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Ai-sampler: Adversarial Learning of Markov kernels with involutive maps

ICML 2024 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Hamiltonian Monte Carlo · 0 citations (OpenAlex)

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Ai-Sampler learns MCMC transition kernels built from time-reversible neural network involutions (Hénon maps) and trains them adversarially with a C2-equivariant discriminator by minimizing a TV-distance upper bound via Pinsker's inequality.

Problem

Learning efficient, well-mixing MCMC transition kernels parameterized by neural networks, where the objective for sample quality and exploration is hard to define and deterministic involutive proposals must satisfy detailed balance.

Approach

Parameterize deterministic involutions as R∘L_θ with L_θ = R∘g⁻¹∘R∘g (time-reversible networks built from Hénon maps) so detailed balance holds by construction, and train the proposal against a discriminator whose log-output is constrained to be C2-equivariant, minimizing an upper bound on TV distance between the target and one-step-transformed target.

Key results

• On 2D densities, Ai-sampler achieves ESS of 1000 on mog2 and mog6 versus 0.8 and 2.4 for HMC, but lower ESS on ring (378 vs 981.3 for HMC).
• On Bayesian logistic regression (Australian), Ai-sampler reaches ESS 1746.5 vs 1113.4 for HMC and 1015.8 for A-NICE-MC.
• ESS/s per chain on mog2 is 1052.6 for Ai-sampler vs 0.4 for HMC, while HMC is faster on ring (2725.8 vs 402.1).

Measured effects reported in the paper

Surprising findings

• HMC achieves higher ESS and ESS/s than the Ai-sampler on the ring density, which the authors note as an exception to their overall advantage.
• Log-predictive posterior and mean matching are nearly identical between Ai-sampler and HMC on german and australian, with HMC slightly better on heart mean matching.

Limitations

• Experiments are limited to 2D synthetic densities and three small logistic regression posteriors.
• The bound's tightness depends on KL[p(x); p(RL(x))], which vanishes only if the proposal is nearly density preserving.
• Adversarial training requires careful tuning of discriminator-to-kernel update ratio and can be unstable.

Experimental setup

Datasets: mog2 mog6 ring ring5 heart german australian

Benchmarks: mog2 mog6 ring ring5 Bayesian logistic regression posteriors (heart, german, australian)

Metrics: effective sample size effective sample size per second average log predictive posterior mean matching error

Baselines: Hamiltonian Monte Carlo A-NICE-MC

Builds on: involutive MCMC A-NICE-MC time-reversible neural networks Hamiltonian Monte Carlo

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