Multilevel Generative Samplers for Investigating Critical Phenomena
Authors: Ankur Singha
Elia Cellini
Kim Andrea Nicoli
Karl Jansen
Stefan Kühn
Shinichi Nakajima
Venue: ICLR 2025 Poster (2025)
Keywords: Generative models
Multilevel sampling
Criticality
Renormalization group
Sequential training
Primary area: generative models
TL;DR: Novel generative sampler specialized for near-critical regime

Abstract: Investigating critical phenomena or phase transitions is of high interest in physics and chemistry, for which Monte Carlo (MC) simulations, a crucial tool for numerically analyzing macroscopic properties of given systems, are often hindered by an emerging divergence of correlation length---known as scale invariance at criticality (SIC) in the renormalization group theory.  SIC causes the system to behave the same at any length scale, from which many existing sampling methods suffer: long-range correlations cause critical slowing down in Markov chain Monte Carlo (MCMC), and require intractably large receptive fields for generative samplers. In this paper, we propose a Renormalization-informed Generative Critical Sampler (RiGCS)---a novel sampler specialized for near-critical systems, where SIC is leveraged as an advantage rather than a nuisance. Specifically, RiGCS builds on MultiLevel Monte Carlo (MLMC) with Heat Bath (HB) algorithms, which perform ancestral sampling from low-resolution to high-resolution lattice configurations with site wise-independent conditional HB sampling.  Although MLMC-HB is highly efficient under exact SIC, it suffers from a low acceptance rate under slight SIC violation. Notably, SIC violation always occurs in finite-size systems, and may induce long-range and higher-order interactions in the renormalized distributions, which are not considered by independent HB samplers.  RiGCS enhances MLMC-HB by replacing a part of the conditional HB sampler with generative models that capture those residual interactions and improve the sampling efficiency. Our experiments show that the effective sample size of RiGCS is a few orders of magnitude higher than state-of-the-art generative model baselines in sampling configurations for $128 \times 128$ two-dimensional Ising systems.  SIC also allows us to adopt a specialized sequential training protocol with model transfer, which significantly accelerates training.

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MULTILEVEL GENERATIVE SAMPLERS FOR INVESTIGATING CRITICAL PHENOMENA
Ankur Singha! 22, Elia Cellini Kim A. S Karl Jansen', Stefan Kühn", Shinichi Nakajima 1.2,8
"IIOLD,, Germany, Technische Universität Berlin, Germany
Univerriità degli Studi di Torino, Italy, INFN Torino, Italy, UUniversity of Bonn, Germany Helmholtz Institute for Radiation and Nuclear Physics (HISKP) Deutsches Secktronen-Synchroree (DESY), Germany, *RIKEN Center for AIP, Japan
ABSTRACT
Investigating critical phenomena or phase transitions is of high interest in physics and chemistry, for which Monte Carlo (MC) simulations, crucial tool for numer- ically analyzing macroscopic properties of given systems, are often hindered by an emerging divergence of correlation length- cality (SIC) in the renormalization group theory. SIC causes the system to behave the same at any length scale, from which many existing sampling methods suf- fer: long-range correlations cause critical slowing down in Markov chain Monte Carlo (MCMC), and require intractably large receptive fields for generative sam- plers. In this paper, propose Renormalization-inooone Sampler (RiGCS) novel sampler specialized for near-critical systems, where SIC is leveraged an advantage rather than a nuisance. Specifically, RiGCS builds on MultiLevel Monte Carlo (MLMC) with Heat Bath (HB) algorithms,, which perform ancestral sampling from low-resolution to high-resolution lattice configurations with fife-wiis-independder ML.MC-BB highly efficient under exact SIC, it suffers from low acceptance rate under slight SIC violation. Notably. SIC violation always occurs in finite-size systems, and may induce long-range and higher-orde interactions the renormal- ized distributions, which are not considered by independent HB samplers. RiGCS enhances MLMC-HB by replacing a part of the conditional HB sampler with gen- erative models that capture those residual interactions and improve the sampling efficiency. Our experiments show that the effective sample size of RiGCS is a few orders of magnitude higher than state-of-the-ar generative model baselines in sampling configurations for 128 x 128 two-dimensional Ising systems. SIC also allows adopt specialized sequential training protocol with model transfer, which significantly accelerates training.
