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AutoStep: Locally adaptive involutive MCMC
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Proposes AutoStep MCMC, a locally adaptive step-size selection scheme for involutive MCMC that refreshes the step size per iteration on an augmented space, with proofs of invariance, irreducibility, aperiodicity and cost bounds.
Problem
Step size tuning in involutive MCMC (RWMH, MALA, HMC) is hard, and a single global step size fails for multiscale targets; existing adaptive methods lack irreducibility guarantees or get stuck in tails/near mode.
Approach
Augment the state with random acceptance thresholds (a,b) and a step size theta drawn from a state-dependent distribution given by a doubling/halving selector that brings |log acceptance ratio| into a symmetric range (|log b|, |log a|); this keeps the chain pi-invariant, irreducible and aperiodic, unlike autoMALA's asymmetric criterion.
Key results
- AutoStep RWMH/MALA are competitive with NUTS, slice sampling, adaptive RWMH/MALA and delayed-rejection HMC across five benchmarks, notably strong on funnel targets.
- Performance per unit cost of AutoStep decays roughly as O(|log theta_0|^-1) versus O(exp(-|log theta_0|)) for fixed-step methods when the initial step size is poorly chosen.
- Symmetric step size criterion keeps acceptance probability above 10% across state norms from 1e-5 to 1e2, whereas the asymmetric criterion yields very low acceptance near the mode and tails.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use symmetric thresholds on the absolute log acceptance ratio instead of the asymmetric criterion (ablation) | AutoStep RWMH · Normal(0,1), Laplace(0,1), and Cauchy(0,1), across state norms 10^{-5} to 10^2 | acceptance probability | Helped |
Surprising findings
- The asymmetric autoMALA-style step size criterion can get stuck for long periods near the mode and in the tails.
Limitations
- Competitive but not uniformly best; AutoStep MALA is slightly lower in some settings.
- Theoretical efficiency (expected squared jump distance, asymptotic variance) not analyzed.
- Evaluation metric KSESS relies on a gold-standard target approximation and is not recommended for practical use.
Experimental setup
Datasets: Neal's funnel (2D, 100D) mRNA transfection model Kilpisjarvi temperature regression Gliese 229 orbit model
Benchmarks: Neal's funnel 2D Neal's funnel 100D mRNA transfection model Kilpisjarvi regression Gliese 229 orbit model
Metrics: KSESS per unit cost min KSESS per second effective sample size per second acceptance probability energy jump distance
Baselines: NUTS hit-and-run slice sampler adaptive RWMH adaptive MALA delayed rejection HMC fixed-step RWMH fixed-step MALA asymmetric autoMALA step size criterion
Builds on: autoMALA involutive MCMC random walk Metropolis-Hastings MALA HMC
Closest papers by meaning
- Involutive MCMC: a Unifying Framework (ICML 2020 · similarity 0.91)
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