Transformed Samplers with Variance Reduction

Siran Liu; Michalis Tisias; Petros Dellaportas

stat.MLOct 7, 2026

Transformed Samplers with Variance Reduction

Authors: Siran Liu, Michalis Tisias, Petros Dellaportas

Organizations: Department of Statistical Science, University College London, UK. · Google DeepMind, UK. · Department of Statistical Science, University College London, UK, and Department of Statistics, Athens University of Economics and Business, Greece.

Abstract

Markov chain Monte Carlo (MCMC) methods are the standard tool for computing expectations under complex probability distributions. Control variates reduce the variance of the resulting estimates, but a good control variate requires solving the Poisson equation of the sampler, which rarely admits a closed-form solution. Exact solutions are available when the sampler's kernel has a known spectral decomposition on a simple reference density. In our work, we extend these solutions to general targets through a learned change of variables. A bijection, such as a normalizing flow, is trained so that the target becomes close to the reference in a latent space, and we show that Markov kernels and their Poisson solutions are transformed by any bijection. Running such samplers in the latent space then yields explicit control variates, and the estimator is consistent under mild tail conditions on the map and target. Importance sampling (IS) from the flow is the limiting case of the same construction and the control variates apply to it as well. Experiments on synthetic targets and real posteriors compare the procedure against state-of-the-art samplers and control variates.

Figures & tables

Figure 1: When to sample instead of reweighting. Each point is one target of Table 1 . The ratio is the precision per second of the best chain-based estimator on the main test function divided by that of SNIS+CV, and the marker shows which chain is best; above the dotted line a chain is more efficient than SNIS+CV. The horizontal axis is the largest normalized importance weight of the flow times

N

, over the

100

replicates of

N=10^{6}

draws.

Target	SNIS	TGI-RWM	TGI-MALA	TESS	NeuTra
GARCH	9.7e7 † (29)	8.4e7 † (37)	2.2e7 (64)	2.4e7 (10)	5.0e7 (109)
One-comp.	8.9e5 (1.6)	6.7e5 (1.7)	4.6e5 (12)	4.7e5 (1.6)	5.8e5 (5.9)
Lotka–Volterra	1.0e8 (28)	2.3e7 (14)	5.6e6 (57)	8.9e6 (6.4)	2.1e7 (176)
Banana	4.1e7 (26)	3.4e7 (35)	1.1e7 (117)	5.9e6 (13)	1.7e7 (104)
Dugongs	2.0e7 (5.1)	5.4e6 (3.5)	5.1e4 (0.6)	9.2e6 (4.9)	1.3e6 (3.3)
Mixture	3.9e6 (3.3)	1.5e6 (2.3)	–	1.6e6 (2.4)	1.6e4 (1.9)

Table 1: Precision per second on the main test function of each target (Tables S8 and S9 ), with the protocol of Appendix S4 . In parentheses, the variance reduction factor of the control variates, the mean squared error without them over that with them on the same run. Targets are ordered by the largest normalized importance weight of the flow times

N

. Entries within

10\%

of the best in a row are in bold.

\dagger

: the Monte Carlo error of the reference value accounts for at least half of the mean squared error; –: RMSE above one posterior standard deviation.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Sampler	Mean	Covariance	Derivatives	Cost per step
TGI-RWM	$\beta z$	$(1-\beta^{2})I$	none	$1$
TGI-MALA	$\beta z+\gamma\nabla\log w(z)$	$(1-\beta^{2})I$	gradient	$2$
TGI-pMALA	$z+\gamma A_{z}\nabla\log\pi_{z}(z)$	$(1-\beta^{2})A_{z}$	diagonal Hessian	$2+2d$

Appendix

Table S1: The TGI samplers on the Gaussian reference. The proposal is

\mathcal{N}(\text{mean},\text{cov})

in the latent space. Cost counts one evaluation of

\log\pi

as one unit and a joint evaluation with its gradient as two.

