Transformed Samplers with Variance Reduction

1 INTRODUCTION

• Exact Poisson solutions are rarely available, and Gaussian-based approximations lose their effect as targets depart from Gaussian.Even Gaussian targets with simple test functions lack known closed forms for common samplers such as RWM and MALA.Source: §1 · p. 1
• A bijection transforms a kernel and its Poisson solutions, extending exact reference-kernel solutions to targets reached by the map.The construction uses kernels with known spectra and computable conditional expectations; the Gaussian reference with GI kernels is the paper’s focus.Source: §1 · p. 1
• A trained flow supplies control variates for any test function, with coefficients computed from reference draws rather than fitted to the chain.The approach needs no additional target evaluations and, for the random-walk sampler, no gradients at sampling time.Source: §1 · pp. 1–2
• The estimator is consistent under mild tail conditions on the map, so a poor map increases variance rather than invalidating correctness.The construction also includes importance sampling from the flow as a limiting case.Source: §1 · pp. 1–2

2.1 Poisson control variates

• When the Poisson solution is known, adding P G − G to the ergodic average removes its variance, but P G is generally intractable for Metropolis–Hastings.The method estimates the correction using proposal-based terms whose stationary means are zero for integrable G.Source: §2.1 · p. 2

2.2 Gaussian-invariant samplers

• On a Gaussian target, GI-MALA, GI-pMALA, and GI-RWM coincide, accept every proposal, and reduce to a Gaussian autoregression in standardized coordinates.The GI-RWM proposal is the preconditioned Crank–Nicolson proposal.Source: §2.2 · p. 2
• Hermite polynomials are eigenfunctions of the Gaussian reference kernel, so expanding F in them solves the Poisson equation term by term.Each degree-k term is divided by 1 − β^k, and the Gaussian proposal makes the needed conditional expectations available in closed form.Source: §2.2 · p. 2

3 RELATED WORK

• Approximate Poisson-solution methods target the sampler on the target distribution, whereas this paper solves exactly for a reference kernel and transforms the solution through the map.The alternatives include weakly approximating chains, diffusion approximations, empirical variance minimization, neural networks, and coupled chains.Source: §3 · p. 2
• Many control-variate methods fit coefficients on draws, while this paper estimates coefficients from independent reference draws without regression.The cited alternatives include Stein-based methods, control functionals, martingale control variates, and importance-sampling control variates.Source: §3 · pp. 2–3
• Earlier Gaussian-prior samplers perform well to the extent that the target resembles a chosen prior; this paper instead trains a map toward a fixed standard Gaussian.HMC on the same flow’s latent space is the paper’s main baseline.Source: §3 · p. 3

4.1 Transformation of Kernels and Poisson Solutions

• Changing variables carries a latent sampler and its Poisson solutions to the target, with the two chains representing the same chain in different coordinates.The control variate can be evaluated in latent space using the forward map, without computing its inverse.Source: §4.1 · p. 3

4.2 Reference Kernels with a Spectral Decomposition

• A reference kernel with an orthonormal eigenbasis and computable conditional expectations supplies a solvable Poisson equation and computable control variates.When the map is exact, the reference kernel matches the latent chain actually run.Source: §4.2 · pp. 3–4
• The transformed kernel retains the reference spectrum for every bijection, and its eigenfunctions are the reference eigenfunctions composed with the inverse map.For the Gaussian reference, nonlinear maps bend Hermite polynomials toward the target, allowing low-degree truncations to represent solutions that polynomials in target coordinates approximate poorly.Source: §4.2 · p. 4
• The Gaussian reference yields transformed GI samplers, including TGI-RWM, TGI-MALA, TESS, and flow-based importance sampling.Their responses to map mismatch differ: through weights for SNIS, rejections for TGI-RWM, slice shrinking for TESS, and drift for TGI-MALA.Source: §4.2 · p. 4

