Bayesian Learning via Stochastic Gradient Langevin Dynamics

Yee Teh & Max Welling

**Concept Graph & Resume using Claude 3.5 Sonnet | Chat GPT4o | Llama 3:**

graph LR
classDef sgld fill:#f9d4d4, font-weight:bold, font-size:14px
classDef theory fill:#d4f9d4, font-weight:bold, font-size:14px
classDef applications fill:#d4d4f9, font-weight:bold, font-size:14px
classDef challenges fill:#f9f9d4, font-weight:bold, font-size:14px
A[Bayesian Learning via

Stochastic Gradient Langevin

Dynamics] --> B[SGLD

Algorithm] A --> C[Theoretical

Developments] A --> D[Applications

and

Extensions] A --> E[Challenges

and

Considerations] B --> B1[Stochastic gradient

descent + Langevin

dynamics. 1] B --> B2[Combines SGD

with Gaussian

noise. 6] B --> B3[Requires decreasing

step

size. 7] B --> B4[Switches from

optimization to

sampling. 9] B --> B5[Uses Fisher scoring

or Riemannian

geometry. 11] B --> B6[SGLD adapted

to Hamiltonian

Monte Carlo. 12] C --> C1[Convergence,

bias-variance

trade-offs. 10] C --> C2[Studies on SGLD

convergence. 14] C --> C3[SGLD converges

slower than

MCMC. 15] C --> C4[Non-asymptotic bounds

for non-convex

settings. 18] C --> C5[Bounds based on

mutual

information. 19] C --> C6[Better performance

with lower-temperature

posteriors. 20] D --> D1[General recipe

for stochastic gradient

MCMC. 13] D --> D2[Stein Variational

Gradient Descent. 27] D --> D3[Handling changing

data

distributions. 28] D --> D4[Transfer learning for

multiple

datasets. 29] D --> D5[Probabilistic ML

importance. 30] D --> D6[Posterior characteristics,

efficient

sampling. 21] E --> E1[Developed during

Bayesian ML

peak. 2] E --> E2[Inefficient MCMC

for large

datasets. 3] E --> E3[Initial MCMC phase

as

optimization. 4] E --> E4[Noise improves neural

network

generalization. 5] E --> E5[Bias-variance

trade-offs. 16] E --> E6[Challenge in incorporating

domain

knowledge. 22] class A,B,B1,B2,B3,B4,B5,B6 sgld class C,C1,C2,C3,C4,C5,C6 theory class D,D1,D2,D3,D4,D5,D6 applications class E,E1,E2,E3,E4,E5,E6 challenges

