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Sampling from multimodal distributions with stochastic localization

Louis Grenioux

Louis Grenioux*, Maxence Noble*, Marylou Gabrié, Alain Oliviero Durmus.

Stochastic Localization via Iterative Posterior Sampling. ICML 2024

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The challenge of multimodal sampling

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

How to sample from ?

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The challenge of multimodal sampling

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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The challenge of multimodal sampling

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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The challenge of multimodal sampling

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MALA

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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The challenge of multimodal sampling

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MALA

NUTS

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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The challenge of multimodal sampling

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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The challenge of multimodal sampling

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MALA

NUTS

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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Handling multimodality with Sequential Monte Carlo

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Target distribution

Base distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

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Handling multimodality with Sequential Monte Carlo

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Target distribution

Base distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

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Handling multimodality with Sequential Monte Carlo

5

Target distribution

Base distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

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Handling multimodality with Sequential Monte Carlo

5

Target distribution

Base distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

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Handling multimodality with Sequential Monte Carlo

5

Target distribution

Base distribution

(1) Jump to the next distribution and reweight

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

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Handling multimodality with Sequential Monte Carlo

5

Target distribution

Base distribution

(1) Jump to the next distribution and reweight

(2) Ressample the particles

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

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Handling multimodality with Sequential Monte Carlo

5

Target distribution

Base distribution

(1) Jump to the next distribution and reweight

(2) Ressample the particles

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

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Handling multimodality with Sequential Monte Carlo

5

Target distribution

Base distribution

(1) Jump to the next distribution and reweight

(2) Ressample the particles

(3) Run MCMC kernel

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

​

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Handling multimodality with Sequential Monte Carlo

5

Target distribution

Base distribution

(1) Jump to the next distribution and reweight

(2) Ressample the particles

(3) Run MCMC kernel

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

​

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Handling multimodality with Sequential Monte Carlo

5

Target distribution

Base distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

​

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Handling multimodality with Sequential Monte Carlo

5

Target distribution

Base distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Dimension 12

Difficult to scale into high-dimension

[1] Del Moral et al. A. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society 2006

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The goal of this talk

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🎯 Give a method to sample multimodal distributions leveraging modern hardware and ideas from generative modeling

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

💡 Using similar ideas as SMC

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The goal of this talk

6

🎯 Give a method to sample multimodal distributions leveraging modern hardware and ideas from generative modeling

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

💡 Using similar ideas as SMC

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A sequence of distribution define by a stochastic process

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The signal part is

increasingly

informative

The signal predominates over the noise part

💡

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Brownian motion

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A sequence of distribution define by a stochastic process

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💡

Stochastic Localization principle

We say that the stochastic process localizes on the target

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Brownian motion

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A sequence of distribution define by a stochastic process

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💡

Target distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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A sequence of distribution define by a stochastic process

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💡

Target distribution

Density of

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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Using stochastic localization for sampling (SLIPS)

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💡 Let’s sample from the stochastic process and use the localization property

✅ It has the same marginal distribution as the solution of this SDE

the denoiser

📍Note that the denoiser is the mean of the posterior distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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Using stochastic localization for sampling (SLIPS)

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💡 Let’s sample from the stochastic process and use the localization property

the denoiser

📍Note that the denoiser is the mean of the posterior distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

✅ It has the same marginal distribution as the solution of this SDE

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Using stochastic localization for sampling (SLIPS)

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💡 Let’s sample from the stochastic process and use the localization property

the denoiser

📍Note that the denoiser is the mean of the posterior distribution

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

✅ It has the same marginal distribution as the solution of this SDE

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Using stochastic localization for sampling (SLIPS)

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✅ We use a Markovian projection

💡 We discretize the SDE with an Euler-Maruyama scheme

⚠️ The denoiser is an intractable expectation

✨ MCMC ✨

Let be a grid on . Set

with

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

🤔 Is it really easier than directly running MCMC on the target ?

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Starting the recursion from a specific time

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

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Starting the recursion from a specific time

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

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Starting the recursion from a specific time

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

MCMC estimation

Density of

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Starting the recursion from a specific time

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

MCMC estimation

Density of

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Starting the recursion from a specific time

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

MCMC estimation

Density of

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Starting the recursion from a specific time

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

MCMC estimation

Density of

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Starting the recursion from a specific time

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

MCMC estimation

💡Start recursion from here

Density of

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Starting the recursion from a specific time

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

MCMC estimation

💡Start recursion from here

Density of

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Starting the recursion from a specific time and point

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Tweedie’s formula

“known” when

“known” when

💡 Sample the initial point using the Langevin algorithm

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Duality of log-concavity

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

Density of

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Duality of log-concavity

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

Density of

Density of

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Duality of log-concavity

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

📍 Not too soon

Density of

Density of

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Duality of log-concavity

14

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

📍 Not too soon

📍 Not too late

Density of

Density of

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Duality of log-concavity

14

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

📍 Not too soon

📍 Not too late

Density of

Density of

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Duality of log-concavity

14

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

📍 Not too soon

📍 Not too late

Density of

Density of

✅

✅

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Using stochastic localization for sampling (SLIPS)

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💡 Let’s sample from the stochastic process and use the localization property

✅ It has the same marginal law as the solution of this SDE

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

💡 We discretize the SDE with an Euler-Maruyama scheme

Let be a grid on .Set

with

log-concave

log-concave

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Using stochastic localization for sampling (SLIPS)

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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Experiments on Gaussian mixtures

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Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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Experiments on a field system

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The model [2]

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  • 1D model on a grid of size 100 (i.e., dimension is 100)
  • Two modes and
  • Parameter

A continuous relaxation of the Ising model

[2] Gabrié et al. Adaptive Monte Carlo augmented with normalizing flows. PNAS, 2022

Approximation of the relative weight

0th order

2nd

order

Louis Grenioux - Sampling from multimodal distributions with stochastic localization

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Thank you for your attention !

Reference: Stochastic Localization via Iterative Posterior Sampling, arXiv:2402.10758, ICML 2024

🚀 Play with the code : https://github.com/h2o64/slips

Louis Grenioux - Sampling from multimodal distributions with stochastic localization