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Langevin algorithms: ULA bias and MALA exactness — Examples

1 · Prerequisites

2 · Summary

The examples exhibit three complementary outcomes: the quadratic ULA chain has no invariant probability at its critical step size $γ = 2$ ; at the interior step size $γ = 1$ it has invariant law $N (0, 2)$ instead of the standard Gaussian target; and MALA corrects the same Gaussian proposal with the forward/reverse Hastings ratio while preserving the target exactly.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-adaptedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

At the critical quadratic ULA step size, no invariant law exists

Example

Let $U (x) = x^{2} / 2$ on $R$ and set $γ = 2$ in ULA. Then the update is $X^{'} = - X + 2 Z$ for a fresh independent $Z \sim N (0, 1)$ . This Markov kernel has no invariant probability law.

Facts & Assumptions

Given: AC; the ULA update for $U (x) = x^{2} / 2$ at $γ = 2$ .

[A1]

AC is assumed in the standard-normal construction. The Axiom of Choice Standard normal and normal laws

[F1]

For $U (x) = x^{2} / 2$ , $\nabla U (x) = x$ , so the ULA update at $γ = 2$ is $X^{'} = - X + 2 Z$ , with fresh independent $Z \sim N (0, 1)$ . Unadjusted Langevin algorithm (ULA) Standard normal and normal laws

[F2]

Affine maps and sums of independent variables transform characteristic functions by $φ_{a X + b} (t) = e^{i t b} φ_{X} (a t)$ and by multiplication. Characteristic functions under affine maps and independent sums

[F3]

Every characteristic function has $φ (0) = 1$ , is continuous, and satisfies $φ (- t) = \overline{φ (t)}$ . Characteristic function of a real random variable Basic properties of characteristic functions

[F4]

The standard normal characteristic function is $e^{- t^{2} / 2}$ . Characteristic function of a normal law

Verification

Given: The update and facts above.

1.1A1F1F2F4given

Suppose for contradiction that an invariant probability $μ$ exists and write $φ$ for its characteristic function. Invariance of the kernel in [F1], together with the affine characteristic-function identity [F2] and the standard-normal characteristic function [F4], gives for every real $t$ , $φ (t) = φ (- t) e^{- 2 t^{2}} .$

2.1F3step 1.1algebra∎

By [F3], $∣ φ (- t) ∣ = ∣ φ (t) ∣$ . For each $t \neq 0$ , the displayed identity implies $∣ φ (t) ∣ = ∣ φ (t) ∣ e^{- 2 t^{2}}$ , and $e^{- 2 t^{2}} < 1$ forces $φ (t) = 0$ . Thus $φ$ vanishes at every nonzero $t$ , contradicting continuity at zero and $φ (0) = 1$ from [F3]. No invariant probability can exist.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

ULA has a biased invariant variance for a standard Gaussian target

Example

Assume AC. Take the standard Gaussian target $π = N (0, 1)$ , potential $U (x) = x^{2} / 2$ , and ULA step size $γ = 1$ . Its invariant law is $N (0, 2)$ , so the unadjusted chain does not preserve the target variance.

Facts & Assumptions

Given: AC, the standard Gaussian target, and ULA with $γ = 1$ .

[A1]

AC is assumed in the standard-normal target and fresh ULA-noise construction. The Axiom of Choice

[F1]

Under this potential, $\nabla U (x) = x$ and the update is $X^{'} = \sqrt{2} Z$ , independent of the current state, where $Z \sim N (0, 1)$ . Unadjusted Langevin algorithm (ULA) Standard normal and normal laws

[F2]

For every $0 < γ < 2$ , the Gaussian ULA invariant law is $N (0, 2 / (2 - γ))$ , and its variance is $2 / (2 - γ)$ . The Gaussian ULA invariant law has variance $2 / (2 - γ)$

[F3]

The target $N (0, 1)$ has variance $1$ . Characteristic function of a normal law

Verification

Given: AC; the standard Gaussian target and $γ = 1$ .