	$K$	1	2	4	6	8	10
$\gamma=0.3$	Predicted	15.6	270	$1.28\times 10^{5}$	$1.03\times 10^{8}$	$1.30\times 10^{11}$	$2.39\times 10^{14}$
	Empirical	15.1	268	$1.29\times 10^{5}$	$1.17\times 10^{8}$	$1.77\times 10^{11}$	$4.58\times 10^{14}$
$\gamma=1$	Predicted	8.35	102	$3.35\times 10^{4}$	$2.27\times 10^{7}$	$2.63\times 10^{10}$	$4.66\times 10^{13}$
	Empirical	8.10	97.5	$3.24\times 10^{4}$	$2.35\times 10^{7}$	$3.03\times 10^{10}$	$8.13\times 10^{13}$

Appendix

Table S2: VRF of TGI-RWM on

\mathrm{LogNormal}(0,0.5^{2})

with the exact map,

F(x)=x

and

\theta=(1,-1)

. Predicted: Proposition 5.1 . Empirical: ratio of mean squared errors over

1000

stationary chains of

5000

steps. At

\gamma=1

the chain draws independently from

\rho

$k$	1	2	3	4	5	6
$\lambda_{k}=\mathbb{E}[\cos^{k}\vartheta]$	0	0.5	0	0.375	0	0.3125
Autocorrelation of $H_{k}(Z_{t})$	0.001	0.499	$-0.002$	0.368	$-0.009$	0.274
$K$	1	2	4	6	8	10
VRF, Proposition 5.1	3.58	117	$3.94\times 10^{4}$	$2.71\times 10^{7}$	$3.17\times 10^{10}$	$5.62\times 10^{13}$
VRF, simulation	3.62	109	$3.81\times 10^{4}$	$2.91\times 10^{7}$	$4.69\times 10^{10}$	$1.09\times 10^{14}$

Appendix

Table S3: Elliptical slice sampling on

\mathrm{LogNormal}(0,0.5^{2})

with the exact map,

1000

stationary chains of

5000

steps. Top: eigenvalues of the candidate kernel on

H_{k}

and the lag-one autocorrelation of

H_{k}(Z_{t})

. Bottom: VRF of

F(x)=x

with the dictionary

\{H_{1},\dots,H_{K}\}

and

\theta=(1,-1)

, from Proposition 5.1 and from the ratio of mean squared errors.

Reference	Map	$\beta$	Acceptance	ESS $/n$	VRF	ESS $\times$ VRF $/n$	Tail gap
Gaussian	moment-matched	0.5	0.78	0.11	1.9	0.20	14.4
Gamma	moment-matched	0.1	0.92	0.60	16	9.7	0.37
Gaussian	exact	0	1.00	0.97	$9.3\times 10^{3}$	$9.0\times 10^{3}$	0
Gamma	exact	0.2	1.00	0.60	$1.1\times 10^{4}$	$6.9\times 10^{3}$	0

Appendix

Table S4: Gaussian and Gamma references on

\tfrac{1}{2}\mathrm{Gamma}(2,1)+\tfrac{1}{2}\mathrm{Gamma}(5,1)

F(x)=x

K=6

, over

1000

chains of

5000

steps started from exact draws of

\pi_{z}

. For each row,

\beta\in[0,0.95]

is chosen on a grid to maximize ESS

\times

VRF. The last column is the largest value of

\log w

minus its

99.9\%

quantile on

2\times 10^{5}

target draws; for the Gaussian moment-matched map

\log w

is unbounded and this gap grows with the number of draws.

				ESS $/n$		VRF
Target	Map	Step	Acceptance	observed	predicted	observed	predicted
Circle	exact	$\tau=0.5$	1.00	0.15	0.15	1900	1810
Circle	exact	$\tau=2$	1.00	0.52	0.53	502	518
Circle	concentrations $\times 0.5$	$\tau=2$	0.80	0.31	–	5.4	–
Circle	concentrations $\times 2$	$\tau=2$	0.77	0.39	–	2.8	–
Sphere	exact	$\varphi=0.6$	1.00	0.11	0.12	783	678

Appendix

Table S5: The MH sampler proposing from

Q

on the circle (

0.7\,\mathrm{vM}(0.5,4)+0.3\,\mathrm{vM}(3.5,2)

F=\cos\theta

, Fourier degree

6

) and on the sphere (

\mathrm{vMF}

with

\kappa=3

F=x\cdot\mu

, harmonics of degree at most

4

), with

1000

chains of

5000

steps. Each map is the quantile map of a model

\pi^{\prime}

, so that

T_{\#}\rho=\pi^{\prime}

and

w=(\pi/\pi^{\prime})\circ T

: on the circle,

\pi^{\prime}

is the target mixture with both concentrations scaled; on the sphere,

\pi^{\prime}=\mathrm{vMF}(\kappa^{\prime})

. Predicted values are from Proposition 5.1 for the exact maps.