as scale invariance at criti-
Generative Critical
conditional HB sampling. Although
INTRODUCTION
Monte Carlo (MC) simulations, where samples from Boltzmann distribution are used estimate macroscopic properties, are ubiquitous in many fields of science, ranging from chemistry (Metropo- lis et al. 1953). statistical physies (Hastings 1970 Creutz et al., 1983). , and quantum field the- ory (Creutz et al. 1979. Wilson 1980) to biology (Huelsenbeck et al. (201)) and financial analy- sis (Doucet al. 2001). In MC simulations, the ability to efficiently sample from unnormalized distributions in a high-dimensional space poses crucial challenges. Standard algorithms such as Monte Carlo Markov Chain (MCMC) methods (Metropolis Ulam 1949, Robert & Casella 2004)
often plagued by, e.g., slow convergence (Cowles & Carlin 1996). energy barriers and local
minima (Cérou et al. 2012). and critical slowing down (Wolff) 1990, 22044] SShaafeer et al.) 2011). This paper specifically tackles the problem of critical slowing down around critical regimes. In the broader context of physical sciences, the term criticality refers to situations where a system
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undergoes sharp behavioral change, often associated with phase transitions NNishimori & Ortiz 2011]. At criticality, physical systems typically exhibit self siiilaiii with respect to the change of scale, i... the physics of the coarse-grainet system is similar to that of the fine-grainee one. This phenomenon, called scale invariance at criticality (SIC), requires us to deal with arhitrar- ily long-range correlations, for which standard MCMC samplers with local moves undergo critical slowing down with arbitrarily long integrated auto-correlation time. Although many highly spe- cialized cluster algorithms, leveraging non-local moves, have been developed in the context of spin systems (Wolfi, 1989b a), critical slowing down still represents one of the major shortcomings of MCMC approaches.
For efficient sampling around criticality, MulriLevel (or Multiscale) Monte Carlo with Heat Bath (MLMC-HB) algorithms (Schmidt) 1983; Faas & Hilhorst, 1986, Jansen et al. 2020) were devel- oped. based on Renormalization Group Theory(RGT) (Kadanoff, 1966 Wilson, 1971, Wilson & Kogut, 1974, Cardy 1996). RGT systematically analyzes how macroscopic features emerge when the system is coarse- coorse-grained grained to larger length scales by marginalizing the degrees of freedom corre- sponding to the fine lattice grid, and provides crucial insights into critical phenomena of physical systems (Wilson 1971 Fisher 1973; Shankar 1994. Cardy [996). An important outcome of RGT is the emergence of SIC over the coarse coarse-grained and fine- mne-grained lattices. Adopting the block-spin transformations (Kadanoff 1966) for partitioning of lattice sites, (reesi,Shmidd (1983) proposed MLMC- HB that performs ancestral sampling from the coarsest lattice sites to the finer ones. Crucially, con- ditional distributions between consecutive resolution levels can thus be factorized into independent distributions under SIC, for which sampling can be efficiently performed by HB algorithmss In MLMC-HB, the long-range correlations are captured in the low resolution lattice, which is much easier than capturing them in the original high resolution lattice.
Machine learning techniques are also seen as potential candidates to, either partially or fully, over- come the shortcomings of MCMC algorithms. In particular, generative models with accessibility to exact sampling probabilities- -scch as normalizing flows (Rezende & Mohamed, 2015 Kobyzev et al. 2020 Papamakarios et al. (022)) and autoregressive models (van den Oord et al. 2016c bj Salimans et al.
efficient independent sampling and unbiased MC estimation via im- portance sampling, showing notable success across various domains. Such applications include statistical physics (Wu et al. 2019] Nicoli et al. 2020), quantum many-body systems (Hibat-Alle et al. 2020), quantum chemistry (Noé et al. 2019 Gebauer et al. (2019), string theory (Caselle et al., 2024 and lattice field theory (Albergo et al. 2019 Nicoli et al. |
Caselle et al.] [202]] Cranmer al. 2023 Abbott et al., (2024). However, capturing long-range interactions in large lattice systems may require intractably large receptive fields. Generative models therefore tend to struggle to gen- erate samples around criticality, except for a few recent works whose goal was to mitigate critical slowing down (Pawlowski & Urban 2020 Bialas et al. 2023).
