Dual-averaging step-size adaptation for Hamiltonian samplers
Algorithms AIMech:DualAveragingHamiltonianStepSize DRAFT
A warmup procedure that tunes the logarithm of a Hamiltonian sampler’s integration step size using cumulative discrepancy from a target acceptance statistic, a shrinkage point and a smoothed sequence of iterates.
Identity Notes & Concept Scope
This is Hoffman and Gelman’s adaptation of dual averaging to HMC/NUTS step-size tuning. It is not the general Nesterov optimization algorithm, trajectory-length adaptation, mass-matrix estimation, or an acceptance test for retaining each NUTS output. The HMC and original NUTS acceptance statistics differ. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 153 prior native IDs/names/aliases. Both checkout text searches included ignored and hidden files and returned no matches for the selected exact IDs, full labels, aliases and source titles. Existing HMM and Markov random field records describe probability models, not these sampling or tuning procedures.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
At warmup iteration m form Hbar_m=(1−1/(m+t0))Hbar_(m−1)+(delta−a_m)/(m+t0), then log epsilon_m=mu−sqrt(m)Hbar_m/gamma. Smooth with log epsilonbar_m=m^(−kappa)log epsilon_m+(1−m^(−kappa))log epsilonbar_(m−1). Here gamma>0, t0≥0 and 1/2<kappa≤1; initialize Hbar_0=0 and the step-size/shrinkage values as specified by the chosen implementation. HMC uses its proposed trajectory’s Metropolis acceptance probability a_m. Original NUTS uses the mean hypothetical Metropolis acceptance probability over states explored in the final doubling, not a binary acceptance of its selected output. Adapt during a finite warmup, then use the averaged step size without continued adaptation for retained sampling.
⚠️ Curated Limitations & Negative Results
- The convergence discussion assumes suitable regularity, including the relevant expected acceptance statistic decreasing with step size. The update formula alone does not prove convergence on every target.
- The target delta, warmup duration, initialization and shrinkage controls remain choices. The paper’s experimental defaults are not universal optimal values or current-library defaults.
- For NUTS the statistic averages potential HMC acceptances over explored states in the final doubling; it is not the proportion of returned positions that changed.
- Adapting throughout retained sampling requires a separate adaptive-MCMC argument. This record follows the paper’s finite-warmup/freeze practice.
- The procedure tunes epsilon, not a mass matrix or the complete distribution geometry, and does not guarantee independent samples or adequate posterior exploration.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | The procedure tunes the integration step size used by a Markov-chain sampler during warmup, after which the fixed sampling transition is used. AIO MarkovChain describes the stochastic process/model, not its tuning procedure. The relation does not assert that a history-adaptive warmup sequence itself is a homogeneous Markov chain on positions alone. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Matthew D. Hoffman, Andrew Gelman
|
primary paper
full_text
|
ID: https://www.jmlr.org/papers/v15/hoffman14a.html
sha256:bf2a5592015b…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T06:48:02Z
Updated: 2026-10-11T06:48:02Z
Source File:
data/curated/DualAveragingHamiltonianStepSize.yaml