Stochastic Gradient Langevin

Dynamics] --> B[SGLD

Algorithm] A --> C[Theoretical

Developments] A --> D[Applications

and

Extensions] A --> E[Challenges

and

Considerations] B --> B1[Stochastic gradient

descent + Langevin

dynamics. 1] B --> B2[Combines SGD

with Gaussian

noise. 6] B --> B3[Requires decreasing

step

size. 7] B --> B4[Switches from

optimization to

sampling. 9] B --> B5[Uses Fisher scoring

or Riemannian

geometry. 11] B --> B6[SGLD adapted

to Hamiltonian

Monte Carlo. 12] C --> C1[Convergence,

bias-variance

trade-offs. 10] C --> C2[Studies on SGLD

convergence. 14] C --> C3[SGLD converges

slower than

MCMC. 15] C --> C4[Non-asymptotic bounds

for non-convex

settings. 18] C --> C5[Bounds based on

mutual

information. 19] C --> C6[Better performance

with lower-temperature

posteriors. 20] D --> D1[General recipe

for stochastic gradient

MCMC. 13] D --> D2[Stein Variational

Gradient Descent. 27] D --> D3[Handling changing

data

distributions. 28] D --> D4[Transfer learning for

multiple

datasets. 29] D --> D5[Probabilistic ML

importance. 30] D --> D6[Posterior characteristics,

efficient

sampling. 21] E --> E1[Developed during

Bayesian ML

peak. 2] E --> E2[Inefficient MCMC

for large

datasets. 3] E --> E3[Initial MCMC phase

as

optimization. 4] E --> E4[Noise improves neural

network

generalization. 5] E --> E5[Bias-variance

trade-offs. 16] E --> E6[Challenge in incorporating

domain

knowledge. 22] class A,B,B1,B2,B3,B4,B5,B6 sgld class C,C1,C2,C3,C4,C5,C6 theory class D,D1,D2,D3,D4,D5,D6 applications class E,E1,E2,E3,E4,E5,E6 challenges

**Resume: **

**1.-** SGLD (Stochastic Gradient Langevin Dynamics): Algorithm combining stochastic gradient descent with Langevin dynamics for scalable Bayesian inference.

**2.-** Historical context: SGLD developed during Bayesian machine learning's peak, as deep learning was emerging.

**3.-** Big data challenge: Traditional MCMC methods inefficient for large datasets, while SGD scales well.

**4.-** Burn-in as optimization: Initial phase of MCMC is essentially optimization, wasting resources on precise updates.

**5.-** Noise for generalization: Adding noise (e.g., dropout) improves generalization in neural networks.

**6.-** SGLD algorithm: Combines SGD with injected Gaussian noise, transitioning from optimization to sampling.

**7.-** Annealing step size: SGLD requires decreasing step size over time to converge to correct distribution.

**8.-** Metropolis-Hastings step: As step size decreases, acceptance probability approaches 1, allowing skip of accept-reject step.

**9.-** Automatic transition: SGLD naturally switches from optimization to sampling as injected noise dominates gradient noise.

**10.-** Theoretical developments: Subsequent work analyzed SGLD's convergence, bias-variance trade-offs, and relationships to continuous-time processes.

**11.-** Preconditioned SGLD: Extensions using Fisher scoring or Riemannian geometry to improve sampling efficiency.

**12.-** Stochastic Gradient Hamiltonian Monte Carlo: Adaptation of SGLD to Hamiltonian Monte Carlo for better exploration.

**13.-** Unified framework: Ma et al. provided a general recipe for stochastic gradient MCMC algorithms.

**14.-** Convergence analysis: Studies on weak vs. strong convergence, consistency, and central limit theorems for SGLD.

**15.-** Convergence rate: SGLD converges at m^(-1/3) rate, slower than standard MCMC's m^(-1/2) rate.

**16.-** Fixed step size analysis: Investigating trade-offs between bias and variance with constant step size.

**17.-** Wasserstein distance: Used to bound convergence of SGLD to diffusion limit and excess risk.

**18.-** Excess risk bounds: Non-asymptotic bounds derived for SGLD in non-convex settings.

**19.-** Generalization error: Bounds based on mutual information between dataset and SGLD iterates.

**20.-** Cold posteriors: Theoretical and practical evidence suggesting better performance with lower-temperature posteriors.

**21.-** Bayesian deep learning: Growing research area with open questions about posterior characteristics and efficient sampling methods.

**22.-** Prior specification: Challenge of incorporating meaningful domain knowledge as priors in Bayesian deep learning.

**23.-** Simplicity and generalizability: SGLD's success attributed to its simple implementation and room for extensions.

**24.-** Theoretical analysis by others: Mathematical community provided rigorous analysis post-publication.

**25.-** Data-dependent bounds: Recent work produces bounds that depend on specific dataset characteristics.

**26.-** Tall data algorithms: Developments in MCMC for datasets with many samples but low dimensionality.

**27.-** Stein Variational Gradient Descent: Alternative method using deterministic particle interactions for posterior approximation.

**28.-** Online and adaptive SGLD: Potential extensions for handling changing data distributions over time.

**29.-** Hierarchical Bayesian modeling: Approach for transfer learning and relating multiple datasets or tasks.

**30.-** Inference algorithms' importance: Crucial for probabilistic machine learning, especially with latent variable models.

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