In this work, we propose a Renormalizaaion-infoonee Generative Critical Sampler (RiGCS), which enhances MLMC-HB algorithms by mitigating their major weakness-iaa -ie., the HB samplers in MLMC-HB ignore long-range and higher-order interactions that may exist renormalized sys- tems when SIC does not hold exactly. Instead, RiGCS uses generative models with sufficiently large receptive fields to approximate renormalized distributions, effectively capturing the majority of those residual interactions. In our experiments, RiGCS drastically improves the sampling ef- ficiency of MLMC-HB, and achieves an effective sample size a few orders of magnitude higher than the previous state-of-the-art generative sampler (Bıalas et al. 2023) for the two-dimensional Ising model. Furthermore, we propose a specialized sequential training procedure with warm starts, which transfers model parameters between different resolution levels, substantially improving the training efficiency. Our contributions include:
Renormalization-nffrnee Multilevel Sampling: We propose novel method that lever- ages both SIC and generative modeling to efficiently draw samples from Boltzmann distri- butions around criticality.
Sequential Training with Warm Starts: We propose a sequential training procedure that initially samples scall-scale system and progressively transitions to the target large-scale system. Model parameters trained on smaller systems are transferred to larger ones for initialization, enabling warm start that significantly accelerates training.
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As with most generative neural samplers for simulating lattice-based physical systems, we do not claim that our method surpasses state-of-thh-aar MCMC samplers, such as cluster methods for the Ising model, which remain unmatched for general observable estimation in large-scale systems.
Wilson & Kogut 1974; Kadanoff 1966) has profoundly influenced the study of critical behavior in statistical systems and quantum field theory. Leveraging results of RGT, MLMC-HB for near-critical systems was pro- posed (Schmidy 1983), showing notable improvements in sampling efficiency for one- and two- dimensional Ising models. Adopting particular partitioning of lattice sites, called block-spin transformations (Kadanoti, 1966). MLMC-HB draws samples hierarchically by the site-wise in- dependent conditional HB sampling based on the renormalized systems different scales. [Faas & Hilhorst (1986) further enhanced MLMC-HB by incorporating long-range interactions. Recently Jansen et al. (2020) introduced low variance MC estimator by leveraging the correlations between the lattices with consecutive resolution levels, which further advanced MLMC-HB.
Related Work Renormalization Group Theory (RGT) (Wilson 1971:
A variety of generative models, including Generative Adversarial Networks (GANs) (Pawlowski & Urban 2020; Sinnhaa et al. 2022), Variational Auto Encoders (VAEs) DD'Angelo & Böttcher, 2020). and energy-based models (D Angelo & Böttcher 2020, Torlai & Melko 2016), have been used as independent MC samplers for lattice systems. Generative models with accessibility to the
exact sampling probability are particularly useful for MC simulations, because they allow for un- biased MC estimation by importance sampling or neural MC (Nicoli et al.) 2020). i.e., sampling with the Metropolis- Hastingg rejection. Specifically, Variational Autoregressive Networks (VANs) (uu et al.] 2019. Nicoli et al.) 2020) and normalizing flows (Albergo al. 2019) have proven to be highly effective for discrete and continuous systems, respectively. Recent works (Singha et al.) 2023a by Gerdes et al., 2023) introduced conditional normalizing flows for scalar field and gauge the- ories, showing that models trained away from criticality can interpolate (or extrapolate) for drawing samples near criticality. Nicoli et al.) (2021) and Bulgarelli et al. (2024) demonstrated that genera- tive models are particularly useful for estimating thermodynamic observables, e.g., free energy and entropy, which cannot be directly estimated with standard MCMC methods.
Recently, hierarchical sampling approaches have been integrated with generative modeling both for discrete systems (Li & Wang 2018 Bialas et al. 2022) and continuous lattice field theories (Finken- (004 Abbott et al. 2024). Neural Network Renormalization Group (NeuraIRG) (Li & Wang, 2018) uses hierarchical bijective map to learn renormalization transform, and was applied to the Ising model using a continuous relaxation technique. Hierarchical Autoregressive Network (HAN) (Bialas et al., 2022)- state-of-the- art generative sampler for discrete physical systems uses a recursive domain decomposition (Cè et al.) 20166, and performs independent conditional sampling for separate regions with trained VANs. The HAN approach has shown improved sam- pling efficiency compared to MLMC-HB in two-dimensional Ising models. We refer to Appendix A for an extended review of related works.