4.3 Control Variates and Estimator

• Control-variate coefficients can be computed from reference draws before the run, without evaluating the target density.For SNIS, the dictionary corrects both averages in its ratio; SNIS also requires weights with a finite second moment, which a lighter-tailed flow can violate.Source: §4.3 · pp. 4–5
• With the stated assumptions and a fixed estimator choice, the estimator is almost surely consistent and asymptotically normal for the supported samplers.For the chains, the stated alternatives give geometric ergodicity or, with independence steps, uniform ergodicity.Source: §4.3 · p. 5

4.4 Learning the Map

• Flow objectives depend on target structure, and the map is trained and then frozen before evaluation chains run.A one-sided tail penalty targets overly large importance weights, which can make chains stick and dominate control-variate variance; the penalty is needed only when the flow is lighter-tailed than the target.Source: §4.4 · pp. 5–6

5.1 Choosing the Estimator

• When max normalized weight times N is modest, use SNIS; otherwise choose a latent chain based on whether mismatch involves modes, tails, or many coordinates.A large Fisher divergence indicates that the flow should be improved rather than the estimator changed.Source: §5.1 · p. 6

5.2 High Dimensions: Dictionary Growth and Pruning

• Oversized Hermite dictionaries can silently increase variance because coefficient-estimation noise grows with dictionary size, despite each control variate having mean zero.Structural pruning restricts total degree and the number of coordinates; magnitude pruning screens coefficients using pilot reference draws before re-estimation.Source: §5.2 · p. 6

5.3 Choosing the Step Size: Mixing versus Estimation

• Choose the step size by balancing effective sample size against variance reduction, since optimizing mixing alone can select a step too large for the estimator.The proposed ESS × VRF objective models rejection loss and is a heuristic because that loss is only approximated; it is optimized during burn-in, then the step size is fixed.Source: §5.3 · pp. 6–7

6 EXPERIMENTS

• Where the flow is accurate, control variates reduce SNIS and TGI-RWM variance by one to two orders of magnitude.With necks or seams between modes, gains fall to a factor of a few because low-degree Hermite polynomials poorly represent the pulled-back test functions.Source: §6 · p. 7
• Across all twelve targets, SNIS+CV is effective while max normalized weight times N stays below a few thousand; beyond that, a latent chain takes over.Figure 1 compares the best chain-based estimator’s precision per second with SNIS+CV against the flow’s largest normalized importance weight.Source: §6 · p. 7, Figure 1 · p. 7
• With the independence step set to δ = 0.01, TGI-RWM+CV precision changes by a factor between 0.55 and 3.1 across all test functions.Precision per second is evaluated on each target’s main test function; Table 1 reports variance reduction factors in parentheses.Source: §6 · p. 8, Table 1 · p. 8
• TGI-RWM is rarely best but stays within a small factor of the best on every target, giving it the smallest worst case among the methods.SNIS+CV is best most often, but falls furthest behind when its weights are heaviest.Source: §6 · p. 8

7 CONCLUSION

• A bijection transforms a Markov kernel and its Poisson solutions together, carrying exact Gaussian-reference solutions to targets reached by a flow.A Hermite-polynomial dictionary serves importance sampling from the flow, latent random walk, and elliptical slice sampling without further target evaluations or target gradients.Source: §7 · p. 8
• The estimators are consistent and asymptotically normal under their respective conditions, and accurate flows yield substantially more precise control-variate estimators.The flow’s largest importance weight guides which estimator to run; training encourages bounded tail weights but does not guarantee them.Source: §7 · p. 8
• Gains depend on transport quality: necks or seams between modes yield little control-variate gain, and targets with tails as heavy as the t mixture lie beyond the theory.When mismatch is spread over many coordinates, HMC on the same flow remains the stronger choice.Source: §7 · p. 8

S1.1 Notation

• The notation defines the pushforward reference, pullback target, importance weight, and composition operator used to express expectations in latent space.The composition operator is an isometry between the stated Lp spaces, and target expectations can be written as weighted reference expectations.Source: S1.1 Notation · p. 13

