Metropolis-adjusted Langevin algorithm
Algorithms AIMech:MetropolisAdjustedLangevinAlgorithm DRAFT
Aliases: MALA
A Metropolis–Hastings sampler that proposes a Gaussian step with drift proportional to the target log-density gradient and corrects the asymmetric proposal with an accept/reject test.
Identity Notes & Concept Scope
This record covers the fixed-step, identity-diffusion formulation of Roberts and Tweedie. It differs from ULA by its Metropolis correction, from random-walk Metropolis by its gradient-directed mean, and from multi-step Hamiltonian trajectories. Preconditioned and manifold variants are not silently included in the displayed formula. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 159 prior native IDs/names/aliases. Ignored-inclusive full-checkout searches found prior discovery leads for Gibbs and slice sampling plus unrelated environment substring hits, not existing concept records. Existing HMM/MRF models and general Metropolis–Hastings were inspected to distinguish model, general transition and specialized algorithm scope.
Claims & Grounded Evidence
2 Claims
Mechanism
#method
For a positive differentiable target pi on Euclidean space, choose h>0 and propose y from q(y|x)=Normal(x+(h/2)grad log pi(x),hI). Accept with min(1,pi(y)q(x|y)/(pi(x)q(y|x))) and otherwise keep x. Evaluate the reverse Gaussian using its mean at y; the state-dependent gradient generally makes q asymmetric, so its ratio cannot be dropped. This correction makes the transition reversible with respect to the intended target under the stated support conditions, despite the discretized proposal.
Limitation
#mala-convergence-boundaries
The paper separates conditions for geometric convergence from counterexamples and non-geometric cases. Its rejection and tail analysis shows that correcting the invariant target does not universally supply fast convergence.
⚠️ Curated Limitations & Negative Results
- The full target gradient and both proposal densities must be correctly evaluated. Unknown common target normalization cancels, but the reverse proposal factor does not.
- Target invariance and correction of proposal discretization do not certify rapid mixing, independent draws or accuracy of a finite run.
- The source gives tail-dependent failures of geometric ergodicity; adding a Metropolis step is not a universal guarantee of exponential convergence.
- Large steps, problematic geometry and numerical gradient or density errors can make practical sampling ineffective. This record asserts no universal step size or acceptance target.
- Truncated drift, stochastic gradients, variable metrics and constrained-state extensions need separate definitions and validation.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | The procedure constructs Markov transitions for sampling a specified target, with the invariance and convergence qualifications stated in this record. AIO MarkovChain denotes the stochastic process/model and is exported beneath Network. This is a method-to-process related match, not equivalence, subclass membership or a claim that the algorithm is a trained network. |
Evidence Sources & Primary Literature
2 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Gareth O. Roberts, Richard L. Tweedie
|
primary paper
full_text
|
ID: doi:10.2307/3318418
sha256:410e26b08534…
|
mala-convergence-boundaries
|
Gareth O. Roberts, Richard L. Tweedie
|
primary paper
full_text
|
ID: doi:10.2307/3318418
sha256:410e26b08534…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T07:00:31Z
Updated: 2026-10-11T07:00:31Z