2 BACKGROUND
This paper focuses on Monte Carlo (MC) simulations of hypercubic lattice systems around critical- ity. We refer to (row vector) sample 8 SV to be a configuration on the V = N grid points in the D) dimensional lattice, where S denotes the domain of the random variable at each site, and N denotes the lattice size (per dimension). Given a Hamiltonian (or energy) (s) describing the interactions between the lattice sites, MC simulations draw samples from the Boltzmann distribution
p(s) 2 -BH((a)
where B is a constant inversely proportional to the temperature, and Z is the (typically unknown) normalization constant called the partition function. With sufficient number M of samples, phys- ical observables O e.g., energy, magnetization, and susceptibility, can be estimated by averaging over the sample configurations (O) Mf Σm=1 O(sm), thus revealing macroscopic physical prop- erties and phenomena like phase transitions. Below, we introduce common sampling methods with and without machine learning techniques.
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Markov Blanket: RiGCS
Markov Blanket: MLMC-HB
Markov Blanket: True
sampled sites
sampled sites condition sites
sampled sites
condition sites
Figure 1: Site partitionings based on the block-spin transformation (adanoff 1966). The ancestral sampling is performed from the coarsest level (right) to the finest level (left), namely, in the order of s--2 (green), s (blue), and s (red). In the one-dimensional case (upper row), the marginal distribution at each resolution level involves only nearest neighbor (NN) interactions. Consequently, the true Markov blanket (indicated by yellow shadows) of the highlighted site (black circle) includes only NN conditioning sites, enabling precise and independent HB sampling. In the two-dimensional case (lower row), marginal distributions generally include long-range and higher-order interactions. As a result, except at the finest level, the true Markov blanket (yellow shadows) possibly encom- passes all other sites. While MLMC-HB with NN Markov blankets (red shadows) as an approximate trial sampler, its locality leads to low acceptance rates for large lattice sizes. In contrast, our RiGCS, with wider Markov blankets (purple shadows) induced by the receptive field of generative models, effectively captures long-range and higher-order interactions, providing more accurate samples. Note that the Markov blankets of RiGCS used for the intermediate resolutions in our implementation are of the size 7 x 7 (i.e., much larger than the one shown in this figure). Sites at SL-2 (green) can be further partitioned until the dimension of S gets sufficiently small.
still serve
2.1 MARKOV CHAIN MONTE CARLO (MCMC) METHODS AROUND CRITICALITY
Markov Chain Monte Carlo (MCMC) methods class of algorithms used to sample from unnor- malized distributions. Since the partition function is not analytically computable in most physical systems, MCMC methods are fundamental tools for performing MC simulations in, e.g., statistical mechanics and lattice quantum field theory. A crucial challenge for MCMC sampling around crit- icality cope with long-range correlations. When distant regions in the lattice become strongly correlated, the standard MCMC methods that rely on local updates struggle to move from a low- energy state to another low-energy state. This because local updates generally ignore the corre- lations, and thus, the proposed trials tend to be rejected in the Metropolis-Hastings rejection step due to the increased energy. This results in a long autocorrelation time for the Markov chain-a phenomenon known critical slowing down (Wolf 1990). Two approaches were developed to mitigate critical slowing down.
Cluster algorithms Cluster algorithms (Wolf 1989a Swendsen Wang 1987) perform global updates by identifying clusters of correlated lattice sites and flipping them collectively. These global updates efficiently reduce the autocorrelation time and mitigate critical slowing down. Appendix B introduces two variants of cluster methods, including the Wolff algorithm (Wolff 1989a) which we refer to as the cluster algorithu throughout the rest of the paper. Cluster methods are not seen universal remedies against critical slowing down because they are not generally applicable to arbitrary continuous variable systems, although generalizations to specific continuous systems were proposed (Kent-Dobias & Sethna 2018).
