Accelerated Parallel Tempering via Neural Transports
Authors: Leo Zhang
Peter Potaptchik
Jiajun He
Yuanqi Du
Arnaud Doucet
Francisco Vargas
Hai-Dang Dau
Saifuddin Syed
Venue: ICLR 2026 Poster (2026)
Keywords: MCMC
Parallel Tempering
Diffusion
Generative Models
Neural Samplers
Normalising Flows
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Markov Chain Monte Carlo (MCMC) algorithms are essential tools in computational statistics for sampling from unnormalised probability distributions, but can be fragile when targeting high-dimensional, multimodal, or complex target distributions. Parallel Tempering (PT) enhances MCMC's sample efficiency through annealing and parallel computation, propagating samples from tractable reference distributions to intractable targets via state swapping across interpolating distributions. The effectiveness of PT is limited by the often minimal overlap between adjacent distributions in challenging problems, which requires increasing the computational resources to compensate. We introduce a framework that accelerates PT by leveraging neural samplers---including normalising flows, diffusion models, and controlled diffusions---to reduce the required overlap. Our approach utilises neural samplers in parallel, circumventing the computational burden of neural samplers while preserving the asymptotic consistency of classical PT. We demonstrate theoretically and empirically on a variety of multimodal sampling problems that our method improves sample quality, reduces the computational cost compared to classical PT, and enables efficient free energy/normalising constant estimation.

Published as a conference paper at ICLR 2026
ACCELERATED PARALLEL TEMPERING VIA NEURAL TRANSPORTS
Leo Zhang * Peter Potaptchik Jiajun He Yuangi Du
Arnaud Doucet Francisco Vargas Hai-Dang Dau Saifuddin Syed
University of Oxford, University of Cambridge, Cornell University, Xaira Therapeutics, 'National University of Singapore, University of British Columbia
ABSTRACT
Markov Chain Monte Carlo (MCMC) algorithms essential tools in computa- tional statisties for sampling from unnormalised probability distributions, but can be fragile when targeting high-dimensional, multimodal, complex target distri- butions. Parallel Tempering (PT) enhances MCMC's sample efficiency through annealing and parallel computation, propagating samples from tractable reference distributions to intractable targets via state swapping across interpolating distribu- tions. The effectiveness of PT is limited by the often minimal overlap between adjacent distributions in challenging problems, which requires increasing compu- tational resources to compensate. We introduce framework that accelerates PT by leveraging neural inclluins normalising flows, diffusion models, and controlled diffusions- to reduce the required overlap. Our approach utilises neural samplers in parallel. circumventing the computational burden of neural sam- plers while preserving the asymptotic consistency of classical PT. We demonstrate theoretically and empirically on variety of multimodal sampling problems that our method improves sample quality, reduces the computational cost compared to classical PT, and enables efficient free energy/normalising constant estimation.
INTRODUCTION
Sampling from a probability distribution π(x) = ex)(-U(=))/Z defined over a state-space X with a tractable un- normmalllee density क X + R and intractable normalising constant Z = (z dr is a fundamental task in machine learning and natural sciences. Markov Chain Monte Carlo (MCMC) methods are usually employed for such purposes, constructing an ergodic Markov chain (Xt) )EN using local moves leaving π invariant. While MCMC algorithms are guaranteed to converge asymptotically, in practice, they struggle when the target is complex with multiple well-separated modes (Papamarkou et al., 2022; Hénin et al., 2022). To handle such cases, Parallel Tempering (PT) (Swendsen & Wang, 1986; Geyer, 1991; Hukushima & Nemoto, 1996) is a popular class of MCMC methods designed to improve the global mixing of locally efficient MCMC algorithms.
PT works by considering an annealing path x n .T of distributions over X' interpolating between simple reference distribution πO 17 (e.g., Gaussian) and the target n n. PT algorithms construct a Markov chain X, = (X ..... X.N) on the extended state-space X N+1 targeting the joint distribution = 7 .... & π The PT chain X, constructed by alternating between (1) a local exploration phase where the nith chain of X, is updated according to a ""-invariant MCMC algorithm; and (2) communication phase which proposes sequence of swaps between neighbouring states accepted according to a Metropolis- Hastings correction ensuring invariance (see Figure (a)). Crucially, PT offsets the additional computation burden of simulating the extended N chains through parallel computation, allowing for similar effective computational cost as a single chain when implemented in a maximally parallelised manner.
N
Typically, the chains X mix faster when closer to the reference and struggle closer towards the target. Therefore, communication between the reference and target, facilitated through swaps, can
First Authors. Corresponding to <leo. zhangestx ox..oxx.oo ww.............................. wx.ac.uk> uk> FFllowing the PT literature, we also refer components of X, chains.

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induce rapid mixing between modes of the target component (Woodard et al., 2009; Surjanovic et al., 2024). While the importance of the swapping mechanism for PT has led to a literature dedicated to optimising communication between the reference and target (Syed et al., 2021; 2022; Surjanovic et al., 2022), such works still rely on the original swapping mechanism (Geyer, 1991).
StndeedPT
APT
n
APT
standent
Nuttee of of taps
n
150
100
n
50
Local explomaion
Swap Accepted Swap
n n-1
20000
40000
60000
50000
100000
Stegs
Figure (Left) An illustration of the local exploration and communication step for PT APT. (Middle) 1,000 samples of a Gaussian mixture model target obtained using PT vs APT with a standard Gaussian reference. See Appendix 6.1 for more details. (Right) Round trips for PT and APT with N = chains over T = 100, 000 iterations of Algorithm 1.
As an alternative PT, recent work has explored both continuous (Zhang Chen, 2022; Vargas et al., 2023; Berner et al., 2024; Akhound-Sadegl et al., 2024; Vargas et al., 2024; Máté & Fleuret, 2023; Albergo & Finden- 2024; Erives et al., 2025) and discrete (Noé al., 2019; Papamakarios et al., 2021; Midgley al., 2023; Gabrié et al., 2022) flows for sampling, under the umbrella of neural samplers. However, these methods usually incur a bias, foregoing theoretical guarantees of MCMC, and can be expensive to implement and train. Due to these shortcomings, standard PT provides a strong baseline that neural samplers struggle to match (He et al., 2025b).
Recent work (Arbel et al., 2021; Albergo & Vanden Vanden-Eijnden, 2024; Phillips al., 2024; Chen et al., 2025) has explored approaches to debias neural samplers using Sequential Monte Carlo (SMC)- based ideas (Del Moral al., 2006). Despite their consistency guarantees, these approaches do not address the mode-collapsing nature of many modern neural samplers (He et al., 2025b). PT offers comparable but computationally dual framework to SMC, where the roles of parallelism and time reversed (Syed et al., 2024). In SMC, particles are generated in parallel, and approximate annealing distributions are constructed sequentially. In contrast, in PT, particles are generated sequentially and annealing distributions are built in parallel. This raises the question: just neural samplers have been integrated with SMC, can we integrate neural samplers with PT, combining the consistency of PT with the flexibility of neural samplers?
We answer positively this question by formalising and exploiting the framework introduced by Ballard & Jarzynski (2009; 2012) in physics for designing more flexible swap mechanisms, which we call Accelerated Parallel Tempering (APT), APT preserves PT's asymptotic consistency and allows us easily integrate normalising flows, diffusion models and stochastic control into existing PT implementations. Moreover, APT uses these neural samplers in a parallelised manner, mitigating their high computational burden. Empirically, APT outperforms PT by accelerating the communication between the reference and target states, even when controlling for the additional computation incurred by neural samplers.
relevant prior work integrating PT with normalising flows is Invernizzi et al. (2022), which trains a flow to directly map configurations from the highest-temperature reference the lowest-temperature target, effectively bypassing the intermediate annealing distributions. Abbott et al. (2024) also combines PT with normalising flows accelerate sampling in lattice quantum chromodynamies; their approach is equivalent to the NF-APT scheme of Section 5.1. Our framework is strictly more general, accommodating any learned or approximate transport map between neighbouring distributions.
PARALLEL TEMPERING
Let m _ ..... N be annealing path of probability distributions X' where T² r) is the reference and n = π is the target. We assume the 7-th annealing distribution admits density (x))) * (x())/n, with respect base measure dæ, where # X IR is the un-normalised
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density which evaluate, with normalising constant Zn := fxx x* (z)))))) Our goal is to estimate T[ʃ] = ((x))((x)) the expectation of X I with respect T> and the normalising constant 2 = =N..
There is considerable flexibility in choosing annealing distributions provided T" with Zo- 1 and 7- т with ZN Z. Without loss of generality, we can assume л²" = T where for 8 E [0, 11. T continuously interpolates between the reference and target as / increases from according to some annealing schedule 0 = Bo < ... < 3N = 1. A common choice is the geometric path, n (x) x -(x))1-Pr(( (x)),, which linearly interpolates between reference and target in log-space. See Masrani et al. (2021), I); Syed et al. (2021); Máté & Fleuret (2023); York (2023) for alternative non-geometric annealing paths.
2.1 NON-REVERSIBLE PARALLEL TEMPERING
The PT algorithm constructs Markov chain X, (Xp...., X on the extended state-space N invariant to the joint distribution π == T² 8... 8 &T". We construct X, from X,-1 by doing (1) local exploration step followed by (2) communication step seen in the top of Figure I (a). For n = o,.... N, the n-th local exploration move (I) updates the n-th component of X,-1 using -"-invariant Markov kernel ("x dx') ) on X' corresponding to MCMC move targeting n",
X K (X dz"),,
We additionally assume that (z, dz') = 17 (dxz) corresponds to an independent sample from the reference. The communication step (2) applies a sequence of swap moves between adjacent components of X,, where the n-th swap move illustrated in Figure I exchanges components X
and X2 in X, = (XP,.. X,") with probability a" (Xx-1, X?"), where for x, x' E X.
2 (x')
(z, x') = min
(1)
Where w" : - IR is the incremental weight equal to the un normolslldeed Radon-Nikodym derivative:
w" (x: = 7n dr (1) お" (x)
Zn dr
(2)
(x)
In practice, it is advantageous to use a non-reversible communication (Okabe et al., 2001; Syed et al., 2022), where the n-th swap move proposed only at iterations with matching parity: n = mod? 2. Both local exploration and communication steps can be done in parallel, allowing distributed implementations to leverage parallel computation to accelerate sampling (Surjanovic al., 2023).
Round trips. While the effective sample size (ESS) of samples generated by a Markov chain is the gold standard for evaluating the performance of MCMC algorithms, in setting, we mainly interested in improving the swap kernel within PT. As ESS measures the intertwined performance of the local exploration and swap kernels, instead evaluate the performance of PT and APT in terms of the communication between reference and target, in order to disentangle the influence of the swap kernels from the local exploration. This is empirically measured by counting the total number of round trips Rr which tracks the number of independent reference samples transported to the target after T iterations of PT (Katzgraber et al., 2006; Lingenheil et al.. 2009): see Appendix for formal definition. In particular, the mixing time of PT is related to the time it takes for a round trip to occur and the total number of round trips is a measure of the particle diversity generated by PT and strongly correlates with the ESS (Surjanovic et al., 2024).
3 ACCELERATED PARALLEL TEMPERING
A limitation of PT is its inflexible swap move, which only proposes directly exchangeable samples between distributions, considering just the relative change in likelihood -- and π. Low acceptance probability occurs when these distributions have minimal overlap. Addressing this requires increasing the number of parallel chains which may not always he possible. We propose Accelerated Parallel Tempering (APT), expanding the framework developed in Ballard & Jarzynski (2009; 2012), to improve distributional overlap and accelerate communication for fixed number of chains.
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Algorithm 1 Accelerated Parallel Tempering
1: Initialise X = (Xo:.... X6";
2: for / = 1, ,..., T do
3: X; = ,. ); X;" ~ (x
> Local exploration move
4: for 11= mod 2 do
Non-reversible communication
5:
X X + _ X?