1.1A1F1F2givenalgebra

Substituting $γ = 1$ in [F2] gives the invariant law $N (0, 2)$ and variance $2$ . Equivalently, [F1] shows every transition row equals $N (0, 2)$ , so this law is invariant directly.

2.1A1F3step 1.1algebra∎

By [F3], the target has variance $1$ . Since the invariant law in step 1.1 has variance $2$ , the two laws differ. The choice $γ = 1$ is interior to $0 < γ < 2$ , the regime of the invariant-law theorem.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

MALA corrects the Gaussian ULA proposal with a Hastings ratio

Example

Let $U (x) = x^{2} / 2$ and $π = N (0, 1)$ on $R$ . For every $γ > 0$ , the MALA proposal is $q_{γ} (y ∣ x) = N ((1 - γ) x, 2 γ)$ , and its logarithmic Hastings ratio is $\log \frac{p (y) q_{γ} (x ∣ y)}{p (x) q_{γ} (y ∣ x)} = \frac{γ}{4} (x^{2} - y^{2}) .$ Thus the acceptance probability is $\min {1, e^{γ (x^{2} - y^{2}) / 4}}$ , and the resulting rejected-self-transition kernel preserves $N (0, 1)$ .

Facts & Assumptions

Given: AC, the standard Gaussian target, and the MALA construction.

[A1]

AC is assumed for the Gaussian law and the general Metropolis–Hastings construction. The Axiom of Choice Metropolis-adjusted Langevin algorithm (MALA)

[F1]

The ULA Gaussian proposal has density $q_{γ} (y ∣ x) = (4 π γ)^{- 1 / 2} \exp (- (y - (1 - γ) x)^{2} / (4 γ))$ . The ULA Gaussian proposal is a probability kernel with a positive density

[F2]

The standard Gaussian target density is proportional to $e^{- x^{2} / 2}$ . Standard normal and normal laws

[F3]

The MALA accept/reject kernel uses $\min (1, p (y) q (x ∣ y) / (p (x) q (y ∣ x)))$ and places every rejected proposal at its current state; this common-density ratio is a version of the accepted-flow derivative. Metropolis-adjusted Langevin algorithm (MALA) The common-density Metropolis–Hastings ratio

[F4]

The MALA kernel is reversible with respect to its target and therefore invariant. MALA preserves its target probability exactly

Verification

Given: $U (x) = x^{2} / 2$ , $γ > 0$ , and [F1]–[F4].

1.1F1F2algebra

Write $a = 1 - γ$ . The Gaussian normalizing constants cancel, and direct expansion gives $\log \frac{q_{γ} (x ∣ y)}{q_{γ} (y ∣ x)} = \frac{(y - a x)^{2} - (x - a y)^{2}}{4 γ} = \frac{2 - γ}{4} (y^{2} - x^{2}) .$ The target log-density ratio is $\log (p (y) / p (x)) = - (y^{2} - x^{2}) / 2$ , so adding the two terms yields $γ (x^{2} - y^{2}) / 4$ .

2.1F1F3step 1.1

By [F3], the acceptance probability is $\min {1, e^{γ (x^{2} - y^{2}) / 4}}$ and rejected proposals remain at $x$ . At $γ = 1$ , the proposal is independent $N (0, 2)$ , but its reverse-to-forward density ratio is still $q_{1} (x) / q_{1} (y)$ and is included in the computed exponent.

3.1A1F1F3F4step 2.1∎

The proposal is a probability kernel with positive density by [F1], and [F3] identifies the displayed acceptance rule with the MALA specialization of Metropolis–Hastings. Applying [F4] proves exact invariance of $N (0, 1)$ for every $γ > 0$ .

Langevin algorithms: ULA bias and MALA exactness — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

At the critical quadratic ULA step size, no invariant law exists

Example

Facts & Assumptions

Verification

ULA has a biased invariant variance for a standard Gaussian target

Example

Facts & Assumptions

Verification

MALA corrects the Gaussian ULA proposal with a Hastings ratio

Example

Facts & Assumptions

Verification

Sources