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Multilevel Monte Carlo with Heat Bath (MLMC-HB) The MultiLevel Monte Carlo with Heat Bath (MLMC-HB) algorithm (Schmidi 1983) a protocol inspired by RGT was developed for sampling physical systems around criticality. We denote by JNN E RVVV homogeneous 2 D nearest neighbor (NN) interaction matrix that satisfies
if (o,,") are 2 D nearest neighbor pairs, otherwise, Ju {
Jo,
(2)
for J IR, where (.),,.. denotes the (u,...)-thl entry of vector, matrix, tensor. For some im- portant physical systems, including the Ising model (Onsager 1944) and the XY model (Kosterlitz) 1974). the Hamiltonian can be written
H(s) NN T
(3)
with JAN RVVV ! In MLMC-HB, one partitions the lattice sites into L - 1 levels 8 (s*, 8"-1........ ;")),,)))))............."))))))) so that all 2 D nearest neighbors of each entry of s (for = 1,.... L) be- long to the coarser level partitions (.-1......) Figure shows examples of site partitionings based on the block-spin transformations (Kadanofi 1966) for (D = 1)- and (D 2) 2)-dimensional lattices, where the sites with the same color (red, blue, or green) belong to the same partition (st, 8h-1, or 2L-2). For compact descriptions, we use inequalities to express subsets of the parti- tions, c.g., gكو = (8) .......,8) ))), and g>\ = (st (sf..... g/t)). We denote by Vi the dimension of أك i..., IZ E RV and g²و E ER' Vi, The marginal distribution of the (-th level lattice is given by
p(s = p((s)D[s>]] - 3 8H1(s
(4)
where the corresponding Hamiltonian ((( (i.e., a scaled negative log-marginal probability) is called a renormalized Hamiltonian. An important result in RGT is that, around the criticality,
the renormalized Hamiltonian for 1 = O...., I, where L is a few levels smaller than L, can be approximated as a Hamiltonian with NN interactions, i.e.
Hi(s ~ i!)(
where
Hi(s = - -
(5)
with the NN interaction matrix JN E R VI p(s" p(s (ااا can be decomposed as
lf
holds exactly, the conditional probability
p(s Is S111
V p( s5-1))
(6)
because the Markoy blanke Bishop 2006) of si, does not contain sy for any o' * v (see yellow shadows in the one-dimensional example Figure [ upper row). This makes the sampling from the conditional distribution (6) extremely easy and efficient-one can apply the HB conditional sampling exactly for the discrete domain (with probability table of size S)) and approximately for the continuous domain (with, e.g., a one-dimensional Gaussian mixture). Therefore, starting from drawing samples from p(su) (which can be efficiently performed by HB or MCMC if L is sufficiently large and hence Vo is small), the ancestral sampling of the full lattice according to
p(s) = (II' p( ((5((15(-1)) p(s")
can be efficiently performed. Intuitively, MLMC-HB captures the long-range correlations by the coarse level marginals, i.e., p(s5') for small I, which avoids two major difficulties-lare laage lattice size and long-range correlations aorrelations-ariinnn at the same time.
'The whole discussion in this paper can be applied to slight generalization with the Hamiltonian in the form of Hss) = En..r (J )o. (88, (,,)), where (s, (() is similarity function between two states, e.g., (s)s = &.,er for Potts model (Wul 1982].
"The corresponding NN interaction coefficient-J Eq. 2 which we refer to as Ji-can be analytically computed in RGT (see Appendix C). When L + co, Ji converges to a limiting value as / decreases, and thus the scale invariance at criticality (SIC) emerges. Since we consider finite lattices, exact SIC never holds. Nevertheless, in one- dimensional finite lattices, the renormalized Hamiltonians consist only of NN interactions, which is sufficient for MLMC-HB to he accurate, explained in F
'The Markov blanket of a random variable su is a set of other random variables B., S (s Foree that have sufficient information to determine the conditional distribution of given the other random variables, i.c. p( )B..) p(sol(s,) be).)).
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For the (D dimenssiill lattice (see Figure 1 upper row), it is known that Eq. (5), and thus Eq. (6) hold exactly, and therefore, MLMC-HB generates accurate samples from the target Boltz- mann distribution. For D 2 2 (see Figure I therefore, MIMMCHB should be combined with importance sampling or Betropolis-Hastinge rejec- tion. Namely, one should draw samples according to
lower row), Eq. ( holds only approximately, and
q (s) pls (Ili pat (d"(((11)))))) (s")
and compensate the sampling bias by using the sampling probability (s)(s). where (q"N(s (8|1 and (so) are approximate distributions with NN interactions to the true conditionals {u(s*|e ( and the true marginal p(5o). respectively. Unfortunately, the ap- proximation errors accumulate through ancestral sampling, leading to a significantly low acceptance rate for large lattice sizes. We will show in Section 4 that MLMC-HB is not very efficient for D = 2. Further details on RGT and MLMC-HB are given in Appendix C and Appendix D respectively.
