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Asymptotically exact variational flows via involutive MCMC kernels
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Constructs invertible, measure-preserving iterated random function flows from involutive MCMC kernels (via auxiliary uniform variables and accept/reject inversion) and proves total variation convergence of three new variational flow families (IRF, backward IRF, ensemble IRF MixFlows).
Problem
Variational families with practical asymptotic exactness: existing MixFlows need an exactly invertible, ergodic, measure-preserving map for continuous targets, which is hard to build with discretized ODEs or MH corrections that break invertibility.
Approach
Represent general involutive MCMC kernels (including MH accept/reject) as invertible, pi-measure-preserving iterated random function systems by augmenting the state with uniform auxiliary variables, and invert the accept/reject step using the involution's self-inverse property; the resulting IRFs define flows whose TV convergence needs only invariance and irreducibility rather than ergodicity.
Key results
- IRF flows reach near-zero ELBO and log Z error on synthetic targets consistently across runs, whereas RealNVP and NSF show high variability and training divergence (10 of 15 RealNVP runs diverged on Funnel; all NSF runs diverged).
- On real-data benchmarks, exact flows match or modestly improve on best-tuned RealNVP and NSF in ELBO and log Z accuracy, at a fraction of normalizing-flow training cost.
- IRF flows outperform NUTS on two of four real models and are slightly worse on the other two, while generally running faster.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add the Metropolis-Hastings correction to the HMC flow (ablation) | homogeneous MixFlow · Banana, Funnel, Cross, and Warped Gaussian | TV distance | Helped |
| add the Metropolis-Hastings correction to the HMC flow (ablation) | homogeneous MixFlow · Banana, Funnel, Cross, and Warped Gaussian | stability at larger step sizes | Helped |
| use IRF flows instead of RealNVP or Neural Spline Flow | variational flows · synthetic targets | per-sample importance sampling ESS | Helped |
| use IRF flows instead of RealNVP or Neural Spline Flow | variational flows · synthetic targets | ELBO and log Z estimate quality | Helped |
| use IRF flows instead of RealNVP or Neural Spline Flow | variational flows · TReg, Brownian, SparseReg, and LGCP | ELBO and log Z accuracy | No clear effect |
| use IRF flows instead of the mean-field Gaussian baseline | variational flows · TReg, Brownian, SparseReg, and LGCP | ELBO | Helped |
| use IRF flows instead of normalizing flows | variational flows · TReg, Brownian, SparseReg, and LGCP | computation time | Helped |
| use IRF flows instead of NUTS | posterior approximation · TReg, Brownian, SparseReg, and LGCP | maximum absolute error of coordinate-wise posterior estimates | Mixed |
| use IRF MixFlows instead of homogeneous or backward IRF MixFlows | IRF MixFlows · synthetic targets | flow length needed to reach a given accuracy | Helped |
Surprising findings
- The uncorrected HMC MixFlow frequently diverges (NaNs) at larger step sizes, whereas the MH-corrected version remains stable.
Limitations
- Inversion accuracy degrades with flow length due to floating-point round-off, with RWMH IRF invertible up to about T=1000 and HMC/MALA up to about T=200.
- IRF MixFlow density evaluation costs O(T^2) and requires parallel hardware to be practical.
- Ensemble IRF MixFlow with M=1 fails to converge, and its convergence is limited by variance in M.
Experimental setup
Datasets: Banana Neal's funnel cross-shaped Gaussian mixture warped Gaussian Student-t regression sparse linear regression latent Brownian motion log-Gaussian Cox process
Benchmarks: Banana Neal's funnel TReg SparseReg Brownian LGCP
Metrics: ELBO log Z estimation error importance sampling ESS total variation distance posterior std error vs NUTS computation time
Baselines: RealNVP Neural Spline Flow NUTS mean-field Gaussian VI uncorrected HMC MixFlow (Hamiltonian-MixFlow)
Builds on: MixFlow involutive MCMC Hamiltonian monte carlo random-walk metropolis-hastings
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