S1.2 Proof of Proposition 4.1

• Under a measurable bijection, the transformed kernel is a Markov kernel, and reversibility and the Poisson solution transfer between latent and target spaces.The proof uses measurability of the map and its inverse; the transformed Metropolis–Hastings proposal preserves acceptance probabilities.Source: S1.2 Proof of Proposition 4.1 · pp. 13–14
• For affine maps, transformed Gaussian-invariant kernels recover earlier Poisson solutions, while an approximate latent solution retains the same residual norm after transformation.The affine-map examples include linear and quadratic functions; the approximation result states equal residual quality for every Gaussian target.Source: S1.2 Proof of Proposition 4.1 · p. 14

• The transformed eigenfunctions form an orthonormal basis, with the same eigenvalues and spectral gap as the reference kernel.The transformed kernel is positive semidefinite exactly when every reference eigenvalue is nonnegative.Source: S1.3 Proof of Theorem 4.2 · pp. 14–15
• Expanding the transformed function in that basis gives a Poisson solution by dividing each nonconstant coefficient by 1 − λ_j.The resulting series converges in L2 under the stated spectral-gap bound.Source: S1.3 Proof of Theorem 4.2 · p. 15
• The centered Poisson solution is unique because any difference between two centered solutions has zero coefficients in every nonconstant basis direction.Source: S1.3 Proof of Theorem 4.2 · p. 15

S1.4.1 SNIS

• The SNIS estimator with fitted control-variate coefficients is consistent and has the fixed-coefficient asymptotic limits.The variance-minimizing coefficient combination is c* = Eρ[w(F ◦ T − μ)ϕ].Source: S1.4.1 SNIS · p. 15

S1.4.2 TGI

• Under the stated drift and small-set conditions, the TGI Metropolis–Hastings kernel is V-uniformly geometrically ergodic.The drift bound and small-set property establish the ergodicity result.Source: S1.4.2 TGI · pp. 15, 17
• Under (A2) and (A3), the TGI estimator converges almost surely to π(F) and satisfies a central limit theorem.The proof requires finite moments of order 2 + ϵ for the summands under the stationary law.Source: S1.4.2 TGI · pp. 18–19
• Under (A2′), mixing an invariant TGI kernel with an independence proposal yields uniform ergodicity.The minorization argument applies for all three TGI samplers and does not require (A4) or (A5).Source: S1.4.2 TGI · p. 19
• Under (A2′), the TGI estimator has almost-sure convergence and a central limit theorem when the polynomial degree and proposal-tail parameter satisfy the stated moment condition.The condition given is ν > Kr(2 + ϵ).Source: S1.4.2 TGI · p. 20

S1.4.3 TESS

• For TESS, the chain is geometrically ergodic under (A2) and the stated tail condition, and uniformly ergodic under (A2′) with an independence-step mixture.Source: S1.4.3 TESS · p. 20
• The augmented TESS chain supports the estimator analysis, which establishes almost-sure convergence and a central limit theorem under the stated assumptions.The proof checks finite 2 + ϵ moments for the estimator terms.Source: S1.4.3 TESS · p. 20

• Expanding the estimator in the reference basis expresses the asymptotic variance through the coefficients of F and the Poisson-solution approximation.Source: S1.5 Proof of Proposition 5.1 · p. 21
• Coefficient-estimation noise contributes variance terms, and the expectation bound does not require independence among the estimated coefficients.Source: S1.5 Proof of Proposition 5.1 · p. 21
• With a budget |K| ≤ k, expected variance is minimized by retaining indices with the largest positive scores; for the Gaussian reference, vκ ≤ (2 − γ)/γ.For γ ∈ (0, 1], vκ decreases as |κ| increases.Source: S1.5 Proof of Proposition 5.1 · p. 21

S2.1 Gaussian Reference

• At K = 10 and γ = 0.3, the empirical-to-predicted VRF ratio drops from 1.9 with 1,000 chains to 1.3 with 80,000, as right-skewed single-chain squared errors make moderate-chain averages usually too small.Source: S2.1 Gaussian Reference · p. 23
• For γ = 0.3 (β = 0.7) and γ = 1 (β = 0, independent draws from ρ), empirical and predicted VRFs agree over fourteen orders of magnitude; the estimator is unbiased.The estimator’s mean deviates from the exact value by at most 2.0 standard errors for γ = 0.3 and 1.2 standard errors for γ = 1.Source: S2.1 Gaussian Reference · p. 23