Initialise forward/backward paths
6:
for -,.... . K do
7:
X Pk (X Xx-1 dr)
1,k-1
B Accelerate forward
8:
K ~ - (X - dz)
Accelerate backward
9:
end for
10:
Wkk Ket f w/ (Xt,0:)) wk (X
Work of forward/backward paths
11:
U Uniform [0, 1])
12:
log U < log wk.t - log w Kit
then
b Accelerated swap move
X X² X 1,00 t,K
X
end
end for end for Output: Return: X,,..., XT
15:
16:
3.1 FORWARD AND BACKWARD ACCELERATORS
For n ....,, N, let Ph Ph (xx-1- dzz) and Q (Ikk, KK. K transition kernels which we call forward accelerators and backward accelerators. They induce Pk- and Q two time-inhen meme--ihooogeeoouu Markov processes obtained by propagating g- forward in time and Th backward in time respectively using the forward and the backward accelerators:
K, be two families of
K
K II km
PK (dzo:k) := (dzo)
II 1=
(ZK-1, Qk (dzo::):==r) (dzzk
dxR
We assume Pn-1 and Q* are mutually absolutely continuous and we can evaluate w : x + R. the incremental weights between the forward and the backward paths, extending w" in Equation (2),
Zn dQk Zn-1 dlp
w (FO:K)
(zu:K).
(3)
3.2 NON-REVERSIBLE ACCELERATED PARALLEL TEMPERING
We construct the Markov chain X, = (Xp, ..... X.) for i = 1,..., T using the same local explo- ration and non-reversible communication as classical PT, but we use the accelerated PT swap move as described below and summarized in Algorithm 1. Given PT state X, (X?...., X E x after the local exploration move, we define the n-th accelerated swap move as follows: generate paths Xn-1- and X obtained by propagating X and X? forward and backward in time using the forward and backward transitions respectively,
X =x-',
X pa- P (X dzæ),
X X
X Qh (Xi, dx
The n-th accelerated swap move illustrated in Figure replaces components X and X3 in X, with X and X respectively with probability a'x (mn11. Xg where for zo.k, 2o.K E x
w (2b.x)
a (Zo. x6.x) = min
wik (Zo.k)
When K = 0, the accelerated swap coincides with the traditional PT swap in Equation (1). However, accelerated swap obtain an acceptance of 1 even if n- * n² provided Ip--1 Qk-
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Therefore, aim to choose the forward and backward accelerators to make the laws of simulated forward and backward paths as close to each other as possible. Theorem shows we can quantify this discrepancy through the (theoretical) rejection rate, r(Pk (x) == |PP & Q PP ||TV
which by Pinsker inequality is controlled by the symmetric KL divergence between IP and QK (for the proof, see Appendix B.1):
r(Pk K QR S 1 1 log + 200 log w SKI((P" [K) 2
L
(4)
Theorem The APT Markov chain X, generated by Algorithm 1 is ergodic and π x-invariant. Moreover, the probability the n-th accelerated swap is rejected at stationarity equals r r [kk-
Expectation and free energy estimators. As a -y-oruuutt of Algorithm I, we obtain a consistent
estimator [ + Ei= f(X{' for the expectation of f X' - IR with respect to x² by taking the can average over WK,t == w (X and Wx, == # (o),),, the weights of the forward and backward accelerated paths generated during the n-th accelerated swap at time t during the communication phase of Algorithm 1, to obtain consistent estimators Zr and Zr for Z respectively as T - oc,
Monte Carlo average over the n-th chain Xf. To also obtain free energy estimates,
N
Z
2 T
Σ
WK,
1
II
2
Σ
1
mod
ZT
T
mod
2 K.t
We then obtain consistent estimator 2n = (z. 1/2 for Z by taking the geometric mean of the forward and backward estimators. Proposition provides the formal statement of these results; for the proof, see Appendix B.1.3.
Proposition 1. The estimators R and LT a.s. converge to [f] and Z respectively as T oo. Moreover, if P pn- = QK for all n, then 2r Z.
When we consider the classical PT with K = 0, these estimators degrade back to free energy perturbation (FEP) (Zwanzig, 1954); when the path is deterministic (c.g., implemented through normalising flow), these estimators recover the target FEP (Jarzynski, 2002); and when the path is stochastic, the estimators can be viewed as a form of Jarzynski equality (Jarzynski, 1997) or escorted Jarzynski equality (Vaikuntanathan & Jarzynski, 2008). Additionally, following He et al. (2025a, Proposition 3.2), we can also apply the Bennett acceptance ratio using the weight for the paths (Bennett, 1976; Shirts et al., 2003; Hahn & Then, 2009; Minh & Chodera, 2009; Vaikuntanathan & Jarzynski, 2011) to achieve reduced variance without additional functional calls.
4 ANALYSIS OF ACCELERATED PT
We analyse how the performance, measured by the round trips Ri observed after iterations, scales with increasing parallel chains N and acceleration time K. This is equivalent to analysing the round trip rate 7 := limy- E[Rr]/T /T defined as the expected percentage of PT iterations where a round trip occurs (Katzgraber et al., 2006; Lingenheil et al., 2009). We use the slope in Figure (Right) to illustrate this quantity. To derive theoretical insights independent of the problem-specific local exploration moves, employ efficient local exploration assumption analogous Syed al. (2021; 2022); Surjanovic et al. (2024), which assumes the weight forward and backward accelerated paths is independent across chains and PT iterations.
Assumption 1 (Efficient local exploration). For all n = 1, N, (w) oo are iid in t and equal in distribution to (w² (X ), wx (Xi.x)) where (XD. .x.x) ~ Pk & Qk-
Proposition 2 relates the round trip rate to the rejections, extending Syed et al. (2021, Corollary 2); for the proof, see Appendix B.2.
Proposition If A'sssmpttion holds, then T = T T(PU- (PU-N-1
@K&) where,
r Q3) N
TPP
TPD:N-L 1...
2 + 2
1
(Pk Qk)
n=1
We that [t] represents expectation with respect the measure the left.
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Asymptotic scaling with acceleration time. We now fix N and use Proposition 2 to study how the round trip rate scales with K in Proposition 3; for the proof, see Appendix B.3. We focus on a special case where Pp- and QK arise K Ktstep discretisations of an underlying stochastic differential equation (SDE) which bridge x x--1 and x" with path measure 1 and Q as K 0c.
Proposition 3. Under appropriate conditions on the drifts of the SDE (Appendix B.3), as K + 00
T (** converges to T -1, Q1:N) and r(Ppk [Kk S r(P (%)) + (LIVVK).
The main appeal of APT comes from the fact that for well-designed pu:N Qk that aim to induce approximately the same measure, we have
r(P 0%) = < -(r=""(,,""
Thus, increasing K can dramatically reduce the rejection rate and increase the round trip rate in challenging scenarios. In practice, we find that for moderate values of K, this benefit outweights the increased computational cost as seen in Section 6.1.
Asymptotic scaling with increased parallelism. Following the analysis of PT in Syed et al. (2022), we consider how the performance of APT scales as N + co. In particular, Theorem 2 shows that N + oc, the round trip rate is controlled by the global barrier AK for APT and this can be estimated by Σm= r(P Q (this can be empirically estimated by approximating r (IPM-1 QR) by the empirical rejection rates observed from Algorithm 1); for the proof, see Appendix B.4.2. This provides a direct generalisation of the analysis of the global barrier for PT from Syed et al. (2022).
Theorem 2. Suppose pA..' and 3.9 are sufficiently regular and satisfy Assumptions 2-4 in Appendix B.4. If ma 18, 1-n-Ban -B--1l = ONN-)) as N - oc, then Σ E [kk converges to Ag and T(PRN-1 Q2 converges to Fk= = (2 + 2Ax) where Ag equals,
r(Pk
K
AK =
- (X ( 11d,,
~ P8.8 @ Q8.8
2
and uik X x + IR is the partial derivative with respect to 3' oflog w B,8' at 8 = B.
Notably, APT observes sharp deterioration in round trip when N < AR. begins to stabilise when N 2 Ax and observes a marginal improvement when N >> Ax. Intuitively, we can consider Ax to be a statistical invariant of APT which quantifies the difficulty of the sampling problem. As discussed in Proposition 3, for reasonable choice of accelerators, we expect Ag to decrease with K. This suggests APT can substantially improve compared to classical PT (K" = 0) for challenging problems by having Ag < Ao.
Schedule tuning. Another important consequence of Theorem 2 is that this allows us to import the schedule tuning algorithm from (Syed et al., 2022, Section 5) by directly following the same reasoning. This provides a simple and robust method for optimising the annealing schedule {&,, === which is one of the most important hyper-paramen parameters impacting the performance of APT. We provide further details in Appendix C.
N
Additional analysis. Finally, provide a theoretical equivalence of how APT can be viewed an instance of PT and explore the trade-off between increasing parallelism and increasing acceleration time in Appendix B.5.
5 DESIGN SPACE FOR ACCELERATED PT
In this section, we explore the design space of APT where X' = Rdd. We provide several variants of APT algorithms which utilise different neural transports in order to improve the round trip rate of APT.
5.1 NORMALISING FLOW ACCELERATED PT
Normalising flows (Tabak & Handen-Eijnden 2010; Rezende & Mohamed, 2015) provide flexible framework for approximating complex probability distributions by transforming a simple base distribution (e.g., a Gaussian) through sequence of invertible, differentiable mappings.

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Let T" . X be such diffeomorphism. We define NF-APT as APT with forward and backward accelerators defined by the deterministic transport given by the normalising flow T and its inverse respectively,
(oo, dæ1) = (do),)
(T1, = å(T") !(x1) (dzo).
Then the incremental weight equals
オ" (21) det JT (2o)); 31 (xoo).
wf ( (zo, 21)
(xo)
We can parametrise the normalising flow 7 by a neural network which is trained to minimise ((T) En=1 SKL(PTT-1 QT) following Equation (4); see Appendix D.1 for further details. We note that NE-APT is partly motivated by prior work (Arbel et al., 2021; Matthews et al.. 2022) which utilise normalising flows to map between annealing distributions in the context of SMC samplers (Del Moral et al., 2006). Interestingly, NF-APT is able to mitigate mode dropping issues associated with these works, as we are able train with the symmetric KL divergence, since NF-APT provides access to samples from both x" and x"
5.2 CONTROLLED MONTE CARLO DIFFUSIONS ACCELERATED PT
Another common transport for sampling is based on the escorted Jarzynski equality (Vaikuntanathan & &Jazyynski 2008). One way to realise this in practice with neural networks is via Controlled Monte Carlo Diffusions (CMCD) (Vargas et al., 2024).
[0, 1], have 82 E Rd, of R4, and U (1 ".) log # o log nn where の E [0, 1] monotonically increasing in s and Øb = 0r = 1. For a fixed K, let Sk. = k/K and Ask = 1/K K be the uniform discretisation of the unit interval. We define CMCD-APT as APT with k-th forward accelerator,
Suppose for
P = (xxx (ske--Ass((0)) Ask(o" (VU (xK-1) Askb SK_ (#k- 24 (o""" 12,,
and the backward accelerator,
(x,, = -ar== ASK(O") " V (Ix) Asab (xx), 2Ask(,,, (T).
The corresponding incremental weights for CMCD-APT equals
*" (*x) 11 K (xx) II PE
l Qi (1
(Fo:k)
(5)
(IX- 1, 2.)))
where Pk (xx-1, xx) and 18 -1- are the Gaussian densities of the forward and harkwards transition kernels with respect to the Lebesgue measure.
We parametrise and learn the neural transport b". the time schedule o and the diffusion coefficient of via the objective: L(be. Φz, L(e o.) > En=1 SKL(PR-) Qk). where also learn o of following Geffner & Domke (2023)², While in theory, once the vectorised field by is perfectly learned, any o can used, we found that learning o" can significantly stabilise the training and enhance performance. We note that alternative divergences and losses can also be employed- see Máté & Fleuret (2023); Richter & Berner (2023); Albergo & Hunden-Eijndee (2024). We discuss the form of the transition kernels, the incremental weights and objective in the limit as K in Appendix D.2.
5.3 DIFFUSION ACCELERATED PT
The Variance- Preserving (VP) diffusion model (Ho et al., 2020; Song et al., 2020) is defined by the (forward) SDE: dY, - Y,ds + dW,,WW, with E [0, 1]. Yo n, and rate function (Yi-s)e{(o.) has the form d.X,, I71.XX 1...22277 2-121-177 lXomm""(XX.)) + where π is the density of Y and Xo m close to a standard Gaussian the common scenario that f(] ds > 1. Samples from π can then
Variance-Preserving
7s [0.
R++ The time-reversal SDE (X,)s5(0.1)
We note that in standard CMCD, we cannot use the symmetric KI. divergence in the objective standard CMCD only has access data from side.