Variant	train cost	ESS ${}_{\rm IS}/N$	$\mathcal{F}$	tail $\log w$	VRF	max $\|z\|$	bias
B	$5.1\times 10^{6}$	0.997	0.0287	7.63	4.19 / 2.92	208	$-1.0\times 10^{-4}$
B + tail	$7.4\times 10^{6}$	0.993	0.0361	1	33.2 / 29.4	4	$+5.0\times 10^{-5}$
A	$5.5\times 10^{6}$	0.997	0.0171	5.46	8.95 / 14.5	96	$-9.9\times 10^{-5}$
A + tail (default)	$7.8\times 10^{6}$	0.998	0.0147	0.699	51.5 / 48.2	5	$-4.6\times 10^{-6}$
A + tail, SMALL	$7.8\times 10^{6}$	0.996	0.0245	0.648	40.5 / 39.8	9	$-6.5\times 10^{-5}$
A + tail, 1/10 cost	$7.8\times 10^{5}$	0.950	0.329	1.88	5.66 / 5.76	8	$+3.4\times 10^{-5}$

Appendix

Table S6: Training ablation, banana: every variant trained once for

10^{4}

steps, except the 1/10-cost one, a complete training of

10^{3}

steps with every schedule scaled to its length; flow diagnostics at the last checkpoint (tail

\log w

: 99.9% quantile over the radial band relative to the median over

\mathcal{N}(0,I)

) and TGI-RWM chains at

J

’s

\gamma

(two seeds of 2000 chains, 500 + 5000 steps; main test function). max

|z|

: the largest

|\hat{\mu}_{r}-\mathrm{median}\,\hat{\mu}|/(1.4826\,\mathrm{MAD})

over the chains’ control-variate estimates

\hat{\mu}_{r}

(about 4 for Gaussian spread; larger values mean that a few chains stuck). bias:

(\bar{\mu}-\mu)/\sigma_{\pi}

of the pooled estimate; ∗ : more than four standard errors

\mathrm{sd}(\hat{\mu}_{r})/\sqrt{R}

Variant	train cost	ESS ${}_{\rm IS}/N$	$\mathcal{F}$	tail $\log w$	$q(x_{1}>0)$	VRF	max $\|z\|$	bias
A	$7.8\times 10^{6}$	0.986	101	1.07	0.96	23.5 / 23.6	7	+0.871 ∗
A a without anchor	$8.2\times 10^{6}$	0.920	388	1.7	0.56	3.59 / 3.1	18	$+6.5\times 10^{-5}$
A a (default)	$8.2\times 10^{6}$	0.959	$6.3\times 10^{3}$	1.63	0.52	3.59 / 3.82	10	$-5.1\times 10^{-4}$
A a without tail	$5.9\times 10^{6}$	0.932	$5.6\times 10^{3}$	8.37	0.51	3.2 / 1.42	97	$+1.6\times 10^{-4}$
A a , 1/10 cost	$8.6\times 10^{5}$	0.489	196	4.62	0.64	1.37 / 1.26	38	$+3.8\times 10^{-3}$ ∗
B	$7.4\times 10^{6}$	0.985	85.3	1.12	0.96	9.81 / 8.52	16	+0.871 ∗

Appendix

Table S7: Training ablation, Gaussian mixture: every variant trained once for

10^{4}

steps, except the 1/10-cost one, a complete training of

10^{3}

steps with every schedule scaled to its length (pilot

8\times 10^{4}

); flow diagnostics at the last checkpoint (tail

\log w

: 99.9% quantile over the radial band relative to the median over

\mathcal{N}(0,I)

) and TGI-RWM chains at

J

’s

\gamma

(two seeds of 2000 chains, 500 + 5000 steps; main test function). max

|z|

: the largest

|\hat{\mu}_{r}-\mathrm{median}\,\hat{\mu}|/(1.4826\,\mathrm{MAD})

over the chains’ control-variate estimates

\hat{\mu}_{r}

(about 4 for Gaussian spread; larger values mean that a few chains stuck). bias:

