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Long-time asymptotics of noisy SVGD outside the population limit
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Studies the long-time limit set of noisy SVGD (SVGD with Langevin noise) for finite particles, proves it is well-defined, approaches the target as n grows via McKean-Vlasov analysis, and avoids SVGD's variance collapse.
Problem
Long-time (k to infinity) behavior of SVGD with a finite number of particles n is not understood; SVGD cannot converge to the continuous target with finitely many atoms and is known to collapse variance in high dimension.
Approach
Adds a Langevin noise term to each SVGD update, making the limit set of the particle empirical measures well-defined for fixed n; by interpolating particle trajectories and relating them to McKean-Vlasov processes, shows the limit set approaches the target as n grows, with a KL Lyapunov argument and LSI-based exponential convergence.
Key results
- Noisy SVGD limit set is well-defined (tight particle empirical measures) for fixed n with lambda>0, and the averaged limit set converges in probability to the target pi as n grows.
- Under LSI, the non-averaged limit set also converges to pi as n grows, and KL is a Lyapunov function with exponential convergence rate alpha*lambda for McKean-Vlasov marginals.
- Experiments on standard Gaussian sampling show noisy SVGD's dimension-averaged marginal variance approaches 1 as n increases, while SVGD's DAMV tends to zero with dimension.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add Langevin noise (λ > 0) to SVGD | SVGD · Standard Gaussian sampling; varying dimension and particle count | Dimension-Averaged Marginal Variance (DAMV) | Helped |
| increase the number of particles | Noisy SVGD · Standard Gaussian sampling | Dimension-Averaged Marginal Variance (DAMV) | Helped |
Surprising findings
- Deterministic SVGD's McKean-Vlasov dynamics can be stationary at Dirac measures at stationary points of F, which is the mechanism behind variance collapse.
Limitations
- Results are asymptotic and do not quantify convergence rates in k or n.
- Requires assumptions such as LSI and smoothness/kernel regularity for the full characterization of the non-averaged limit set.
- Experiments are limited to standard Gaussian targets.
Experimental setup
Metrics: dimension-averaged marginal variance
Baselines: SVGD
Builds on: Stein Variational Gradient Descent Langevin algorithm McKean-Vlasov process
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