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Table PT versus APT with different acceleration methods, targeting 40-mode Gaussian Mixture model (GMM-10) target in 10 dimensions and standard Gaussian reference using N = 6. 10, 30 parallel chains for T 100, 000 iterations. For each method, we report the round trips (R), round trips per target evaluation, denoted as compute-normalised round trips (CN-R), the number of neural network evaluations per parallel chain every iteration (Neural Calls), and Ax estimated using = 30
chains (Ak).
# Chain
N N=6
== 10
N
30
Method
Neural Calls (1)
A3 (L)
R (t)
CN-R (1)
R (t)
CN-R (1)
R (1)
CN-R (1)
NE-APT
1
7.198
194
97.0
1655
827.5
2441
1220.5
CMCD-APT (K = 1)
6,911
234
117.0
2126
1063.0
3264
1632.0
CMCD-APT (K 2)
5.932
526
175.3
3287
1092.7
4767
1589.0
CMCD-APT (K- 5)
6
4.822
1743
290.5
5525
920.8
6231
Diff-APT (K-
2
9.025
375
187.5
1551
775.5
2820
1410.0
Diff-APT (K- =
3
7.298
748
249.3
2064
688.0
3480
1160.0
Diff-APT (K== =5)
6
5.795
1565
260.8
3080
513.3
4334
722.3
Di (K- 0)
2
8.932
204
102.0
734
367.0
1586
793.0
8.346
17
681
340.5
1888
944.0
be generated by simulating the time-reversal SDE initialised with samples from (0,, I); the score V log nYP is unknown but can be learned by score matching objectives.
To employ diffusion models for APT, we consider parametrising an energy energy-based -based model (Salimans & Ho, 2021) T which satisfies the boundary conditions: o PN(0, I) and х πτ and which we train to approximate π. We use T to define our annealing path nh by T T for some choice of schedule 0 50
< N 1. Finally, for fixed K, we define Diff-APT as APT with forward and backward accelerator respectively given by some K Ktstep integrator of the time-reversal SDE (where substitute T for ? and the forward SDE between the times Sn-1 and s,₃ for further details, see Appendix D...
To provide concrete example of Ph- and Qi- consider the Buler- Buler- Maruyama discretisation with step-size 6, (s,, - Sn-1)/K and interpolating times s sn-1 + k8 The forward and backward accelerators are given by:
Ph (IR-1, dæx) = (x==1 + 8n 0n 1 + 2V log log (10x-1)),26m1))) 7 II)
Qh-1( (IR, d -N(xx 8n Uk, 26, 21 7
and the resulting formula for the incremental weights w' is the same as Equation (5).
Regarding training, as we initially do have access samples from n, iteratively train, using the score matching objective, on approximate samples from obtained by sampling from Diff-APT. Furthermore, we stress that a parallelised implementation of Diff-APT allows for generating a sample from with similar effective cost as K discretisation steps solving the time-reversal SDE.
EXPERIMENTS
In this section, evaluate our proposed variants of APT on variety of targets. We defer further experimental details to Appendix F.
6.1 COMPARISON OF ACCELERATION METHODS
We evaluate the three variants APT introduced in Section 5: NF-APT, CMCD-APT and DifT-APT, against the baseline of classical PT using the geometric path with reference 7 = ~O I), a 40- mode Gaussian mixture model in 10 dimensions (GMM-10) (Midgley et al., 2023). Table compares the performance of these methods in terms of round trips achieved and, in order control for the additional computational cost required by APT, their
round trips; for further
details, Appendix E.3. We choose evaluate the different methods for N 6, 10, 30 chains to
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cover': (1) the smallest N where PT is able to obtain round trips (2) the regime N z A where PT is stable (A is the global barrier for PT), and (3) the regime where > A and PT is close to optimal.
We see all three APT variants substantially improve over PT with increased round trips N and K increase, and even when controlling for the additional computation required by APT. For instance, in the small N regime, (1) resulted in 10x to 100x improvement and highlights the potential of APT for challenging problems where PT struggles and parallel chains are constrained. By contrast, when N >> A and the improvements over standard PT are less dramatic, they might not offset the cost of additional neural evaluations. Notably, CMCD with K = and N = 30 outperforms the theoretical limit (2 + 2A) x 349 of round trips for classical PT (when N - oo) predicted by Theorem
6.2 SCALING WITH DIMENSIONS
To understand the theoretical performance of K-step APT for a fixed number of chains when scale the dimension d, we use the fact that the path of distributions 1e
PT
K-0 DifAAPT
K-2 DIFF
K=1 DifTAPT
K=5 DIT-APT
Round Trip Rate
CN Round Trip Rate
1.0
20
-1-1
(=T'))=eo.1] induced by a VP diffusion pro- cess initialised at GMM-d is analytically tractable, in order to run K step Diff-APT with the true diffusion path. We compare this against the same PT setup as previously. In Figure 2, we plot the resulting round round trip rate of the different algorithms when fix N = =30; for further details, see Appendix E.4. For all values of d, the round trip rate
10-2
10
10-2
10-1
0
50 70
100
25
50
100
Dimension
Dimension
Figure 2: Round trip metries for K-step Diff-APT
(K = 1, 2. 5) and Diff-PT using the true diffusion path, and PT targeting GMM-d for d = 10, 50, 100 when using 30 chains. (Left) Round trip rate against d. (Right) Compute-normalised round trip rate against d. of Diff-APT monotonically increases over PT as we increase K, with greater gains for larger values of d, demonstrating that the accelerated swap improves the communication of states over the standard PT swap. We see a substantial improvement in round trip rate (right) with acceleration compared to PT (K = and minor difference between algorithms K increases when normalised for compute (Right), suggesting the extra computation for APT is justified.
trip rate and compute-normalle
6.3 LOG-NORMALISHN CONSTANT (FREE-ENERGY) ESTIMATION
As described in Section 3.2, by-product of Algorithm is the estimation of change in free energy AF = -oog Z. We compare free energy estimates from CMCD-APT and Diff-APT against PT on two targets: DoubleWell-4 (DW-4), particle system in Cartesian coordinates; and ManyWell-32 Well-32 (MW-32), a highly multi-modal density introduced by Midgley et al. (2023). Figure presents box-plots of 30 free energy estimates for DW-4 and MW-32 using 1,000 samples each; for further details, see Appendix E.5.
K=1 CNCDDAPP
K-2 CHCD-AFT
K+5 CNLD-APT
K-2 DifLAPT
K=1 DiIAAPT
K-2 DIR-APT
K=5 D.APPT
DW4
ManyWell-32
30.25
30:25
155.0
156.0
165.5
165.5
29.75
29.75
185.0
185.0
25.50
29.50
164.5
164.5
25.25
29.25
153.5
153.5
29.00
29.00
163.0
163.0
Number of Chains
Figure Estimates of AF for DW4 and ManyWell-32 by PT, CMCD-APT (K 1,2, 5) and Diff-APT (K = 0,1, 2) using 1,000 samples. Each box consists of 30 estimates. The black dashed lines denote the reference constant AF Rs 29.660 estimated with PT using 60 chains and 100,000 samples and AF xe 164.696 from Midgley et al. (2023) for MellWel3-32.
"We refer the discussion in Section and 6.3 of Syed et al. (2024) for this classification of regimes.

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Several key observations emerge: (1) Both CMCD-APT and Diff-APT exhibit markedly lower variance and bias than PT across both targets. (2) For CMCD/Diff-PT, the variance and bias reduce steadily as K increases. (3) For both PT and CMCD/Diff- the variance and bias markedly decrease as the number of chains N increases, thereby empirically validating Proposition 1.
6.4 COMPARING APT WITH NEURAL SAMPLERS
significant advantage of APT is its asymptotic consistency-unie most neural samplers, the Metropolis correction ensures APT does not incur bias if the neural transport is poorly trained. To demonstrate this advantage, for CMCD-APT and Diff-APT on DW-4, we take the trained forward transition kernels PT- and map samples directly from the reference distribution the target; denote these respective new samplers by CMCD and Diffusion respectively and we compare the performance of these neural samplers to CMCD-APT and Diff-APT in Figure 4; for further details, see Appendix E.6. To aid with visualisation, we plot the interatomic distances of samples in DW-4
As see, directly using the learnt neural samplers dramatically drops performance, especially for small K. Out of the two, diffusion performs better than CMCD but still misallocates probability mass across modes. On the contrary, all variants of CMCD-APT and Diff-APT recover the correct mode weights and align closely with the ground truth.
K-1
K+2
K-5
Ground Truhh
A6
2.6
Derecty
0.4
Denecty
0.4
Dereity
2.4
Dereftr
0.2
0.2
0.2
CMCD
CMCD-APT
Diffusion
DIP-PPT
Figure 4: Interatomic distance dej of 5,000 samples by CMCD, CMCD-APT, Diffusion, Diff-APT with 30 chains, K 1,2 on DW4, take 100,000 samples by PT with 60 chains as ground truth.
6.5 ALANINE DIPEPTIDE
Finally, to provide realistic target which to evaluate APT, we take the Boltzmann distribution of Alanine Dipeptide in Cartesian coordinates at 300K, and we compare PT against our best performing APT variant: CMCD-APT with N 4. This a small molecule with 22 atoms (66 dimensions sample
Table PT versus CMCD-APT targeting Ala- nine Dipeptide. We use the same metrics as defined in Table 1.
Method
Åg (L)
R (t)
CN-R (t)
in total) which is highly challenging from due to many prohibitive energy barriers and high multimodality. Following the conventions of molecular dynamics, we take the reference distribution be a tempered version of the target corresponding to 1200K. We run each algorithm for 7' 50, 000 iterations and report the results in Table 2; for further details, see Appendix E.7. We can see that CMCD-APT is able to provide significant acceleration even in this difficult setting with increased round trips, as well computed-normaliss round trips, over PT.
CMCD-APT (K = 1)
3,23
465
232.5
CMCD-APT
3.15
597
199
CMCD-APT
3.09
627
104.5
PT
3.38
199
99.5
CONCLUSION
In this work, we formalise the framework of Accelerated Parallel Tempering which integrates neural transports to improve the efficiency of PT. However, several limitations, common to neural samplers, remain, including the burden of well optimising a neural network (Grenioux & Noble, 2026) which has a large impact performance-is i.e. poorly trained neural transport may result in underperformance Further, while our approach accelerates PT by increasing the round trip rate, it incurs additional neural network evaluations. Future work should therefore develop principled criteria for deciding when to rely on PT or APT for robustness. On a different note, an interesting avenue for future work developing APT for inference- time control of generative models (He et al., 2025c), following the similar application of SMC for these purposes.
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ACKNOWLEDGEMENTS
The authors are grateful to George Deligiannidis for helpful discussions. LZ and PP are supported by the EPSRC CDT Modern Statistics and Statistical Machine Learning (EP/S023151/I). JH ac- knowledges support from the University of Cambridge Harding Distinguished Postgraduate Scholars Programme. SS acknowledges support from the EPSRC CoSInES grant (EP/R034710/1) and the NSERC Postdoctoral Fellowship.
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A FURTHER DETAILS ON ROUND TRIPS
Consider the APT Markov chain X = (XP,... X constructed by Algorithm Assume there are ....., N parallel threads, each storing component of X, at time t. During the communication step, it is equivalent to swap indices rather than states. This has computational advantages a distributed implementation of PT since swapping states across machines incurs cost of (d)) compared to the O(1) cost of swapping indices. By tracking how the indices are shuffled, we can recover the PT state and also track the communication between reference and target through the dynamics of the permuted indices. See (Syed et al., 2022; Surjanovic et al., 2023, Algorithm 5) for details on implementing PT with a distributed implementation.
Summarising Syed et al. (2022), we define the index process (I,, E:) where 1 = ( 4 is sequence of permutations of [N] {0, .O,..., NN that tracks the underlying communication of states in X,, and = (c.,...,e'" v with c {-1.1} tracks the direction of the swap proposal on machine m.
We initialize 1 = m and $ = -1 if m is even and cD' = if m is odd. The subsequent values of I, are then determined by the swap moves. iteration of PT, we apply the same swaps to the components of I, that are proposed and accepted during the communication phase. As discussed in Syed et al. (2022), for non reverssere communication, 1 and ex" satisfy the recursion:
if S [I =1
1 + mL
1mTh
if S m Veritte
[-1
4-11
if TH 1 c
--11
if i I
Here S/ if the n-th swap time accepted and St 0 otherwise.
Hence, for a realization of X, with 7 steps, we define the round trips for index n E [N] as the number of times the index in I, completes the journyy from the 0-th component the N-th component and then back to the O-th component-i.c.. completes the round trip from the reference to the target and back again.