(\bar{\mu}-\mu)/\sigma_{\pi}

of the pooled estimate; ∗ : more than four standard errors

\mathrm{sd}(\hat{\mu}_{r})/\sqrt{R}

Target	$d$	Scheme	Train. cost	$\max w{\cdot}N$	Main $F$	$L$	$\gamma$
Banana	8	$\mathrm{A}$	$7.8\times 10^{6}$	$65$	$x_{2}$	1	0.90 / 0.20
Funnel	10	$\mathrm{A}$	$7.8\times 10^{6}$	$1.6\times 10^{3}$	$v$	2	1.00 / 0.10
Gaussian mixture	10	$\mathrm{A}^{\mathrm{a}}$	$8.2\times 10^{6}$	$1.2\times 10^{3}$	$x_{1}$	2	0.95 / 0.20
$t$ mixture	10	$\mathrm{A}^{\mathrm{a}}$	$8.2\times 10^{6}$	$1.3\times 10^{5}$	$x_{1}$	2	1.00 / 0.10

Appendix

Table S8: Settings for the synthetic targets. All flows use the LARGE network,

10^{4}

training steps, batch size

256

and the tail penalty; the multi-start pilot of the mixtures costs

4\times 10^{5}

and is included in the training cost.

\max w\cdot N

is the largest normalized importance weight of the flow times

N=10^{6}

, the largest over

100

replicates.

L

is the number of leapfrog steps of NeuTra and

\gamma

the step size of TGI-RWM / TGI-MALA, both chosen by pilot runs.

Figure S1: Analytic transports of the banana, main test function

x_{2}

, from the chain experiments of Appendix S4 . From left to right: the best ESS per unit cost over the step size of TGI-RWM and TGI-MALA, and

\mathrm{ESS}_{IS}/N

, against the Fisher divergence of the shear map

T_{c}

; the ESS, the VRF and their product against the step size

\gamma

, relative to their maxima, for TGI-RWM and TGI-MALA on the accurate member

c=0.01

(dashed: the step chosen by

J

; dotted: by the speed measure); and the VRF (solid) and the number of kept Hermite terms (dotted) against the maximal degree

K

for each pruning rule, with TGI-MALA on the same member.

Figure S2: The banana with the training of the flow charged. Left: RMSE of the main test function in posterior standard deviations against the total number of target evaluations, with bootstrap $95\%$ bands; each flow-based method shows the lower envelope over training checkpoints and sampling budgets. Right: a grid of the latent space pushed forward by the flow after $100$ , $1000$ and $10^{4}$ training steps, over the target density (grey; contours hold $50$ , $90$ and $99\%$ of the mass).

Posterior	$d$	Scheme	Network	Tail	Train. cost	$\max w{\cdot}N$	Main $F$	$L$	$\gamma$
GARCH(1,1)	4	$\mathrm{A}$	SMALL	yes	$7.8\times 10^{6}$	$3$	$\beta_{1}$	1	0.90
One-compartment PK	4	$\mathrm{A}$	SMALL	yes	$7.8\times 10^{6}$	$9$	$\log K_{m}$	1	0.99
Dugongs	4	$\mathrm{B}$	SMALL	yes	$7.4\times 10^{6}$	$3.5\times 10^{2}$	$\beta$	1	0.80
Lotka–Volterra	8	$\mathrm{A}$	SMALL	yes	$7.8\times 10^{6}$	$42$	$\alpha$	1	1.00
GP Poisson	13	$\mathrm{A}$	LARGE	no	$5.5\times 10^{6}$	$7.7\times 10^{3}$	$\alpha$	2	0.99
Eight schools	10	$\mathrm{A}$	LARGE	no	$1.6\times 10^{7}$	$6.8\times 10^{3}$	$\theta_{1}$	1	0.90

Appendix

Table S9: Settings for the real posteriors: dimension, training scheme, network, tail penalty, training cost in target evaluations, importance sampling effective sample size of the flow, main test function, leapfrog steps

L

selected for NeuTra and step size

\gamma

of TGI-RWM, both chosen by pilot runs. All flows are trained for

10^{4}

steps with batch size

256

, except eight schools (

3\times 10^{4}

steps) and the sparse logistic regression (

3\times 10^{5}

steps, batch size

1024

, a LARGE network of width

256

, marked ∗ ); for the

\phi^{4}

field the multi-start pilot uses

64

NUTS chains and

4\times 10^{6}

evaluations, included in the training cost.

