paths: single path example Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

  43. Friendster network contd. Stability of sample paths: single path example Varying super-node size Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

  44. Friendster network contd. Stability of sample paths: single path example Varying super-node size Varying step size Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

  45. Conclusions Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

  46. Conclusions  Rand Walk based estimators of Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

  47. Conclusions  Rand Walk based estimators of  Numerical and theoretical study of Mean Square Error & Asymptotic Variance of  Metropolis-Hastings sampling  Respondent Driven sampling (RDS)  New Reinforcement Learning based sampling (RL) Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

  48. Conclusions  Rand Walk based estimators of  Numerical and theoretical study of Mean Square Error & Asymptotic Variance of  Metropolis-Hastings sampling  Respondent Driven sampling (RDS)  New Reinforcement Learning based sampling (RL)  Reinforcement Learning technique:  Tackles disconnected graph  A cross between deterministic iteration and MCMC  Can control the stability of the algorithm with step sizes Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

  49. Conclusions  Rand Walk based estimators of  Numerical and theoretical study of Mean Square Error & Asymptotic Variance of  Metropolis-Hastings sampling  Respondent Driven sampling (RDS)  New Reinforcement Learning based sampling (RL)  Reinforcement Learning technique:  Tackles disconnected graph  A cross between deterministic iteration and MCMC  Can control the stability of the algorithm with step sizes  RDS works better. RL technique comparable, yet more stable and no burn-in ! Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

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