Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a...

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Lecture : Variational Inference Riashat Islam Slides courtesy of David Blei Reasoning and Learning Lab McGill University 31st October 2018

Transcript of Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a...

Page 1: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Lecture : Variational Inference

Riashat IslamSlides courtesy of David Blei

Reasoning and Learning LabMcGill University

31st October 2018

Page 2: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Probabilistic Machine Learning

Page 3: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Approximate Inference

Page 4: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Computational Challenge

Page 5: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Probabilistic Model

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Variational Inference

Page 7: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Variational Bound

Page 8: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Variational Bound

Page 9: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Motivation : Topic Modelling

Page 10: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Example : Latent Dirichlet Allocation

Page 11: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Example : Latent Dirichlet Allocation

Page 12: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Example : Latent Dirichlet Allocation

Page 13: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

LDA as Graphical Model

Page 14: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Posterior Inference in LDA

Page 15: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

The Evidence Lower Bound (ELBO)

Page 16: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Stochastic Gradients of ELBO

Page 17: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Variational Inference Recipe

Page 18: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Variational Inference Recipe

Page 19: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Variational Inference Recipe

Page 20: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Variational Inference Recipe

Page 21: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Variational Inference Recipe

Page 22: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Variational Inference Recipe

Page 23: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Variational Inference Recipe

Page 24: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Black Box Variational Inference

Page 25: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Problem of Classical VI

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New VI Recipe

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Computing Gradients of Expectations

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Score Function Gradients of ELBO

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Score Function Estimator

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Noisy Unbiased Gradients

Page 31: Lecture : Variational Inference - GitHub Pages · VI turns inference into optimization. Posit a variational family of distributions over the latent variables, q(z; v) Fit the variational

Black Box Variational Inference

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Pathwise Gradients of ELBO

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Pathwise Estimator

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Pathwise Estimator

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Score Function vs Pathwise Estimator