Figure S3: The

100

estimates of the main test function of every target, in posterior standard deviations from the reference value (dashed line). Panels are ordered by

\max w\cdot N

; circles are estimates beyond

1.5

interquartile ranges from the box.

Posterior	$F$	NUTS+ZV	SNIS+CV	TGI-RWM+CV	TESS+CV	NeuTra+ZV
GARCH	$\beta_{1}$	2.1e-3	1.9e-4	2.0e-4	3.5e-4	2.9e-4
One-compartment	$\log K_{m}$	1.3e-2	1.3e-3	1.3e-3	1.5e-3	1.3e-3
Lotka–Volterra	$\alpha$	1.6e-3	1.9e-4	3.1e-4	4.7e-4	2.7e-4
Dugongs	$\beta$	5.0e-3	5.0e-4	8.5e-4	5.8e-4	2.0e-3
GP Poisson	$\alpha$	1.5e-2	1.8e-3	1.1e-3	1.0e-3	7.5e-4
Eight schools	$\theta_{1}$	1.4e-2	1.6e-3	1.4e-3	1.2e-3	2.1e-3

Appendix

Table S10: RMSE in posterior standard deviations of the main test function at

B=10^{6}

target evaluations, without the training of the flow, over

100

estimates for the flow-based estimators and

50

for NUTS. NUTS+ZV was more accurate than plain NUTS on every posterior where both were run.

\dagger

: extrapolated from the reference run as its Monte Carlo standard error times

\sqrt{C_{\rm ref}/B}

;

\ddagger

: NUTS stays in one mode. Entries within

10\%

of the best in a row are in bold.

		SNIS		TGI-RWM		TGI-RWM δ		TGI-MALA		TESS		NeuTra
Target	$F$	plain	+CV	plain	+CV	plain	+CV	plain	+CV	plain	+CV	plain	+ZV
GARCH	$\beta_{1}$ $\star$	3.3e6	9.7e7 †	2.3e6	8.4e7 †	1.6e6	6.3e7 †	3.4e5	2.2e7	2.3e6	2.4e7	4.6e5	5.0e7
	$\mu$	3.3e6	5.3e8	2.3e6	2.6e8	1.6e6	1.8e8	3.4e5	3.6e7	2.5e6	3.6e7	4.4e5	5.7e8
	$\alpha_{0}$	3.1e6	3.2e7 †	2.2e6	3.2e7 †	1.5e6	2.6e7 †	3.5e5	1.4e7	2.1e6	1.7e7	4.7e5	3.3e7
	$\alpha_{1}$	3.4e6	4.8e8 †	2.3e6	2.4e8 †	1.6e6	1.5e8 †	3.4e5	2.8e7	2.3e6	3.2e7	4.6e5	8.1e7
One-comp.	$\log K_{m}$ $\star$	5.7e5	8.9e5	4.0e5	6.7e5	3.9e5	6.4e5	4.0e4	4.6e5	2.9e5	4.7e5	9.9e4	5.8e5

Appendix

Table S11: Precision per second on every test function, extending Table 1 ;

\star

marks the main test function. For the sparse logistic regression we give

\log\tau

w_{1}

and the median, maximum and minimum over the

25

coefficients

w_{j}

. Entries within

10\%

of the best in a row are in bold.

\dagger

: the Monte Carlo error of the reference value accounts for at least half of the mean squared error;

\ddagger

: the errors of this target are limited by a bias of about

0.01

posterior standard deviations in

\log\tau

shared by the flow-based estimators; –: RMSE above one posterior standard deviation.

