Presentation is loading. Please wait.

Presentation is loading. Please wait.

Sampling algorithms and Markov chains László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052

Published byFrederick Daniel Modified over 8 years ago

Similar presentations

Presentation on theme: "Sampling algorithms and Markov chains László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052"— Presentation transcript:

1 Sampling algorithms and Markov chains László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052 lovasz@microsoft.com

2 Sampling: a general algorithmic task Applications: - statistics - simulation - counting - numerical integration - optimization - …

3 L: a language in NP, with presentation polynomial time algorithm Find: - a certificate Given: x certificate - an optimal certificate - the number of certificates - a random certificate (uniform, or given distribution)

4 One general method for sampling: Markov chains (+rejection sampling, lifting,…) Construct ergodic Markov chain with states: V stationary distribution: p Want: sample from distribution p on set V Simulate (run) the chain for T steps Output the final state ???????????? mixing time

5 Given: poset 1 2 54 3 State: compatible linear order 1 3 54 2 Transition: - pick randomly label i<n ; - interchange i and i+1 if possible

6 Mixing time Bipartite graph?! : distribution after t steps Roughly: (enough to consider )

7 Conductance in sequence of independent samples: frequency of stepping from S to K\S conductance: in Markov chain:

8 Jerrum - Sinclair But in finer analysis? In typical sampling application: polynomial

9 Key lemma: Proof for l=k+1

10 L – Simonovits Dyer – Frieze Simple isoperimetric inequality: Improved isoperimetric inequality: Kannan-L After appropriate preprocessing,

11 Lifting Markov chains Diaconis – Holmes – Neal

Download ppt "Sampling algorithms and Markov chains László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052"

Similar presentations

Slow and Fast Mixing of Tempering and Swapping for the Potts Model Nayantara Bhatnagar, UC Berkeley Dana Randall, Georgia Tech.

Slow and Fast Mixing of Tempering and Swapping for the Potts Model Nayantara Bhatnagar, UC Berkeley Dana Randall, Georgia Tech.
On allocations that maximize fairness Uriel Feige Microsoft Research and Weizmann Institute.

On allocations that maximize fairness Uriel Feige Microsoft Research and Weizmann Institute.
=0.01010001011.

=
Approximate List- Decoding and Hardness Amplification Valentine Kabanets (SFU) joint work with Russell Impagliazzo and Ragesh Jaiswal (UCSD)

Approximate List- Decoding and Hardness Amplification Valentine Kabanets (SFU) joint work with Russell Impagliazzo and Ragesh Jaiswal (UCSD)
Slice Sampling Radford M. Neal The Annals of Statistics (Vol. 31, No. 3, 2003)

Slice Sampling Radford M. Neal The Annals of Statistics (Vol. 31, No. 3, 2003)
Gibbs sampler - simple properties It’s not hard to show that this MC chain is aperiodic. Often is reversible distribution. If in addition the chain is.

Gibbs sampler - simple properties It’s not hard to show that this MC chain is aperiodic. Often is reversible distribution. If in addition the chain is.
Counting Algorithms for Knapsack and Related Problems 1 Raghu Meka (UT Austin, work done at MSR, SVC) Parikshit Gopalan (Microsoft Research, SVC) Adam.

Counting Algorithms for Knapsack and Related Problems 1 Raghu Meka (UT Austin, work done at MSR, SVC) Parikshit Gopalan (Microsoft Research, SVC) Adam.
1 The Monte Carlo method. 2 (0,0) (1,1) (-1,-1) (-1,1) (1,-1) 1 Z= 1 If  X 2 +Y 2  1 0 o/w (X,Y) is a point chosen uniformly at random in a 2  2 square.

1 The Monte Carlo method. 2 (0,0) (1,1) (-1,-1) (-1,1) (1,-1) 1 Z= 1 If  X 2 +Y 2  1 0 o/w (X,Y) is a point chosen uniformly at random in a 2  2 square.
Randomized Algorithms Kyomin Jung KAIST Applied Algorithm Lab Jan 12, WSAC 2010 1.

Randomized Algorithms Kyomin Jung KAIST Applied Algorithm Lab Jan 12, WSAC
Topological nonexistence results in complexity theory and combinatorics László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052

Topological nonexistence results in complexity theory and combinatorics László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052
11 - Markov Chains Jim Vallandingham.

11 - Markov Chains Jim Vallandingham.
10/11/2001Random walks and spectral segmentation1 CSE 291 Fall 2001 Marina Meila and Jianbo Shi: Learning Segmentation by Random Walks/A Random Walks View.

10/11/2001Random walks and spectral segmentation1 CSE 291 Fall 2001 Marina Meila and Jianbo Shi: Learning Segmentation by Random Walks/A Random Walks View.
Markov Chains Ali Jalali. Basic Definitions Assume s as states and s as happened states. For a 3 state Markov model, we construct a transition matrix.

Markov Chains Ali Jalali. Basic Definitions Assume s as states and s as happened states. For a 3 state Markov model, we construct a transition matrix.
Graduate School of Information Sciences, Tohoku University

Graduate School of Information Sciences, Tohoku University
6.896: Probability and Computation Spring 2011 Constantinos (Costis) Daskalakis lecture 2.

6.896: Probability and Computation Spring 2011 Constantinos (Costis) Daskalakis lecture 2.
BAYESIAN INFERENCE Sampling techniques

BAYESIAN INFERENCE Sampling techniques
1 Perfect Simulation and Stationarity of a Class of Mobility Models Jean-Yves Le Boudec (EPFL) Milan Vojnovic (Microsoft Research Cambridge)

1 Perfect Simulation and Stationarity of a Class of Mobility Models Jean-Yves Le Boudec (EPFL) Milan Vojnovic (Microsoft Research Cambridge)
Computational statistics 2009 Random walk. Computational statistics 2009 Random walk with absorbing barrier.

Computational statistics 2009 Random walk. Computational statistics 2009 Random walk with absorbing barrier.
Computational statistics, course introduction Course contents  Monte Carlo Methods  Random number generation  Simulation methodology  Bootstrap  Markov.

Computational statistics, course introduction Course contents  Monte Carlo Methods  Random number generation  Simulation methodology  Bootstrap  Markov.
Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,

Christian Sohler 1 University of Dortmund Testing Expansion in Bounded Degree Graphs Christian Sohler University of Dortmund (joint work with Artur Czumaj,

Similar presentations

Ads by Google