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Metropolis Adjusted Microcanonical Hamiltonian Monte Carlo
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Introduces MAMS, a Metropolis-adjusted microcanonical (MCLMC-style) sampler with derived MH acceptance and automatic tuning, outperforming NUTS by 2-7x in gradient calls on benchmarks.
Problem
MCLMC is efficient for high-dimensional sampling but produces biased samples because its integration error is not corrected by a Metropolis-Hastings step; NUTS remains the main unbiased black-box alternative.
Approach
Derives the Metropolis-Hastings acceptance for microcanonical (velocity-norm-preserving) dynamics, showing the MH weight equals the discretization energy error plus a divergence term, and adds an automatic tuning scheme (step size, preconditioning, trajectory length) so the sampler is asymptotically unbiased and out of the box.
Key results
- MAMS needs up to about 7x fewer gradient calls than NUTS on benchmarks (e.g., Gaussian 3,249 vs 19,652; StochVol 430,088 vs 843,768), and is typically at least 2x better.
- On 100-d Gaussians, MAMS scales as condition number^(-1/2) like NUTS with roughly 4x better proportionality constant, and both scale as d^(1/4) with dimension.
- Automatically tuned trajectory length performs close to grid-search-optimal L on all benchmarks; NUTS fails to converge on Neal's Funnel while MAMS converges with acceptance 0.99.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace NUTS with MAMS | sampling · Gaussian | gradient calls to squared error < 0.01: 19,652 -> 3,249 | Helped |
| replace NUTS with MAMS | sampling · Banana | gradient calls to squared error < 0.01: 95,519 -> 14,078 | Helped |
| replace NUTS with MAMS | sampling · Bimodal | gradient calls to squared error < 0.01: 210,758 -> 139,418 | Helped |
| replace NUTS with MAMS | sampling · Rosenbrock | gradient calls to squared error < 0.01: 161,359 -> 94,184 | Helped |
| replace NUTS with MAMS | sampling · Cauchy | gradient calls to squared error < 0.01: 171,373 -> 110,404 | Helped |
| replace NUTS with MAMS | sampling · Brownian | gradient calls to squared error < 0.01: 29,816 -> 13,528 | Helped |
| replace NUTS with MAMS | sampling · German Credit | gradient calls to squared error < 0.01: 88,975 -> 55,748 | Helped |
| replace NUTS with MAMS | sampling · ItemResp | gradient calls to squared error < 0.01: 76,043 -> 45,371 | Helped |
| replace NUTS with MAMS | sampling · StochVol | gradient calls to squared error < 0.01: 843,768 -> 430,088 | Helped |
| replace NUTS with MAMS | sampling · Funnel | gradient calls to squared error < 0.01: $> 10^8$ -> 2,346,899 | Helped |
| use Langevin noise instead of trajectory length randomization (ablation) | MAMS · Gaussian | gradient calls to squared error < 0.01: 3,249 -> 3,172 | Helped |
| use Langevin noise instead of trajectory length randomization (ablation) | MAMS · Banana | gradient calls to squared error < 0.01: 14,078 -> 14,818 | Hurt |
Surprising findings
- NUTS failed to converge on Neal's Funnel, whereas MAMS converged (requiring acceptance rate 0.99).
Limitations
- Gradient calls used as proxy for wall-clock time rather than full runtime comparisons on expensive models
- Not applicable to stochastic gradients / Bayesian neural networks
- Tuning constants (e.g., trajectory-length proportionality) determined numerically on Gaussians; method needs broader real-world testing
Experimental setup
Datasets: Inference Gym benchmark problems (Gaussian, Banana, Bimodal, Rosenbrock, Cauchy, Brownian, German Credit, ItemResp, Stoc
Benchmarks: Gaussian Banana Bimodal Rosenbrock Cauchy Brownian German Credit ItemResp StochVol Funnel
Metrics: gradient calls to reach b^2_max < 0.01 squared error of second moment effective sample size per gradient
Baselines: NUTS MCLMC MAMS (Langevin) MAMS (Grid Search)
Builds on: Microcanonical Langevin Monte Carlo No-U-Turn Sampler MALT Hamiltonian Monte Carlo
Closest papers by meaning
- Practical and Scalable Hamiltonian Monte Carlo Without the Metropolis Test (ICML 2026 · similarity 0.91)
- Entropy-based adaptive Hamiltonian Monte Carlo (NeurIPS 2021 · similarity 0.90)
- A Gradient Based Strategy for Hamiltonian Monte Carlo Hyperparameter Optimization (ICML 2021 · similarity 0.89)
- Microcanonical Langevin Ensembles: Advancing the Sampling of Bayesian Neural Networks (ICLR 2025 · similarity 0.88)
- Parallelizing MCMC Across the Sequence Length (NeurIPS 2025 · similarity 0.88)
- Tuning Sequential Monte Carlo Samplers via Greedy Incremental Divergence Minimization (ICML 2025 · similarity 0.87)
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