Method	mean gap	factor	sd	worst row	within $2\times$	best	failed
(A) main test function, $12$ functions
SNIS	$-0.36$	2.3	0.65	$-1.74$ ( $t$ mixture, $x_{1}$ )	9/12	8/12	0
TGI-RWM	$-0.28$	1.9	0.20	$-0.65$ (Lotka–Volterra, $\alpha$ )	6/12	1/12	0
TGI-MALA	$-1.98$	96	2.49	$-6.46$ (Mixture, $x_{1}$ )	3/12	2/12	3
TESS	$-0.39$	2.4	0.31	$-1.06$ (Lotka–Volterra, $\alpha$ )	5/12	1/12	0
NeuTra	$-0.83$	6.8	0.87	$-2.40$ (Mixture, $x_{1}$ )	5/12	2/12	0

Appendix

Table S12: Stability of the estimators: the gap

\log_{10}(\text{precision}/\text{best of the row})

of each method with its control variates to the most precise one, over all test functions of Table S11 (

0

: the best;

-1

: ten times less precise). Mean gap, the corresponding factor

10^{-\text{mean}}

, standard deviation and worst row; number of rows within a factor

2

of the best and within

10\%

of it (the bold rule); failures (RMSE above one posterior standard deviation) are counted at RMSE

=1

. (A) The main test function of each target. (B) All 70 test functions; the mean and standard deviation are over the

12

per-target means, so that every target has the same weight, the worst row and the counts over all rows.

			TGI-RWM					TGI-MALA
Target	Member	$\mathcal{F}$	$\gamma_{J}$	$\gamma^{\star}$	acc. $\gamma_{J}$ / $1$	$\mathrm{R}(1,\gamma_{J})$	$\mathrm{R}(\gamma_{J},\gamma^{\star})$	$\gamma_{J}$	$\gamma^{\star}$	acc. $\gamma_{J}$ / $1$	$\mathrm{R}(1,\gamma_{J})$	$\mathrm{R}(\gamma_{J},\gamma^{\star})$
Banana	$c=0.01$	$7\times 10^{-4}$	0.95	1.00	0.99/0.99	1.01	0.99	0.57	0.50	1.00/0.99	0.60	0.99
	$c=0.3$	$0.63$	0.91	0.60	0.81/0.81	0.88	0.87	0.37	0.35	0.92/0.69	0.01	1.00
	$c=1$	$7.0$	0.48	0.45	0.60/0.55	0.62	0.82	0.14	0.07	0.82/0.28	0.12	0.40
Funnel	$\varrho=0.01$	$1.3\times 10^{-3}$	0.99	0.85	0.99/0.99	1.00	1.00	0.40	0.30	0.99/0.98	0.22	0.90
	$\varrho=0.3$	$1.2$	0.99	0.10	0.70/0.70	1.00	0.44	0.22	0.30	0.94/0.45	0.08	0.99

Appendix

Table S13: The step

\gamma_{J}

selected by

J

against

\gamma=1

and against the best step

\gamma^{\star}

on a grid of

25

values, on the analytic families: acceptance rate at

\gamma_{J}

and at

1

, and the ratios

\mathrm{R}(\gamma,\gamma^{\prime})

of ESS

\times

VRF of the main test function at

\gamma

to that at

\gamma^{\prime}

, from

500

chains of

5000

steps with common random numbers.

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Gaussian Invariant Markov Chain Monte Carlo

Michalis K. Titsias, Angelos Alexopoulos, Siran Liu, Petros Dellaportas

Google DeepMind, UK · Athens University of Economics and Business, Greece · UCL, UK

Markov Chain Monte Carlo Variance Reduction

Open paper AskPDF arXiv

We develop sampling methods, which consist of Gaussian invariant versions of random walk Metropolis (RWM), Metropolis adjusted Langevin algorithm (MALA) and second order Hessian or Manifold MALA. Unlike standard RWM and MALA, we show that Gaussian invariant sampling can lead to ergodic estimators with improved statistical efficiency. This is due to a remarkable property of Gaussian invariance that allows us to obtain exact analytical solutions to the Poisson equation for Gaussian targets. These solutions can be used to construct efficient and easy to use control variates for variance reduction of estimators under any intractable target. We demonstrate the new samplers and estimators in several examples, including high dimensional targets in latent Gaussian models where we compare against several advanced methods and obtain state-of-the-art results. We also provide theoretical results regarding geometric ergodicity, and an optimal scaling analysis that shows the dependence of the optimal acceptance rate on the Gaussianity of the target.

Abstract

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Gaussian Invariant Markov Chain Monte Carlo

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