S2.2 Elliptical Slice Sampling

• Elliptical slice sampling fits the spectral construction because its candidate kernel has Hermite eigenfunctions, with nonnegative eigenvalues and a largest nonconstant eigenvalue of 1/2.When w ≡ 1, the initial point on the ellipse is accepted, satisfying the stated kernel condition.Source: S2.2 Elliptical Slice Sampling · p. 24
• For elliptical slice sampling, λ2 = 1/2 gives a spectral gap of 1/2; measured eigenvalues agree within 0.04, and VRF predictions agree within 7% for K ≤ 6.The estimator is unbiased; for K ≥ 8, rare tail draws again make the average squared error across 1,000 chains underestimate variance.Source: S2.2 Elliptical Slice Sampling · p. 24

S2.3 Gamma Reference

• For positive parameters with power-law behavior near zero and exponential tails, a matched Gamma reference offers an alternative to a log transform and Gaussian reference.The reversible Cox–Ingersoll–Ross transition has generalized Laguerre eigenfunctions and eigenvalues β^|κ|.Source: S2.3 Gamma Reference · pp. 24–25

S2.4 Circle and Sphere

• On the circle, a Gaussian-angle random-walk kernel has Fourier eigenfunctions; on the sphere, zonal kernels act on spherical harmonics, though some eigenvalues can be negative.Using the spherical heat kernel instead restores nonnegative eigenvalues.Source: S2.4 Circle and Sphere · pp. 25–26
• On the tested circle and sphere targets, exact maps yield ESS and VRF matching Proposition 5.1; a sphere map with M/m = e6 ≈ 400 produces low ESS as chains stall in underweighted regions.The maps are quantile maps, and the sphere case varies the concentration κ′.Source: S2.4 Circle and Sphere · p. 26, Table S5 · p. 26

S3.1 General Procedure

• The procedure requires a reference pair, its eigenfunctions and eigenvalues, and conditional expectations, then prepares a map and selects an estimator before building the control variate.The selected estimator can be SNIS, TGI-RWM, or TESS, based on importance-sampling and pilot-chain diagnostics.Source: S3.1 General Procedure · p. 27
• The algorithm estimates basis coefficients on reference draws, prunes the candidate set, re-estimates retained coefficients on fresh draws, and constructs G using ηj = ĉj/(1 − λj).Selection and estimation use independent reference draws, so K does not depend on the coefficients it retains.Source: S3.1 General Procedure · p. 27
• Coefficient estimates depend on the map and test function, not step size, so they can be reused across step sizes and runs; latent-space computation avoids evaluating T^-1.Lines 1–3 require forward passes of T and evaluations of F, but no evaluation of π.Source: S3.1 General Procedure · p. 27

S3.2 Implementation for the Gaussian Reference

• For the Gaussian reference, dictionary pruning limits candidate Hermite indices, while recurrence-based evaluation costs O(dK + p|K|).The step-size procedure winsorizes per-chain gradients at the 10% and 90% quantiles before averaging, because a few chains at mode seams otherwise drive γ to its lower bound.Source: S3.2 Implementation for the Gaussian… · p. 28

S4 EXPERIMENTAL DETAILS

• For each target, the flow is trained once and frozen; R = 100 independent replicates each use B = 10^6 target evaluations, include burn-in in cost, and exclude flow training.For real posteriors, reference values come from long NUTS chains with second-order zero-variance control variates.Source: S4 EXPERIMENTAL DETAILS · pp. 29, 31

