Variational inference using implicit distributions
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Variational inference using implicit distributions aims to relax the strong parametric assumptions of traditional variational inference. Implicit distributions can model more complex distributions and are defined by their ability to sample from and take derivatives of samples. The key challenges with implicit distributions are that the evidence lower bound (ELBO) must be approximated differently, using density log-ratios. The paper proposes methods to optimize the ELBO for implicit distributions using prior-contrastive forms, adversarial training similar to generative adversarial networks, and learning from denoiser gradients.
![[1] Prior contrastive form of ELBO
[2] Joint contrastive form of ELBO
const
prior](https://image.slidesharecdn.com/variationalinferenceusingimplicitdistributions-181004122721/85/Variational-inference-using-implicit-distributions-7-320.jpg)
![[1] Gradient of L (so r and s) w.r.t. 흍
forward model has to
be explicit
p(z) and q
can be implicit
this we can optimize by
reparametrizing
z = g흍
(x, 휺)
reminder:
this ratio we
approximate with
and learn](https://image.slidesharecdn.com/variationalinferenceusingimplicitdistributions-181004122721/85/Variational-inference-using-implicit-distributions-8-320.jpg)
![[1] Gradient of L (so r and s) w.r.t. 흍
Observation: gradient at some position 흍0
can be simplified so r/s does not depend on 흍:
reminder:
Optimization:
● Update
● Update ELBO](https://image.slidesharecdn.com/variationalinferenceusingimplicitdistributions-181004122721/85/Variational-inference-using-implicit-distributions-9-320.jpg)
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Variational inference using implicit distributions
- 1. Variational Inference using Implicit Distributions by Ferenc Huszar MUPI journal club 2018-10-04
- 2. Sources ● arXiv: Ferenc Huszár: Variational Inference using Implicit Distributions https://arxiv.org/abs/1702.08235 (Feb 2017 version) ● blog posts: https://www.inference.vc/variational-inference-with-implicit-probabilistic-model s-part-1-2/
- 3. Reminder: VI & ELBO approximation to posterior p(z|x) David M. Blei, Alp Kucukelbir, Jon D. McAuliffe: Variational Inference: A Review for Statisticians (https://arxiv.org/abs/1601.00670)
- 4. Explicit vs implicit ● The parametric assumptions we make in VI are often too strong. ● Implicit models would be one way to relax these. ● We can model more complicated distributions. vs https://en.wikipedia.org/wiki/Normal_distribution Implicit distributions: ● can sample from ● cab take derivatives of samples w.r.t. params
- 5. Explicit vs implicit ● The parametric assumptions we make in VI are often too strong. ● Implicit models would be one way to relax these. ● We can model more complicated distributions. vs https://en.wikipedia.org/wiki/Normal_distribution For example 휺 ~ N(0,1) 휺 Neural network Implicit distributions: ● can sample from ● cab take derivatives of samples w.r.t. params
- 6. Notation; what is implicit/explicit we usually would not see these parametrized “If q or p are implicit the ELBO needs to be approximated differently, e.g., in terms of density log-ratios”
- 7. [1] Prior contrastive form of ELBO [2] Joint contrastive form of ELBO const prior
- 8. [1] Gradient of L (so r and s) w.r.t. 흍 forward model has to be explicit p(z) and q can be implicit this we can optimize by reparametrizing z = g흍 (x, 휺) reminder: this ratio we approximate with and learn
- 9. [1] Gradient of L (so r and s) w.r.t. 흍 Observation: gradient at some position 흍0 can be simplified so r/s does not depend on 흍: reminder: Optimization: ● Update ● Update ELBO
- 10. notation: Update log ratio reminder: logistic regression empirical loss: y = -1 ⇔ z ~ p y = +1 ⇔ z ~ q Learn log ratio by optimizing logistic regression loss
- 11. Prior-contrastive adversarial VI one step of ELBO optimization using reparametrization z = g흍 (x, 휺) K steps of fitting
- 12. Translate to GANs ● discriminator ● generator G = g흍 (x, 휺) ● training mode similar to GANs - K steps for D vs 1 step for G ● ⇾ adversarial variational Bayes
- 13. Denoiser-guided learning ELBO: approximate with samples from q reparametrization chain rule ok ok ? derivative
- 14. Learn ● Trick: denoiser solution contains gradient of generating distribution: ● Find denoiser numerically by optimizing: ● Extract gradients: denoiser added noisegenerating distribution
- 15. Denoiser-guided learning ELBO: approximate with samples from q reparametrization chain rule ok ok gradient from denoiser derivative
- 16. Full algorithm one step of ELBO optimization using formula for K steps of learning denoisers
- 17. Results