For m E [N], let T7. == inf : (41", "0,,))))))) (0, 11) and for j 2 1 define T and TV, recursively as follows:
The > - (7)) (N,1))
TJa infst > T (-" (0)-,)))
Notably, 7 represents the j-th time the index for machine m has traversed from 0 to N and back to 0, thus defining a round trip. round trip indicates that a sample on machine m from the reference propagated a reference sample to the target independent of previously visited target states on the same machine. Let RT: = max ThE < T} denote the number round trips that occur on machine m by iteration T. The overall round trips count then defined the sum of round trips over all index values: Rr= = Σm 177, Finally, the empirical round trip rate Tт = Rt /T represents the fraction of PT iterations during which round trip occurred, and our objective is to maximise the expected round trip rate as T D6:
E[RT]
lim
E[
lim
T-xx
7'
T
B THEORETICAL ANALYSIS OF ACCELERATED PT
B. PROOF OF THEOREM 1
B.1.1 PROOF OF ERGODICITY (THEOREM 1, PART 1)
Proof. At stationary, have (Xr-1 X n- n. We next simulate the paths x,ock
(X ..... Xx=') ....,,"T'] and X0OK (Xi,o.... X Xiik as described in Section 3 and calculate the acceptance probability
o (Xn-1 Ko X
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given in Section 3.2. The new states X,-!, Xy determined by
(X Xn-I) iKK (X% X
with probability a,
with probability
We would like prove that the swap operation keeps the target distribution invariant, that is, for any real-valued test function we have
x
One way to do this is to consider the extended target distribution
1,0kK @ Xfo:
P (dxxbxk) @ Qk (drgk))
and the involution
I(ro:KK y0.k) (Bock: To:K)
and apply an instance of MCMC algorithms with deterministic proposal (Tiernyy, 1998), here given by I. For completeness we give a self-contained proof. Write
I [
{7)
(XT)o X
X )a x pXxoo x(xx(( - ai)
(X x;']
pxxxxo xxx)aa+
+ p( X )a
To arrive at Equation (6) need show that
Erso 9 (X L.0:k X.,o, )a] Eseo (p ( xoak)a
(7)
where (To: yo:k) := (zo. 8K) and the notation Ereq means that the expectation taken with respect to Pk P (dxx).) @ Q% (da).k). Noting that
dQ/dP (d.)))) dQ/dP 1Q/dP(XX...)
ax (X
Xn xxxo)) = = 1,0:k)
=
we write
ErsQ
p 1b( Xn-1 X x)xx
dP Look X {(ook)
Egor
po 1/( (X
In
aQ @
dQ/dP/ (Q// (
Egop
p (X X
dQ dQ/dPp (Xn-1)
dQ/dPP A
Egor
po (X
1,0:K Xlook
dQ/dP(X xn-1
(X
EpsQ
o o vb (XTo: X
dQ/dP(XT...k
Ersq
yp th( Xn
Ko xork)a
Thus Equation (7) is established.
Finally, the accelerated-PP Markov chain aperiodic because the swaps can be rejected. The chain is clearly irreducible if each exploration kernel is irreducible. (In fact it can be proved, using more complicated arguments, that the chain is irreducible if the exploration kernel for the reference is irreducible and, for all the two distributions P and Q% mutually absolutely continuous.) By Roberts & Rosenthal (2004, Theorem 4, Fact 5), the Markov chain is ergodic and particular the law of large numbers holds.
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B.1.2 PROOF OF REJECTION RATE (THEOREM 1, PART 2)
Proof. The rejection rate at stationary is defined as
P ( = Eseq ax (X 1,0:Ks (o
(8)
where Breq shorthand for the expectation under PT- & Q and the acceptance ratio o'k is given by
0'k (X K0KKK xook) == dQR (X7(0k)
d(Qk @ PKk
=1
(Xa-1
X(oo
(9)
dQ dPP (X)..)
d(Pk @ Qx)
From Equation (8), Equation (9), and Lemma
below we have
r(Pk (R) = |P 2 Qk
QK & ItV.
1p-1
Lemma 1. Let jn and u2 be two mutually absolutely continuous probability measures. Then
1. duz
dui ]
| pellTV
Eus
min
Proof. Recall the definition of the TV distance
|1 PallTV == sup A
[E [h]
E, [h] |.
First, we remark that the absolute value in the definition can be omitted since
11 HellrV
sup max (Ey [h] 12 [hi] 18 [h]] E,,, [h)] h=,1-
sup max (Ey [M] E2 [M]] Byy [1 11] Eynz [1 1))
h: h=,')
sup Ep, [h] Ep [h).
Therefore
dji2
M1 - ~2/lrv
sup Eyy h.
djt2
1:X -0,11
ddi2
sup
E,,; h-
min
1, "up
(10)
h:x' [0.1]
Epp
1 - min
1,
dpı
On the other hand, let
{x X such that /uz//d/1(1) (x) s 1} and put h²(x)) Lx(x). Using
h²(x) = 1 min (1,
duz (x)
+ du2
(x) (x)]
dup
dp
write
Pellrv 2 Ep,
[h*]
E μz
[7'1]
d/122
Edj2 1
E
Enn
- min
1, duu
E
E,
[1
du
(11)
d/12
E
min
1,
Combining Equation (10) and Equation (11) get the desired result.
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B.1.3 PROOF OF PROPOSITION
Proof. We will show the consistency of the expectation and free energy estimator separately, By Theorem I, the PT chain X; = X²) is ergodic with stationary distribution n. @ ... @ 8x" N By the ergodic theorem, we have
T Σ T
# fl]
f(x))
x (f]
T-000
will now show the consistency of the free energy estimator. By definition of W in Equation (3), we can write AF, in terms of expectations of W and the Radon-Nikodym derivative between Px and QK respectively:
Zn d
Zn dPk -1 Zn d2
K
Zn-1 dlPk
w
By taking expectations of the left and right expressions with respect to 1Px and QK respectively, and taking the product over n I,...,N, N, we obtain the following expressions for Z and Z-1 respectively:
N
N Qk [ww))
Z
I1 P P
[w]),
z- II
(12)
Since X, ergodic, Zr and Zr are consistent estimators for respectively. Therefore 2g ZT) is consistent as well. Finally, note that if [pa-1 = Qk then (xo.k) a-5. Zn/Un-1- 1+ and hence Zr Zr a.s. Z.
B.2 PROOF OF PROPOSITION 2
Let ST be the indicator random variable with S," if the 7ith swap is proposed and accepted at iteration t. and ST = 0 otherwise. By Assumption the random variables Sy are i.i.d. in t with
S2....... ..... = Bernoulli(s,), where s, := 1 - r(Pn-1_ Q"). This implies that the index processes ST.Sy
(ITh, s(") e²") form Markov chains on (0.... N] x { 1, 1] with initial conditions I m and e/" when is even, +0 = 1 when m is odd. For 2 1, each process satisfies the following recursion:
1
with probability 1 te? with probability ST7 (ITL
I
(13)
1
1
if I 1 + if T=m== = 1
(14)
The remainder of the proof follows identically to (Syed et al., 2022, Corollary 2).
B.3 PROOF OF PROPOSITION
We first spell out the full statement Proposition
Proposition. Consider two SDEs
x--1(0) n-1 dx dx- f(u, (u)dd odB, for E [o, 1]
(15)
x²(1)~ ~(11)~ dx" blu, X"))) adB', for [0. 1]:
(16)
where the second equation is integrated backwards time. Let P (dxxx) and Pk (dxx)
be respectively the full and the Euler-discretised path measures of Equation (15), in particular Iiso = (u))
E C(O, 1] and Iœ = [0x(0), =1)KK) [1]] ] E KK+ Similarly let (dr and @ (dax) be respectively the full and the Huler-discretised path measures of
Equation (16). Assume that f(u,-) and f(,,x) are Lipschitz for all and with global con-
stant; and that the same holds for b. Moreover, suppose that there exists a G > 0 such that If, x)]l x)( < G(L ||= Then
lim r(P (%) r(P 1.0%) K+000
(17)
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and
r(Pk [
S sr(P
, 2%0 + O
(18)
K
where r(p. 9) = l1p x - 9 x PlITV.
Proof.
Let pa- and 2n/KK be the restrictions of the path measures 1P--1 and T to the time oolK
discretisation points. (As such, Pa-k and 1o0 defined on the space TP--1 and Qk-)
We have
1r(Pk QK)
r rrPP
ool Q (
< |Px Pork + IIQ 2% (K
(19)
S <0(/VK)
where the first incquality is clementary and the second follows from Proposition 4 below. By data processing inequality
rPPaejk <eKK> (((jk) & r(PP -
(20)
Equation (19) and Equation (20) establish Equation (18). On the other hand, putting
d(OR PR
Rx (#k, yk)
(Tk. ykk
d(Pk x Q
R (To Moo) = d(D2 x PPo-))
(Ioo : yoo);
d(p% x Q%
((a) == m1,)
(21)
we have by Lemma
r(Pk Qk) Epg x |p o Rx].
(22)
Since (pa) E [0, 1] the definition of the total variation distance implies
Bpa- x Q& |p Rk]
Epa xQ2
1p Rkl s ((L/VK).
(23)
In addition define the discretisation operator
Dx C(0,1 1] (R2)) K+1
Foo + Took = (0.)))))))
which takes continuous path and extract its values at K + discretisation points. Let Dx (Zoo, Moo) = (Dx( (B)o)). Then
Epn-1 x x xx
10 o Rx] = -Ep-- x0g 1pp o Rx D&c
(24)
K-+0 Epg- x ap Rao] = r(P Q**
by the dominated convergence theorem, taking into account the pathwise convergence of the Euler- Maruyama schemes (Kloeden & Neuenkirch (2007); Gyöngy (1998), see also Berner et al. (2025, Lemma B.). More precisely, the function yp defined in Equation (21) is continuous and bounded between 0 and 1, and
dQt dlp- - K
dlp--1
R o Dx( Voo)
(D (oc)
d (D (Dxyoo)=
T" (= (1))
N (2
xoo
( K
b
K
x
(x (0))
N (2
*
(yoo (0)) K II N10
K
n" (yoo (1)
Am
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When K oo, the term in the first big square bracket converges to dQk/ / (#oo) and that in the second converges to d /dQK (Vsc), establishing Equation (24).
Finally, Equation (22), Equation (23), and Equation (24) entail Equation (17), finishing the proof. D Proposition Let (X))) be the solution of an SDE of the form
dX, b(l, X,)dt + adB.,
(25)
where we assume that b(t,-) is L-Lipschitz continuous for all t and that b(-,z) is B-Lipschitz continuous for all x. We also assume linear growth condition of the form [[b(t,z)] S G(1 ||=|]).
Consider the K -tep Euler- Maaler-Mruyymma discretization N Nxy...., Xtx where 4g k/ K. Let Poc. be the path-measure of the process (Xt)telo.1) Pk be the law of Xtas ................................ Xtg. and IP be the law of ..o...., Nex- Then
1 VR
PP PKIIIV O
Proof. Consider the continuous-time extension (X of the K-step Euler-Maruyama discretiza- tion of the SDE in Equation (25) given by
dX, b(tx, X,, dt odB,, t [Ex, t&+1).
We use P.x- = p(()) to denote its path-measure. By Pinsker's inequality
|P 1I S (
By the data-pnn processoocsssggg inequality and Girsanov's theorem, we have that
El|((t, b(ta, 111) By the Lipschitz continutity of b, we have the following bound
KL(P [Fx) S KL [Px)
20
El|b(t, Xi) b(tz. xx, 11 < 2E (tt, b(t, xx, )12 + b(t ) - b(tx, x12
< x Xx, 11 + 2B² 1x)
By the Cauchy-Schward inequality, the fact that B, Bi, ~N(00, (t La) and the linear growth assumption, we obtain
X1I - -x = 111"" X.)ds oo b(s,,
x1a11==0
Bia))
$ 2(t th) [ E El(b(s, X.)1 ds 2do (t tt))
<2G (t tk)
E(1 + |xx 11) ds + 2do - tx)
14
f (1+E|X,1 13 + 2t (tttt) tx)
$ 4G tx)
It is well known that under the assumptions in the proposition, there exists a constant M (independent of K) such that 2 < M for
1. Putting of this together, we have shown that
Et]b(t, X₂) b(tx, X2, < SG" (2 + M)(t tx 4do² L tx) + 2B² 1x
Summing over l and integrating, this yields
K-
El|b(,, Xx) b(l2, Xx,) 1ltt
20
k=0
K-1
1
2
1
M))(k+ 41)3 + 2do 12
4x) B² 31(41+1 (x)3
20
3
4
1
1
B 1
<
0
1
M
+
d
+
3
K2
K
302
K2.
