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Learning Latent Variable Models via Jarzynski-adjusted Langevin Algorithm
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Proposes JALA-EM, an SMC-based EM-style method that uses Jarzynski-weighted unadjusted Langevin particles to estimate marginal likelihood gradients and maximum marginal likelihood parameters, with nonasymptotic SGD convergence analysis.
Problem
Maximum marginal likelihood estimation (MMLE) for latent variable models is intractable; existing MCMC or particle-based EM variants are costly, lack marginal likelihood estimates for model selection, or have theory tied to plain gradient descent.
Approach
Runs unadjusted Langevin dynamics on time-varying joint potentials and corrects the bias with a recursively updated Jarzynski log-weight (sequential Monte Carlo instead of Metropolis), giving an importance-weighted gradient estimate for the marginal likelihood and on-the-fly normalising constant estimates.
Key results
- On error model selection with d_y=1500 and N=50, JALA-EM identified the true model class in 100% of 100 trials when the true model is Gaussian and 99% when the true model is Student-t.
- JALA-EM maintains high-quality posterior approximations with fewer particles (N=10, 50) where PGD and SOUL degrade, on Bayesian logistic regression.
- Theory: E[V(theta_k)-V*] bound with geometric decay plus O(1/N) error, so N rather than step size controls accuracy.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use JALA-EM to select between Gaussian and Student-t error models | Bayesian regression error-model selection · synthetic data, true Gaussian | correct model selection rate: 100 % | Helped |
| use JALA-EM to select between Gaussian and Student-t error models | Bayesian regression error-model selection · synthetic data, true Student-t | correct model selection rate: 99\% | Helped |
| use JALA-EM instead of PGD and SOUL | Bayesian logistic regression · Wisconsin Breast Cancer | parameter estimation speed | Mixed |
| use JALA-EM reweighting rather than equally weighted particle clouds | Bayesian logistic regression · Wisconsin Breast Cancer, lower particle counts | posterior approximation fidelity | Helped |
| fit the true Student-t error model with JALA-EM | Bayesian regression error-model selection · synthetic data, true Student-t | parameter recovery | Helped |
| use JALA-EM rather than OLS plus BIC | Bayesian polynomial regression model-order selection · true model orders p_star in [2, 8] | model-order selection performance | Mixed |
| use JALA-EM instead of SOUL | Bayesian neural network · MNIST 4-versus-9 and multi-class digits | predictive performance | No clear effect |
| use JALA-EM instead of PGD | Bayesian neural network · MNIST 4-versus-9 and multi-class digits | wall-clock speed | Hurt |
Surprising findings
- JALA-EM's predictive performance is comparable to SOUL on the BNN, while PGD converges faster in wall-clock time but cannot estimate marginal likelihood.
Limitations
- Weight degeneracy in high dimensions requires resampling and tuning; SMC-based cost grows with particles.
- Theory assumes bounded gradients, bounded exponential moments of weights, and PL or strong convexity conditions.
- Benchmarks limited to PGD and SOUL; Metropolis-based E-step and gradient-free M-step not compared.
Experimental setup
Datasets: Wisconsin Breast Cancer MNIST synthetic Bayesian linear regression data
Benchmarks: Wisconsin Breast Cancer MNIST 4 vs 9 MNIST 2,4,7,9
Metrics: log pointwise predictive density test error log marginal likelihood wall-clock time model selection accuracy
Baselines: PGD SOUL SFLA IPLA BIC with OLS
Builds on: Jarzynski-based sampling of time-evolving measures particle gradient descent stochastic optimisation via unadjusted Langevin (SOUL) Fisher's identity sequential Monte Carlo samplers
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