Unadjusted Langevin algorithm
Algorithms AIMech:UnadjustedLangevinAlgorithm DRAFT
Aliases: ULA
An approximate sampling algorithm that discretizes overdamped Langevin dynamics by an explicit gradient drift and Gaussian noise without a Metropolis acceptance correction.
Identity Notes & Concept Scope
This is the full-gradient Euler discretization for a differentiable continuous target. It is not deterministic gradient descent, stochastic-gradient Langevin dynamics, the exact continuous-time diffusion or Metropolis-adjusted Langevin sampling. A fixed-step chain generally has a different invariant law from its intended target. 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 proper target pi(x) proportional to exp(−U(x)), with differentiable U, update X_(k+1)=X_k−gamma_(k+1) grad U(X_k)+sqrt(2 gamma_(k+1)) Z_(k+1), where the Z values are independent standard Gaussian vectors and gamma is positive. There is no accept/reject step. Constant and predetermined decreasing step-size schedules are both covered by the source. Equivalently, writing h=2 gamma and grad log pi=−grad U gives mean x+(h/2)grad log pi(x) and noise covariance hI. This equivalence is an explicit change of notation, not a separate sampling method.
Limitation
#discrete-bias-and-instability
Roberts and Tweedie distinguish ULA’s generally biased stationary law from the intended continuous-diffusion target and show that discretization can qualitatively change recurrence and convergence behavior. These are conditional results, not a blanket claim that every ULA run is unstable.
⚠️ Curated Limitations & Negative Results
- At a fixed positive step size, a stationary distribution, when it exists, generally differs from pi. Running longer does not by itself remove that discretization bias.
- The paper’s convergence results have regularity, drift and step-size assumptions. Merely taking gamma to zero does not establish convergence for every target or schedule.
- Discrete Euler dynamics can be unstable even when the continuous diffusion is well behaved. Gradients and noise scaling both depend on the step-size convention.
- Roberts and Tweedie’s introductory ULA display on printed p. 345 / PDF p. 6 appears to omit h in the drift. This record uses the unambiguous Durmus–Moulines Equation (2), consistent with the h-scaled drift in the older paper’s later analysis; it does not claim a published erratum was found.
- No minibatch gradient approximation, preconditioning, projection, taming or empirical accuracy bound is included in this record.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | ULA constructs a discrete Markov process intended to approximate sampling from a target. At fixed positive step size its invariant law generally differs from that target. The AIO CLASS denotes the stochastic process/model, so this is a method-to-process related match, not equivalence, subclass membership or an assertion of exact target stationarity. |
Evidence Sources & Primary Literature
2 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Alain Durmus, Éric Moulines
|
author technical report
full_text
|
ID: arxiv:1507.05021v3
sha256:5794a1e4a701…
|
discrete-bias-and-instability
|
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
Source File:
data/curated/UnadjustedLangevinAlgorithm.yaml