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Therefore
PR - PxllTV O(IVK).
We verify this result in practice by plotting the TV distance between Px and Poo for the Orn- stein-Uhlenbeck process in dimension.
confidence interval
Reject rate proxy
2-7
23
24
25
26
27
28
2
2
K
B.4 FURTHER DETAILS ON SCALING WITH PARALLEL CHAINS
B.4.1 REGULARITY ASSUMPTIONS
Suppose T (x) exp(-UUP (xp(-U*(=))/Z where ZB Jx (xp(UU(())) For 3 (3,8") suppose p k (dro.)) and Qfx (dro.k) are mutually absolutely continuous path measures on x with marginals 1o (x0) = 7 (dzo). and (dæk) = x (dæk) with weight functional w x +1
R.
Zz dQ
@ (xo.k)
Z, d (ro.K),
We will make the following regularity assumptions on p9 and @ analogous to Assumptions 5-7 in Syed et al. (2024). Suppose there exists = (F.)ee a nested sequence Fo C Fi C... vector spaces of measurable function K R. such that (1) F, contains the constant functions, and (2) F; closed under domination, i.e., if g is a measurable function such that [g] I [f] for some F,, then F,,
Assumption 2. For i 0, 2, and 9 + [ and B - Q are i-times continuously differen- tiable for all f E F,, there exists signed-measures 0zpS and 04 integrable over F; such that for all f F,,
0yp (' (f)). =
Assumption 3. BB W is 2-times continuously differentiable, fnd i = 0, 1, and .1.....44 21 and i + ..... + in = i, we have W., ... Wi, E F3-i. where W; = =ussup 03
Assumption 4. For all B we have ps = 1 3 9 Lemma 2. Let g : xK+1 XK+1 R,
B dd (zo:k, Ho.k) is continuously differentiable,
.o.K + (20: sbx) and s/Ox d (No:K vo.k are in Fi.
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Tu:K + Ogg ( ko.k and zo.x and 0ggº (zu:k, zo.k) ) are in Fo-
Then ~ pa Bp (P3 Q () 0gP0 2 p & 0g [g] p 10
Q [ is continuously differentiable with derivative.
Jo:k and x' To.k- Given 2 let Le ; + R be the marginal ofd (x') = g(r, and let qr x - R denote the expectation
Proof. For notational convenience denote
of fh with respect to Q i.e. qºd (a) Q By product rule for measure-valued derivatives,
[o
have for all x, B + (xx) is continuously differentiable with derivative,
Ogg (x) 0p((29
by taking expectation of both sides with respect to pa and using Fubini's theorem have 1 p & Q 1 pd & [0
Again, using the product rule for measure-valued derivatives,
(p (g 01
= 03p²(1 [g + (0
The result follows by noting that 0g dgps @ (0 by Fubini's theorem.
B.4.2 PROOF THEOREM
For B = (3, 3¹) recall the rejection rate can expressed as r(IP@, Q)9) = pp @ [a where +1 is the product between the forward and backwards paths, pe @ Q is measure over x x X x
p s0 @ (droo d=f.x) == p& (dzzo: x )d2 (d='o:))
a. xK+ x (0, 1] the acceptance probability for swap proposed,
" ( -xp (min { A. (Iu.K, mbxx)) ( (=(ox)
a (Zo:k 2o.k) min {
where AWB XK+1 x - R is the change in log weight W (Zo:k) ) = log w (2o:k)
AWB (Zo.K ToKK) W (xu.k) W (1b:k)
Lemma 3. Suppose Assumptions 2-4 hold. For all B = (3,3") with AB = 3² 38-32- ->>0 0, exists a
constant C >> 0 independent of B such that.
rTp Q0) - ff
8
Andb < CAB..
where := 1 & Q"AANT and AW3 = 1i 0z AW8. Moreover, Ag is continuously differentiable in 3.
Given annealing schedule 0 So
NN 1, let B,, (8 Bn)) and AB, -nn Bx-1-
follows from Lemma 3 that the sum of the rejection satisfies,
N
Σ
S
C
Σ A < C max
ABal-
n
n=1
Therefore as N and m A we have the sum of the rejection rates converge to A.
For the same annealing schedule define 7 (Bu:N) as
N
r (13., B. rPPB.. C.B..)
-1
7 (8o.N) = 2 2 Σ
n=l
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By Lemma 3 we have,
r(p@h, )..))
r(P (+8.)
r (P Cyan.
r(BB., 1B.)) 1 r(PA,, (ua))
g r(P8., (P.))
sup, Ao max,5N N A
Therefore, by taking the sum over n and using the squeeze theorem, we have in the limit N o0 and maxpgN |A,,, 0,
(PA. Qoo.) A
lim
1=1 r(PP., Q..) 2A)-1
and hence lim N- N. (3D:N)
-1, which completes the proof.
Proof of Lemma 3. Since B3 + W is twice differentiable, we have that 3 + a is twice differen- tiable when AW3 + O, with first-order partial derivatives with respect to S":
a 03 AW (xp(AW")) 0),
and the second-order partial derivative with respect 3':
0z 10 + (xpA") <0..
By Taylor's theorem, for AB >> O, we have
8 = 1 à AB
where d lim a and | sup 10 103.0. a² As2.. Therefore, the rejection rate equals:
r(pa, (+) -p & 16 A p & @
(26)
We will first approximate the first term Equation (26). Note that since AW = and 0g AWS lim - AW / =: AWA, we have that á satisfies:
AW3
8º lim Oy AW A exp(AW
<
3-33
A3
AW IAWi < 0j
-la A <).
We bound 8 uniformly in B in terms of Wi:
j (Fo.K xo.K)
S I (26.k)
- (F): + (1
= Wi @ W (xo: x0.x).
It follows from Lemma 2 that 8 + IP8 @ [a is differentiable in B". By Lemma 2 and the mean value theorem, there exists B = (", with 3 < B B¹ such that 8
pg
8s @ [_[ 0g (P&
3
0 p
3- + p @ 2r 1100
Using the triangle inequality and |d
< W1 + Wi, we have:
p s
Q
(
p
< sup |
Wi + Wi]
A3
8
+ + pe
[ @ Wi]
sup (10
W + |
+ sup ( 110 [ + p8 0
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Since and W are in F1, have that each of the terms the right- side is continuous and hence by the extreme value theorem, there exists C1 such that
p
p @ 16 S C
For the second term in Equation (26), we have
A
pa 80
p
Q
sup
2
8
p
Q
12 W2 (W1 @w)
2
A
sup
p8
Q W W2 (W1 e w))
= C2A
where W2 W2(20. = = W2(xo:k) + W2(1 Assumption guaranteess that the expec-
tation in the second-to-lase line is continuous in B, and hence C2 finite. Next, we note that since A W/ (xo. xxxx)] = A (zb.k: 2o:k)| is symmetric and p8 @9, we have:
pa 69 (a
p 60 1
AW 1 AW OJ]
p8 @ [AW
A
By Lemma 2, B - Ae is continuously differentiable. Since A3 is right Riemann sum for the integral of λø with error:
AoAB - fo A db
1
ddg
A
sup Ø
2
d3
Finally, by the triangle incquality:
8¹
rTP. (TP.8QQ))
d db SCAR
for C = Ci C2 C3.
B.5 ACCELERATED PT VANILLA PT EXTENDED SPACE
In this section we establish the theoretical relationship between accelerated PT and vanilla PT. We state an equivalence between accelerated PT and a particular vanilla PT problem which "linearises" extended space,
N More concretely, we define sequence of distributions T2 Tex Tex supported such that we run vanilla PT on it, we obtain the same round trip rate as if we ran accelerated PT on T ...... ,T
We consider the case K and simplify the notations of forward and backward kernels to pm p"(x--1, dz") and Q="1(x" (=",,=="1) This does not incur any loss of generality since conceptually multiple Markov steps can be collapsed into one.
Given sequence of distributions T ..... . each supported on X, define the distributions T
Tee (dæ Nx = n"" (dxx")
II 12n+1
(i dx') II
(27)
jEn-1
In particular we stress that the distributions xh are supported on N+1+ and not X.
The following proposition establishes the isometry between accelerated PT on ......, and vanilla pr PT on ! "......, Recall that we use r(u,, P2) to denote the rejection between two distributions [n and /22
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Proposition 5. For all S N. we have r(a"=1 x Pm, P=,Qa=1 1 x²)
=(=xx) -Tx)
Proof. Since the rejection rate only depends on the Radon-Nikodym derivative, it suffices to verify that
d
d(==11 x pm)
(x
d(Qn-1 x x (x"-1,, ,,
dr
which is straightforward from Equation (27).
This proposition shows that Accelerated PT outperforms traditional PT in two ways:
First, while traditional PT bridges n² and n",, accelerated PT bridges Tex and xं which can be much closer to each other if the forward and backward kernels good;
In addition, accelerated PT inserts N distributions between Tox and m
B.5.1
PARALLELISM VERSUS ACCELERATION TIME
Given two distributions -ー and π", should we apply K K-step forward and backward kernels; or insert K intermediate distributions (,==1, )== k= kernels instead?
and use only one-step forward and backward
By Proposition 2, the inverse of the local round trip rate between n- and x² for the first method (fime-accelerated) is
r (IP Q
TA
+
1 - r(P% 1%)
The inverse of the local round trip rate between n n-1 and x" for the second method (parallel- accelerated)
T( x pn-1 2h x nt" 11
TP = 2
k
1 x Ph pa-1 Q T-
where we make the convention 7-100 1,0 -ー and n 1,K n²
The following proposition analyses these local rates for both methods. As in Appendix B.5, the main idea to find vanilla PT equivalent for both algorithms. We define
K
r(uk-1.14k)
Tix (No, ,K) 2+
the inverse round trip rate of a vanilla PT algorithm sequence of distributions :o....; Proposition 6. Define the sequence of distributions ($x) K
k as
Se(dzo, dei,...,dx = n 12 (dxxk) x
II
pn- P.- (z dr;) II (7
12k+1
jSk-1
Then the following equalities hold
TIA = TyA (So. Sk)
(28)
Tp Tya (So, .
(29)
Proof. The first point is straightforward. To show the second point, we need to check that
pr Qh- x -ー = r($)-1,54).
Note that the rejection rates only depend on the Radon-Nikodym derivatives, so it suffices to verify that
dS
d(Q X -T
(o I1,
FK)
d( Pm-1) (#k-1,
which is straightforward from the definition of (Sx k=
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This proposition shows that it is preferable to the acceleeeee method, the quantity in Equation (29) is generally greater than that of Equation (28) thanks to the effect of the bridge between So and Sx. However, in practice the acmelaccelerated method consumes less memory and so might be more suitable in certain circumstances.
C SCHEDULE TUNING
Under Assumption 1 (efficient local exploration), by Proposition 2, we wish optimise schedule 0 = 90 < < NN = to minimise
Σ (PK- QR
r(Pk Q3)
in order to maximise the round trip rate of APT. Additionally, by Theorem 2. we have -n=1 rPPk1 K Qk) R Ag. Therefore, reasonable proxy objective is to minimise 2n Tn (n)) under the constraint that En=1 In Aς and nn > 0. Indeed this same reasoning employed in (Syed et al., 2022, Section 5) to derive their schedule tuning algorithm.
Thus, following Syed al. (2022), the solution to the proxy objective to ensure the Yn are constant in n. This provides natural objective to choose our schedule in such way which results in the theoretical rejection rates r(Pk (kk being uniform for all n. Moreover, while we do not have access to rPPk Q3). we can estimate these quantities empirically through computing the average rejection rates f observed in rumning Algorithm 1. Specifically, during sampling, keep track of the average 0 of the computed acceptance probabilities oKK for n I....,N.. N. The estimated are then given by P = a
rejection rates f ~ r(Pk @
Finally, achieve above goal, 2022, Section 5). For completeness, we reproduce this in Algorithm 2.
the same schedule tuning algorithm from (Syed et al.,
Algorithm 2 Schedule Tuning
Input: Initial schedule 0 = Bu .... < BN 1, number of tuning steps m, number of sampling steps Tume:
1: for i 11=1,.. I....,, do
2: Run Algorithm for Time steps and compute the average rejection rates {fR J
3: Set (n( = --ÿ=1 f for n 1, I,...,NN N and Åx (0) O.