S4.1 Training the Map

• On the banana, the tail penalty raises VRF to 29–52 from 3–15 and reduces extreme chain deviations; on hierarchical posteriors it can degrade LARGE flows, so it is switched off there.With the penalty, scheme A reaches VRF 50 versus 31 for B, and LARGE reaches 50 versus 40 for SMALL at the same training cost.Source: S4.1 Training the Map · p. 31
• For multimodal targets, the Scheme Aa anchor keeps the Gaussian-mixture flow’s mode mass between 0.44 and 0.52, versus 0.39 to 0.69 without it, while MH-corrected final estimates remain unbiased.Starting from one mode, schemes A and B miss the other; on the ϕ4 field, 64 pilot starts roughly double TGI-RWM effective sample size per step versus 16.Source: S4 EXPERIMENTAL DETAILS · p. 30, S4.1 Training the Map · pp. 31–32
• With one tenth of the training budget, the banana map remains usable but its VRF falls from about 50 to 6, while the mixture map has ESSIS/N 0.49, VRF 1.3, and a small significant bias.Default training costs 7.8 × 10^6 target evaluations, or 8.2 × 10^6 for the mixture; the real posteriors use 5.5 × 10^6 to 1.6 × 10^7, except sparse logistic regression at 6.1 × 10^8.Source: S4.1 Training the Map · p. 32

S4.2.1 Targets

• The synthetic suite includes analytic transport families for the banana, funnel, and Gaussian mixture, plus a t mixture whose Gaussian-tailed flow has unbounded importance weights.The analytic families have exact-transport members, and their accuracy is measured by Fisher divergence; the t mixture shares the Gaussian mixture’s means and covariances but has heavier polynomial tails.Source: S4.2.1 Targets · p. 33

S4.2.2 Results

• At 10^7 target evaluations, SNIS and TGI-RWM with control variates are about fifty times more accurate than NUTS on the banana; NeuTra+ZV levels off an order of magnitude above them.When flow training is charged, flow-based methods pass NUTS at a total cost of a few times 10^5; ESSIS/N rises from 0.06 to 0.90 within the first thousand training steps.Source: S4.2.2 Results · p. 34, Figure S2 · p. 35

S4.3 Real Posteriors

• For real posteriors, the tail penalty helps SMALL flows on near-Gaussian targets but degrades LARGE hierarchical and multimodal flows, where it is disabled; stated centered parametrizations are retained.The posterior suite also includes a centered eight-schools funnel, a sparse logistic regression with 25 small funnels, and a bimodal ϕ4 field whose symmetry is not used by methods.Source: S4.3 Real Posteriors · p. 35, Targets · p. 36

Results.

• On every real posterior, NUTS with zero-variance control variates is less accurate than the best flow-based estimator and cannot estimate the ϕ4 mode weight.For light-weight targets, SNIS+CV is most efficient on nearly every function, with gains up to about 160; on heavy-weight targets gains fall to a few, and gradient-based chains or TESS+CV lead on most functions.Source: Table S10 · p. 36, Results · p. 36
• On light-weight targets, all four estimators are unbiased, but SNIS+CV has isolated estimates on sparse regression and the t mixture as max w · N grows.NeuTra+ZV spreads widely on the three multimodal targets and is biased on the t mixture, where its chains rarely move between modes; the one-compartment offset is about 10^-3 posterior standard deviations, consistent with reference-value error.Source: Results · p. 36, Figure S3 · p. 38
• TGI-RWM+CV has the smallest mean gap (about a factor of 2), spread, and worst case (4.5 on main functions and below 10 on all), making it the safe default.It is within a factor of 2 of the best on more test functions than any other method; TESS+CV is nearly as stable.Source: Results · p. 38, Table S12 · p. 39
• SNIS+CV is best most often but falls behind by about 50-fold on the t-mixture main function and over 400-fold on a sparse-regression coefficient.Its gaps are most uneven on targets with the heaviest weights.Source: Results · p. 38, Table S12 · p. 39

S4.4 The independence limit

• For TGI-RWM, J selects γ between 0.95 and 1 on accurate members, but on coarse transports it favors the independence limit and reaches only 0.44–0.87 of the best-step ESS×VRF.On the coarse funnel and mixture, a grid search gains up to a further factor of 2.3.Source: Table S13 · p. 39, S4.4 The independence limit · p. 39
• For TGI-MALA, J selects γ well below 1 except on the accurate mixture, achieving ESS×VRF within 10% of the best step except on the Laplace map.At γ = 1, R(1, γJ) is 0.01 to 0.6 because the drift misdirects large steps and acceptance drops; J finds gains at γ < 1 on coarser transports.Source: Table S13 · p. 39, S4.4 The independence limit · p. 39