4: S + [(B,,, (x( )
Ax(a)
5: Fit monotonically increasing interpolation Ax (-) (e.g. spline) of the points in S which satisfies the boundary conditions: Ax(0), = and Ax 1.
6:
Bn + -Ax (N) where Ax (-) denotes the inverse of (x(.).
7: end for
Output: ( 0= B1 .... < BN= =
D FURTHER DETAILS ON DESIGN-SPACE FOR ACCELERATED PT
D.1 FURTHER DETAILS ON FLOW APT
D.1.1 WORK FORMULA FOR DETERMINISTIC FLOWS
Proposition 7. Let X pd and suppose that -- and 'h" admit strictly positive densities 1 and 7n with respect to the Lebesgue measure. Let T .dd - [Rd be a diffeomorphism with Jacobian matrix Jr, (I). we choose the one-step forward and backward kernels pn-1 and Q" such that
p nx dxx")= "0d""")x" = - 1y(dx").
(2" dd==1) d(TT)) 2 (wn)(dx"-1)
then, for all (x--1 2n) such that TT (2"=11 z", we have the following expression for the weight defined Equation (3):
n" (2")
w (2" ",--11=)-=-
(==-12²")= 7n 1n-1( det JTm (2"
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and the acceptance rate o" defined in Section 3.2 becomes
A (x (x"-1) "="
det( (x--1)))
(= x 1==== x..) A
(30)
nn (x (x") det (Jr (2+
Proof. Let S == [( 2 E Idd x Rd such that (===-1 = 2"). Recall the definition of the
extended measures p--1 and :
pp- (d="=) -2z)) n" -=- (d==" pm paappp----0 (===1 (="__,d=" ,d="), Q Q"(d=--1 dz²) ="(d²") (2" dz=-1).
Moreover for (zn-1, x²))= ) S.
pa- (dz"
(dz" (d===1_a ,d=") (T==""))(==)000 Q"" d=",d===!))
Therefore
n 1- 2²" = (T #T (="==)(d="""
n" (d2")
dP
n" 7"-1((1m) ((()) det (Jt(( (z")) x²(2") det JI (1-((=111)
nT" (2
(31)
T" 1 (2n-1)
which justifies the identity for the weight. Applying this at (z--1, , )- = ",=-,,"")) E gives
dQ
=²")) T² (x"") (Jt(JTr (I"
(32)
d
T (2--11
Similarly, applying Equation (31) at (,m-1,²")) ) = (a--1, ,,")) gives
dQ dp=-1 (x 1 xz=)== ("(x)) det (Jr n" (2+
(33)
Together Equation (32) and Equation (33) establish the proposition.
D.1.2 TRAINING
Since APT provides approximate samples from both the target and reference densities for each flow, it enables a range of training objectives. Some choices include maximum likelihood estimation (MLE) (equivalent to forward KL), reverse KL, and symmetric KL (SKL), which averages the two. Each has trade-offs: forward KL promotes mode-covering, reverse KL is more mode-seeking, and SKL balances both. APT's parallel structure makes SKL particularly effective by providing access to samples at each intermediate annealing distribution, feature many other methods lack. For example, sequential Monte Carlo (SMC)-based approaches such as FAB (Midgley al., 2023) and CRAFT (Matthews et al., 2022) rely on samples from only one side, limiting their choice of loss functions.
Additionally, one can explore loss functions based on APT's rejection rates, such the analytic round trip rate, see Proposition 2. Since higher round trip rates indicate more efficient mixing, optimizing for this metric improves sampling performance. Empirically, we found that using SKL yielded the most stable and robust results across different settings. This loss was therefore used in our final experiments. However, we leave it to future work to study other possible losses in more detail.
There are also possible variants to the training pipeline. In each case, we initialize normalizing flows to the identity transformation. One option is run the APT algorithm with the current flows. After every (or several) steps of the APT algorithm, the current samples can be used to update the flow parameters. Since the flows are initialized at the identity transformation, the initial sampling of APT behaves similarly to PT. A second possible training pipeline is to instead use PT directly generate a large batch of samples, and then use this batch of samples to update the flows. The latter approach is more stable and is thus what employed in the final experiments. We provide more details in Appendix E. We note that the first training pipeline has the potential to allow for better exploration, and we therefore leave more detailed exploration of it to future work.
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D.2 FURTHER DETAILS ON CONTROL ACCELERATED PT
At the limit K - x., following (Berner et al., 2025, Lemma B.7), and by the controlled Crooks fluc- tuation theorem (Vaikuntanathan & Jarzynski, 2008; Vargas et al., 2024), we arrive the generalised work functional
W (a) = log woo (2)
v VUZ bb(())) (z,
inducing the corresponding continuous processes (Xx)aejo.1)) (X)) aEjo1) for p Q% respectively, that is for the forward process
Xo x n-1
dx, - -[ V
and for the backwards process
X ~ = (x) V2dB..
The training objective is
((be, = Σ SKL(PP-) QK)
SKL(P-- (%))
K-+00
Note that the discrete version discussed in the main text the only discretisation choice. Other options (Albergo & Vanden-Eijnden, 2024; Máté & Fleuret, 2023) may also be applied.
D.3
FURTHER DETAILS ON DIFF-APT
Annealing path We use the following Variance-Preserne (VP) diffusion process
dY, 7s Y,ds d2,,dW,, YO Ym~m
with the choice schedule 7.s 2(1-s) to define the path distributions )eE())))))) by Y, ~ -i-s'
Due to the singularity at 1, YP is not defined by the path, but standard Gaussian as the path converges to this distribution in the limit as s approaches in order to define the full annealing path (= )=(o,.1)
define this point to be
Accelerators For a given annealing schedule 0 = 8g <.. < = 1, we construct Pm Q through the linear discretisation of the time-reversal SDE on the time interval [8n-1, 8n] with step size 6n (sn, s,-1)/K K and interpolating times s 8p-1 kd
In particular, take Q (Ix, dr to be NV I(VI- an V an,k 1I) where an
VP to x - Furthermore, we exp(-2 5 1-=t-= 7sds) which is the closed-form kernel transporting क of take ph (Ix-1, to the exponential integrator given by WV (Ix 1), an.k-1l) where
P VI an,k + 2(1 VI an,k (2 (x= V log VP (x))
Network parametrisation We parametrise an energy-based model as outlined in Phillips et al. (2024) but modified ensure that =x(x)) & N I). For completeness, we specifically take log T. (x) = log gxr) ă |-] where
log go s) (ra(1) re( re(0)) s),z)
+(1 ro ro(s)) log (v/s),
where (2x) 91 (x) 0, I). Here, is scalar-valued neural network and N is vector-valued function R.. We also note that n (z) x n, hence Th serves as a valid annealing path between ((, I) and n.
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E EXPERIMENTAL DETAILS
E.1 TARGET DISTRIBUTIONS
GMM-d We take the 40-mode Gaussian mixture model (GMM-2) dimensions from Midgley al. (2023) where extend this distribution higher dimensions d, we extend the means with zero padding to a vector in I2 and keep the covariances as the identity matrix but now within R4, This helps to disentangle the effect of multi-modality from the effect of dimensionality we essentially fix the structure of the modes across different values of d. Following previous work (Akhound- -adegh al., 2024; Phillips al., 2024), also scale the distribution GMM-d by factor of 40 to ensure the modes are contained within the range [-1, 1] for our experiments with APT, where we also use the same scaling for the PT baseline to ensure a fair comparison.
performance as
DW-4 We take the DW-4 target from Köhler (2020) which describes the energy landscape for toy system of particles (I1, 23, v4) and 2; E R2 given by
(σx) exp
Σ
a(dij do) b(d,j do)² c(d. dg)² )
2T
$fj
where dej = |24 - أو and we set = 0, b = -4, -, c= 0.9, T accordance with previous work.
MW-32 We take the Many Well-32 target from Midgley et al. (2023) formed from concatenating 16 copies of the 2-dimensional distribution
32) x exp (=x Gx7 2 2
i.e. the distribution (2x) I (322-11, where R Each copy of has modes and hence contains in total 26 modes.
Alanine Dipeptide This is small molecule with 22 atoms, each of which has 3 dimensions. The target energy is defined with the amber14 //144/4/pttin.m ff14SB forcefield in vacuum using the DMFF library in JAX (Wang al., 2023).
E.2 NETWORK AND TRAINING DETAILS
NF-APT For GMM-d, we use 20 RealNVP layers where we employ 2-layer MLP with 128 hidden units for the scale and translation functions (Dinh et al., 2017). We initialize the flow at the identity transformation. We use the Adam optimizer with learning rate of 1e-3, perform gradient clipping with norm 1 and employ EMA with decay parameter 0.99.
We use the same training pipeline as for CMCD-APT. See further details in the CMCD-APT descrip- tion below. Note that we also employ the SKL loss for training and the linear annealing path with standard Gaussian as the reference distribution.
For both GMM-d and MW-32, we use 4-layer MLP with 512 hidden units, and for DW-4, we use a 4-llayer EGNN (Satorras et al., 2021) with 64 hidden units. Recall that in CMCD-APT, we use the geometric path, =xx) = nTx) 1-1 =x)()) which linearly interpolates between reference and target in log-space. Therefore, Adam with a learning rate of 1 and EGNN with learning rate of 1e-4. We additionally use gradient clipping with norm 1 for stability.
CMCD-APT
network is conditional on the 8. We optimise the MLP by
Furthermore, our training pipeline following the 3 stages outlined below:
Tuning PT: We initialise {Ba N uniformly from to 1. Then, we run PT for 600 steps, remove the first 100 steps burn-in, and take the last 500 steps calculate the rejection rate between adjacent chains using this to apply Algorithm 2 to update the schedule. We repeat this process 10 times ensure (3,, Jn=1 stable.
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Collecting data and training: We use PT with the tuned schedule to collect data. For experiments with 5 chains, we run 200K steps to collect 200K samples for each chain. For experiments with 10 and 30 chains, we run 65536 steps to collect data. We then train CMCD for 100,000 iterations with a batch size of 512. In each batch, we randomly select the chain index and samples according the chain train CMCD using SKI.. We repeat this step twice to ensure the network is well-trained until convergence.
Testing: We follow Algorithm 1, running CMCD-APT for 100K steps, and calculate the round trip.
Diff-APT For GMM-d, we use a 3-layer MLP with 128 hidden units, for MW-32, we use a 3-layer MLP with 256 hidden units and for DW-4, we use a 3 layer EGNN with 128 hidden units. For all models, use a learning rate of 5e 4, gradient clipping with norm for stability and EMA with decay parameter of 0.99. For training, we follow the same pipeline as CMCD-APT, but we only retain samples at the target chain in order to optimise the standard score matching objective. At sampling time, we tune the annealing schedule by running Diff-APT for 1,100 steps, discarding the first 100 samples as burn-in and using the last 1,000 steps calculate rejection rates. We then use this to apply Algorithm 2 to update the schedule. We then repeat this 10 times where we initialise from the uniform schedule.
PT For all experiments with PT, use the linear path with standard Gaussian as reference distribution. We tune the annealing schedule in the same manner as with Diff-APT to ensure comparisons with strong baseline.
E.3 FURTHER DETAILS ON COMPARISON OF ACCELERATION METHODS
For both training and testing for all methods, we take a single step of HMC with step size of 0.03 and 5 leapfrog steps as our local exploration step across each annealing chain. We note that while we could have improved performance by tuning the step size for each anncaling distribution, keep this fixed to disentangle the effect of local exploration from our communication steps.
Compute-normalised round trips The compute-normalised round trips, as reported in Table 1, is computed by dividing the original round trips by the number of potential evaluations that a single "machine" required to implement within parallelised implementation of PT/APT i.e. the number of potential evaluations required by the computation of wx (X%.ok) (or equivalently wikk (K)o.k) for a single n and t. Similarly, we count the number of neural calls in the corresponding manner.
NF-APT: We need 2 potential evaluations for -- (Xx%)) and x² (XXTTT) and single network evaluation.
CMCD-APT: For > 0, we need to calculate the potential and score of πh" n- at X and n² at X We also need compute the score of U X for k: 1,..., K t.k We note that can reuse all of the above score evaluations for both the forward and reverse transition kernels of Ph and Qh 1. In total, this requires K + potential and network evaluations.
Diff-APT: For K O, we require K + potential and network evaluations following the same logic as for CMCD. For K = 0, we require 2 potential and network evaluations for x (X-1(X"11) and π (X*=11) as we parametrise our annealing path in terms of the target distribution and neural network.
PT: We need 2 potential evaluations for n. - (x==1) and π "(X"=1) and we do not require any network evaluation.
E.4 FURTHER DETAILS ON SCALING WITH DIMENSION
For all methods, we take a single step of HMC with step size of 0.03 and 5 leapfrog steps as our local exploration step across each annealing chain and take 100,000 samples. Additionally, we report the
'We count this single potential evaluation.
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(compute-nore) normalised) round trip rate which involves dividing the by the number of sampling steps. We note that we use the same methodology as above for computing Compute-normalinn round trips.
round trips
E.5 FURTHER DETAILS ON LOG-NORMALISING CONSTANT (FREE-ENERGY) ESTIMATION
For CMCD-APT, we take single step of HMC with step size 0.22 and leapfrog steps for our local exploration step across each annealing chain for both DW-4 and MW-32, For Diff-APT, we take two steps of HMC with 5 leapfrog steps and step size of 0.2 and 0.22 respectively for DW-4 and MW-32 for our local exploration step across each annealing chain.
For all methods, generate 100,000 samples at the target distribution. For each estimate of AF, we subsample 1,000 samples uniformly without replacement from the 100,000 to compute the APT free energy estimator. This is then repeated 30 times for each method.
We take the ground truth free energy of DW-4 from estimating AF with the APT estimator using 100,000 samples from PT with 60 chains (after tuming). We take the ground truth free energy of MW-32 from Midgley et al. (2023) which calculates the normalising constant of a single copy of numerically allowing for the trivial computation of the overall normalising constant.
E.6 FURTHER DETAILS ON COMPARING APT WITH NEURAL SAMPLERS
For CMCD, we take our CMCD-APT model on DW-4 with 30 chains and K = 2, 5 from Section 6.3, and instead of sampling from these models using APT, we collect 5, 5,000 independent samples from our reference distribution to which we apply our learned kernels II, I P pn-1 which we recall are explicitly trained transport samples from the reference the target distribution. With final samples, we then plot the histogram of dij values for i * j-specificallyy, we note that DW-4 represents distribution over 4 particles in 2D), therefore for each sample from DW-4, compute the pairwise distances d, (i + j) over these particles in the sample; we then collect all of these values plot their histogram.
For Diffusion, we follow the same procedure but we tune our annealing schedule with Algorithm applied in the same way as in Appendix E.2 (this step is not required for CMCD as the annealing schedule is required to be fixed during training) before we collect samples by applying PE
For CMCD-APT and Diff-APT, we simply sample from each method to generate 5,000 samples at the target distribution before plotting the histogram of daj values.
E.7 FURTHER DETAILS ON ALANINE DIPEPTIDE
We run CMCD-APT using chains and K = 2, 5. For the local move, we adopt HMC with a dynamic step size: when the acceptance rate is larger than 0.9, increase the step size by 1.2; if the acceptance rate smaller than 0.8, we divide the step size by 1.2. Other settings are the same as those of other targets.
E.8 LICENSE
Our implementation is based on the following codebases:
https:
attpp:///gttaa..ccooooooolt///// jaaaaaaajaaMMMMlllaaa (Midgley et al., 2023) (MIT License) http::///gtthbb..ccmm//rrrrrgggp (MIT License)
aiffrasioou_cmmpeeeeehehhhhhhhte 2024) (No License)
http://twwbbccoeeeeeeeeeeeeee/ee jax (MIT License)
E.9 COMPUTING RESOURCES
The experiments conducted in this paper are not resource- iesturce-ittensive. We use a mixture 24GB GTX 3090 and 80GB A100 GPU, but all experiments can be conducted single 80GB A100 GPU.
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F ADDITIONAL EXPERIMENTAL RESULTS
F.1 VISUALISATION OF MANYWELL--32
In Figures 5 and 6, we visualise Many Well-32 samples generated by 1,000 consecutive CMCD-APT and Diff-APT steps with N = 5, 10, 30 and K- = 1, 5 from ManyWell-33 Well-32 via marginal projections over the first four dimensions. We also show 1,000 independent ground truth samples in Figure 7 for comparison. We choose to report only 1,000 steps to illustrate how fast our sampler mixes. As we can see, mode weights become more accurate as we increase and K.
(a)
=.
5, 5.
9 N - =0
( N - =
N 30, K
(N 30 K 5.
Figure 5: Visualisation of Many ManyWell-32 samples generated by 1,000 consecutive CMCD-APT steps.
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( -
(b) 5
N =0,
d 10, =
(c) N 30, K =1
( 30, K
Figure 6: Many Well-32 samples generated by 1,000 consecutive Diff-APT steps.
Figure 7: 1,000 independent ground truth samples from Many Well-32
F.2 COMPARISON ACCELERATION METHODS FOR MANYWELL-32 AND DW-4
We take our trained models from Section 6.3 and provide the same comparison with PT as in Table for ManyWell Well-32 and DW-4 below.
Table PT versus APT with different acceleration methods, targeting Many Well-32 in 32 dimensions and standard Gaussian reference using N 5, 10, 30 parallel chains for T 100, 000 iterations. For each method, we report the round trips (R), round trips per target evaluation, denoted as compute- normalised round trips (CN-R), the number of neural network evaluations per parallel chain every iteration (Neural Calls), and Ag estimated using N
30 chains (Âx).
# Chain
5
N
10
N
30
Method
Neural Calls (1)
Åg (1)
R (t)
CN-R (t)
R (t)
CN-R (t)
R (1)
CN-R (t)
CMCD-APE APT (K = 1)
2
4,384
1154
577.0
2802
1401.0
4729
2364.5
CMCD-APT (K= 2)
3
3.827
1587
529.0
3640
1213.3
5544
1848.0
CMCD-APT (K- 5)
6
3.148
2878
479.7
4790
798.3
6678
1113.0
DifF-APT (K 1)
2
6.663
425
212.5
2402
1201
4398
2199
Diff-APT (K =))
3
5.225
1387
462.3
4022
1340.7
5894
1964.7
DIT (K- 5)
6
3.94
3627
604.5
5704
950.7
7634
1272.3
Diff PT (K-
7.423
251
125.5
1561
780.5
3440
PT
0
5.475
550
275
1879
939.5
3733
1866.5
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Table 4: PT versus APT with different acceleration methods, targeting DW-4 in 8 dimensions and standard Gaussian reference using N = 5, 10, 30 parallel chains for T = 100, 000 iterations. For each method, we report the round trips (R), round trips per target evaluation, denoted as compute- normalised round trips (CN-R), the number of neural network evaluations per parallel chain every iteration (Neural Calls), and Ag estimated using N
30 chains (Ak).
Chain
Method
Neural Calls (1)
Ax (L)
CN-R
(1)
CN-R (1)
R (1)
CN-R (t)
CMCD-APT (K
3.173
3020
1510.0
6407
3203.5
9456
4728.0
CMCD-APT (K 2)
2.671
4239
1413.0
7549
2516.3
10538
3512.7
CMCD-APT (K-
2.107
6971
1161.8
9808
1634.7
12634
2105.7
Dift-APT 1)
2
4.565
4331
2165.5
7397
3698.5
7729
DifT-APT
3,810
7187
2395.7
10176
3392
9176
3058.7
DiST-APT 55
4.358
12456
2076
12740
2123.3
8104
1350.7
Pi (K =
2
4.739
2962
1481
5862
2921
7067
3533.5
PT
0
4,016
2329
1164.5
5128
2564
7610
3805
34


Reviews:
[{"review_id": "0ZzwWIckTT", "reviewer": "Reviewer_xVCo", "summary": "This paper introduces Accelerated Parallel Tempering (APT), a new framework that integrates neural samplers into Parallel Tempering (PT) to improve sampling efficiency on complex, high-dimensional, and multi-modal distributions. Classical PT often struggles because adjacent temperature distributions share low overlap, limiting swap acceptance rates and requiring a large number of parallel chains. APT addresses this by using neural transports to “bridge” neighboring distributions before proposing swaps, dramatically increasing swap acceptance and reducing the number of chains needed—while preserving the exact asymptotic correctness of PT.", "questions": "1. If gradient information is already cheap, who won't we just use vanilla PT sampler? \n\n2. Low acceptance probability occurs when these distributions have minimal overlap. Addressing this requires increasing\nthe number of parallel chains N, which may not always be possible. Why? If a problem cannot be simulated using 4-chain PT, it must be a hard problem, I don't expect this algorithm to solve it as well.\n\n3. Could you show the distribution error plot by comparing vanilla PT and CMCD-APT using different computational budgets?", "limitations": "", "rating": 2, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "1. Math derivation is sound.\n\n2. PT is known to be an effective sampler for multi-modal simulations. If diffusion-enhanced sampler really works, PT will surely boost the perfomance of multi-modal simulation.", "weaknesses": "1. I don't like the whole community that utilizes ideas like the diffusion model to do sampling, which is super expensive and doesn't make a lot of sense. I have extensive research experience in sampling and diffusion models but I don't this is a nice combination.\n\n2. The motivation why do and when do we need diffusion models to do sampling is not well-supported. The intuition why a backward process is needed is not explained clearly. \n\n3. for section 6.1, measuring the round trip is not a good idea, the eventual goal is to sample the 40-mode mixture distribution, you can simply measure the empirical TV/ KL/ W1 error.\n\n4. No scalable real-world experiments. Molecular is too small.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9WBJYOzntO", "reviewer": "Reviewer_93BL", "summary": "The paper proposes an extension of Parallel Tempering (PT) called Accelerated Parallel Tempering (APT), in which the standard swap operation is enhanced with deterministic or stochastic transition kernels. These kernels are trained adaptively using samples generated by APT within a data-driven objective. The deterministic variant employs normalizing flows (NF-APT), while the stochastic variants derive from forward–backward discretizations of controlled annealed Langevin dynamics (CMCD-APT) or from diffusion models, where the forward process is exact and the reverse process is obtained by integrating the learned reverse SDE (Diff-APT). The sequence of intermediate potentials is user-defined for NF-APT and CMCD-APT (typically along a tempering path) and learned automatically in Diff-APT. Experimental results on synthetic benchmarks and particle systems demonstrate that APT accelerates sampling compared to standard parallel tempering.", "questions": "* In the introduction, the references to Noé et al. (2019), Midgley et al. (2023), and Gabrié et al. (2022) suggest that these methods \"incur a bias\" similar to the previously mentioned approaches. This interpretation is incorrect. The earlier works (except iDEM) perform variational inference to learn a generative model without data, where the bias arises from optimization and model misspecification. In contrast, the three cited works embedded the models within Monte Carlo schemes - such as IS, AIS, or MCMC - which, in the infinite-particle/chain length limit, correct the bias of the learned model. Their residual bias is purely statistical. Grouping these methods together is therefore misleading.\n* The training procedures are described very vaguely. What are the exact loss functions used, and how are they implemented in practice? How many times is the adaptation loop repeated (i.e., how many optimizations of the data-based loss are performed) ?\n* Could you visualize the learned diffusion potentials path over time for a simple mixture of two one-dimensional Gaussian distributions?\n* Comparisons with standard AIS, SMC, PT (under an equivalent computational budget, including training), and other multimodal samplers (potentially ML-enhanced) on target-specific metrics would be highly informative.\n* For Diff-APT, could you first train an unconstrained energy-based model and then learn an auxiliary function analogous to $r_{\\theta}$ a posteriori? This might help avoid the expressivity constraints imposed by the current formulation.\n* For Diff-APT, why not use the target score matching loss [A], similar to what PDDS employs [B] ?\n* What mechanisms prevent the adaptive procedure from suffering from mode collapse, particularly given the mode-switching phenomenon? This concern is especially relevant since the loss functions include (at least partially) a reverse KL term.\n* According to Appendix D, only a single HMC step is performed for the local exploration move, which seems unrealistically low. Could you provide an ablation study on the swap frequency, i.e., the number of interleaved local steps between swaps?\n* The ManyWell-32 experiment in Appendix E.1 claims to demonstrate recovery of mode weights, yet no quantitative metrics are reported. Why not use the benchmark from [C], which is specifically designed to evaluate this capability?\n* How are the chains initialized in PT/APT?\n* The total number of MCMC steps (50k or 100k) is extremely large. Combined with the adaptive training loop, this implies a substantial computational cost. Could you (i) report actual wall-clock runtimes on a specific hardware setup and (ii) provide an ablation on the total number of MCMC steps?\n* In Fig. 4, how do you explain the large performance gap between Diffusion and Diff-APT? If Diffusion is trained using samples from Diff-APT, this discrepancy seems counterintuitive.\n* For ALDP, how are the chains initialized, and how do you sample from the $T = 1200K$ reference distribution?\n* For ALDP, is the energy parametrization (L1400) rotation-invariant? Additionally, how do you ensure that both the transition kernels and the energy function are defined in the center-of-mass (CoM) space, i.e., the mass-centered coordinate system?\n\n[A] Valentin De Bortoli, Michael Hutchinson, Peter Wirnsberger, & Arnaud Doucet. (2024). Target Score Matching.\n\n[B] Phillips, A., Dau, H.D., Hutchinson, M., De Bortoli, V., Deligiannidis, G., & Doucet, A. (2024). Particle Denoising Diffusion Sampler. In Proceedings of the 41st International Conference on Machine Learning (pp. 40688–40724). PMLR.\n\n[C] Grenioux, L., Noble, M., & Gabrie, M. (2025). Improving the evaluation of samplers on multi-modal targets. In Frontiers in Probabilistic Inference: Learning meets Sampling.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 1, "strengths": "* The paper addresses an important problem and demonstrates clear improvements in round-trip rates for parallel tempering.", "weaknesses": "* The paper is very poorly written. The presentation based on the Jarzynski framework makes it difficult to follow. A formulation directly grounded in the Metropolis–Hastings framework, with clearly defined target distribution and proposal kernel (see [1]), would greatly improve readability and conceptual clarity.\n* The novelty is rather limited. The deterministic case has already been covered in [2] (as acknowledged by the authors), while the stochastic variant represents only a modest generalization of prior work on AIS [3,4] and SMC methods [5], against which no comparisons are provided.\n* The proposed method does not address one of the core limitation of PT in multi-modal settings : mode switching. It is well known that along tempering paths, probability mass tends to shift abruptly between distinct high-probability regions [5,6], severely hindering mixing and performance in both PT and SMC. Even with accelerated swaps in NF-APT or CMCD-APT, the transition kernels remain local refinements (as shown in the left panel of Fig. 1) and cannot overcome this issue. Similarly, Diff-APT is affected because its learned marginal distributions rely on score matching, which is known to be mode-blind [7,8,9], causing the learned path to exhibit mode switching as well. Consequently, the method provides limited benefit in genuinely multi-modal settings. This issue becomes more pronounced in higher dimensions (likely explaining the use of a \"perfect\" path in Section 6.2). It would be informative to report results with a learned path. While increasing $K$ and reducing $N$ might partially mitigate this, it does not fundamentally resolve the scalability issue.\n* The evaluation is quite narrow. The round-trip rate is the primary quantitative metric used (except for Fig. 3, which reports free-energy differences, and Figs. 4 and 6–7, which are purely qualitative). This metric alone may obscure important behaviors such as mode switching and limits meaningful comparison to standard PT. The justification for this metric relies solely on [10], which claims PT outperforms neural samplers. Moreover, the only non-PT baseline compared is CMCD, which is known to perform poorly [10,11,12].\n* The target-informed parametrization (L1400) is known to substantially restrict the expressivity of the energy-based model [13] and to introduce significant computational overhead [10], further reducing the appeal of this approach.\n\n[1] Syed, S., Bouchard-Côté, A., Deligiannidis, G., & Doucet, A. (2022). Non-reversible parallel tempering: A scalable highly parallel MCMC scheme. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 84(2), 321–350.\n\n[2] Invernizzi, M., Krämer, A., Clementi, C., & Noe, F. (2022). Skipping the Replica Exchange Ladder with Normalizing Flows. The Journal of Physical Chemistry Letters, 13(50), 11643–11649.\n\n[3] Zhang, F., He, J., Midgley, L., Antorán, J., & Hernández-Lobato, J. (2024). Efficient and unbiased sampling of boltzmann distributions via consistency models. arXiv preprint arXiv:2409.07323.\n\n[4] Fengzhe Zhang, Laurence I. Midgley, & José Miguel Hernández-Lobato. (2025). Efficient and Unbiased Sampling from Boltzmann Distributions via Variance-Tuned Diffusion Models.\n\n[5] Phillips, A., Dau, H.D., Hutchinson, M., De Bortoli, V., Deligiannidis, G., & Doucet, A. (2024). Particle Denoising Diffusion Sampler. In Proceedings of the 41st International Conference on Machine Learning (pp. 40688–40724). PMLR.\n\n[6] Bálint Máté, & François Fleuret (2023). Learning Interpolations between Boltzmann Densities. Transactions on Machine Learning Research.\n\n[7] Wenliang, L., & Kanagawa, H.. (2021). Blindness of score-based methods to isolated components and mixing proportions.\n\n[8] Zhang, M., Key, O., Hayes, P., Barber, D., Paige, B., & Briol, F.X. (2022). Towards Healing the Blindness of Score Matching. In NeurIPS 2022 Workshop on Score-Based Methods.\n\n[9] Shi, Z., Yu, L., Xie, T., & Zhang, C.. (2024). Diffusion-PINN Sampler.\n\n[10] Jiajun He, Yuanqi Du, Francisco Vargas, Dinghuai Zhang, Shreyas Padhy, RuiKang OuYang, Carla P Gomes, & José Miguel Hernández-Lobato (2025). No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplers. In Frontiers in Probabilistic Inference: Learning meets Sampling.\n\n[11] Junhua Chen, Lorenz Richter, Julius Berner, Denis Blessing, Gerhard Neumann, & Anima Anandkumar (2025). Sequential Controlled Langevin Diffusions. In The Thirteenth International Conference on Learning Representations.\n\n[12] Noble, M., Grenioux, L., Gabrié, M., & Durmus, A. (2025). Learned Reference-based Diffusion Sampler for multi-modal distributions. In The Thirteenth International Conference on Learning Representations.\n\n[13] Jiajun He, Yuanqi Du, Francisco Vargas, Yuanqing Wang, Carla P. Gomes, José Miguel Hernández-Laobato, & Eric Vanden-Eĳnden. (2025). FEAT: Free energy Estimators with Adaptive Transport.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YpdQP0ShGV", "reviewer": "Reviewer_WMqg", "summary": "The present paper proposes to accelerate the parallel tempering (PT) algorithm using learned transport processes between neighboring distributions in the annealing ladder. The approach draws inspiration from the statistical mechanics literature (works by Jarzynski and collaborators) and a single step version with normalizing flows (Invernizzi et al 2022). It bears similarities with previously proposed approaches mixing deep learning and Sequential Monte Carlo, namely the Annealed Flow Transport (Arbel et al 2021), NETs (Albergo et al 2024) and CMCD (Vargas et al 2024). \n\nThe proposed approach is presented in a general framework along with different concrete variants of the algorithm leveraging either normalizing flows, diffusion models or stochastic control. The algorithm is shown to yield consistent estimators of expectations and normalization constants. A theoretical analysis of performances is given under symplifying assumptions. A set of numerical experiments demonstrates a possible advantage of the method over vanilla parallel tempering and compares the different variant proposed. Finally, limitations are discussed in a short conclusion where, in particular, the need to take into account the computational cost of neural networks in future works is acknowledged by the authors.", "questions": "-", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- Although related to recent literature and not unexpected in this respect, the proposed algorithm is novel and the authors do a great job at presenting in general the method before showcasing different possible applications with different types of generative modeling ideas. \n- The article also does a great job at connecting its method to the adjacent literature. \n- The method is justified by proofs of consistency. I have not read in details the proofs, but the result appear reasonable and the adequate physics literature is cited. \n- A theoretical analysis of performance for edge cases is also given. \n- The numerical section spans different examples, notably in relatively high dimension, even investigating the impact of dimension increase in a synthetic experiment.", "weaknesses": "- It would be desirable to insist more on the question of the training of the neural networks in the main text and on the fact that it is not a trivial question in this sampling setting. For instance, for NF-APT, the authors state in Appendix C.1.2 that the retained strategy is to first run PT to be able to train the flows, which is arguably an important limitation. \n\n- The introduction is not always fair to the adjacent literature\n\t- line 072 - “However, these methods usually incur a bias, foregoing theoretical guarantees of MCMC, and can be expensive to implement and train.” Is not true of the discrete flows approaches that are cited by the authors , nor of NETs (Albergo et al 2024).\n\n\t- line 095 - “By contrast, our framework leverages normalising flows to facilitate exchanges between all neighbouring temperature levels, thereby enhancing sampling efficiency across the entire annealing path and providing a more stable training objective” - the ambition of Invernizzi et al. is to drastically simplify the procedure by avoiding the necessity to have many replicas. As such, this last sentence is unclear, and the claim for the increased stability of the training objective needs to be substantiated. If I understand correctly this reference is implementing NF-APT with N = 1? \n\n\nMinor:\n- It would be worth mentioning [Noble2025] in section 5.3, as this reference already proposes to learn energy-based models along a noising process to ease sampling, although this reference was exploiting it in a different way. \n- l103 the definition of $Z_n$ is lacking the exponent $n$ to $U$.\n- l190 “we use using”\n- In Eq 4, maybe restate that the brackets are notations for an expectation. Also this is a common notation in the statistics/Monte Carlo literature, it is not usually encountered in the machine learning literature.\n- l215: “To also obtain free energy estimates, by averaging …” this sentence has no main proposition. \n- The notation $\\tau$ in proposition 250 is not introduced, I am guessing this the round trip rate. \n- l262 “a good neural network approximation” it would be a good idea to state a good approximation of what if the authors want to make this point at this stage, otherwise it can wait for the next section.\n\n[Noble2025] Noble, Maxence, Louis Grenioux, Marylou Gabrié, and Alain Oliviero Durmus. “Learned Reference-Based Diffusion Sampler for Multi-Modal Distributions.” Paper presented at The Thirteenth International Conference on Learning Representations. April 4, 2025. https://openreview.net/forum?id=fmJUYgmMbL.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1zBqxaw8gd", "reviewer": "Reviewer_bR4T", "summary": "This manuscript proposes a framework for the application of neural sampling methods to augment Parallel Tempering (PT) based MCMC. Neural samplers like normalizing flows, CMCD, and diffusion models are used to introduce time-inhomogeneous inner-loop Markov processes that accelerate the outer-loop parallel tempering by learning how to make adjacent chains overlap. This resulting procedure Accelerated Parallel Tempering (APT) increases the efficiency of PT in terms of the rate of round trips which is an indicator for the rate of mixing and it conserves the invariance with respect to the target distribution. Theoretical results underpin the conceptual soundness of this method and experimental results demonstrate that it yields practical benefits in several relevant sampling benchmarks.", "questions": "- What is the intuitive meaning of the global barrier $\\Lambda$?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "- This method appears to be novel.\n\n- The theoretical results are concise and clear and are related to the experimental results (e.g. connection between N and $\\Lambda$)\n\n- The mathematical notation is at an appropriate complexity level, resulting in good readability.", "weaknesses": "- Lack of comparison to neural samplers: the authors argue correctly that MCMC like PT and APT enjoy theoretical guarantees that are not provided in neural samplers.\nThis theoretical distinction is not a sound reason to exclude neural samplers entirely from the experimental comparison. The ultimate goal of these methods is exactly the same as for MCMC, namely to sample unnormalized distribution, and hence neural samplers should not be dismissed entirely based on consideration of asymptotic guarantees. The experiments shown in section 6.4 which compare to neural samplers (CMCD and diffusion samplers) are no fair comparisons since they utilize very different compute budgets (transport maps are concatenated here). \n \n- As argued in the paper the number of round trips is indeed a well-established metric in the context of PT and therefore a well-motivated metric. Comparing to sampling methods more broadly is essential and hence a comparison to other methods should include more general sample quality metrics, like ELBO, Sinkhorn, ESS, log Z. The authors argue “As ESS measures the intertwined performance of the local exploration and swap kernels, we are instead interested in maximising communication between reference and target …”. But such these performance metrics are ultimately crucial to judge the performance of APT relative to other sampling methods.\n\n\n- l147: the distributions $\\pi$ are called potentials\n\n\n- l206: “Theorem 1 shows we can quantify this  discrepancy in through the rejection rate”. The discrepancy in what?\n\n\n- Reproducibility suffers from a lack of a code base and corresponding details of the implementation appear to be scattered across many prior works. However, the reviewer did not attempt to reproduce results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)