2.2 GENERATIVE MODELING FOR MC SIMULATIONS
In recent years, deep generative models have gained significant traction in the field of physies for their efficient modeling of complicated probability distributions. In particular, normalizing flows (Kobyzev et al. (020)) and autoregressive neural networks (van den Oord et al.) (0T6c)) be- came very popular in the context of computational physics due to their intrinsic capability of pro- viding the exact sampling probability (0))), which allows asymptoticalll unbiased MC estima- tion. Notably, well-trained generative models provide independent samples from an approxi- mate distribution, and asymptotically unbiased estimates of physical observables can then be com- puted by importance sampling (Nicoli et al. 2020): (O) & /े MM
E 3M
O(sm). where mm "..""
16m
Wm = -- /go(sm) are the unnormalized importance weights. However, naive generative modeling can be problematic for sampling large lattices near criticality because large receptive fields are required to capture long-range correlations. Improving the scalability of generative samplers for large systems is therefore one of the most crucial challenges to achieving the same level of perfor- mance as sate-of-the-ar MCMC samplers, and ultimately surpassing them in efficiency.
3 METHOD
In this section, we describe our proposed method that enhances MLMC-HB (introduced in Sec- tion 2.1b with generative modeling (introduced in Section 2.2h. We focus on (D 2)-dimensional lattice systems with (V = N x N) grid points.
3.1 RUNORMALIZATIOX-IIOORRR
GENERATIVE CRITICAL SAMPLER (RIGCS)
Higher-order RGT (Maris Kadanoff 1978) shows that, for D 2, the Hamiltonian of the marginal distribution (4) consists not only of the NN interaction terms but also of long-range and higher-order interaction terms:
H((sS = - S. (SSi) SSS - Σ
(THh) Juat
(9)
V
where J.R and J.H0 denote the matrix and the tensor that express the longerange and high-order in- teractions, respectively Instead of simply approximating the renormalized Hamiltonian 9 by the Hamiltonian (5) with NN interactions (as done in MLMC-HB), our method, called Renormalization- informed Generative Critical Sampler (RiGCS), approximates it with generative models that can capture the long-range and higher-order interactions. Specifically, RiGCS performs ancestral sam- pling according
go - 1(5-115--1) (TI qo, (8'105(11) go, (s9),
(10)
'Note the difference hetween the "interactions" and the "correlations". The former means the direct cross dependent terms in the Hamiltonian, while the latter means statistical dependence in the Boltzmann distribution.

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Level 0
0g
+2x
O E
-8f7((a9)
Level
0g
01
HB
4
α e
-AH
Level
0g
01
02
03
X e
3f1,(s
Level
0g
01
02
03
04
0s
HB
16 x 16 OC e -8H(())
Figure 2: Illustration of sequential training for a six-layer (L RiGCS model on 16 x 16 lattice. Training begins with the marginal model (s²) on the smallest 2 x 2 target Boltzmann distribution. Conditional models for progressively larger RiGCS layers are then trained sequentially on target Boltzmann distributions for increasingly larger lattices. Red arrows indicate model transfer initializations, while parameters without incoming red arrows are randomly initialized. For larger lattices, the final step is repeated until the target size is achieved.
where qe, (s"|sS!-1) for 1 L and qo, (50) are conditional and unconditional gener- ative models that approximate p(s* and p(su), respectively. Here, e (0L-1..... 0g) denotes all model's trainable parameters. Similarly to the configuration variable 8, we use inequali- ties to express subsets of the parameters, e.g., Ogt- (f).... Oo). Note that, at the finest level, the conditional distribution p(s" ( has only NN interactions by assumption, and therefore, the exact HB algorithm can be efficiently applied without generative modeling. Therefore, it holds that
0 Ogl OgL-1-as there is no model parameter at the L-th level.
3.2 RECEPTIVE FIELD DESIGN
It is known that long-range and higher-order interactions between lattice sites diminish with in- creasing distance (Maris Kadanoff, 1978). Therefore, the accuracy of approximating the renor- malized Hamiltonian can be controlled by adjusting the receptive field size of conditional generative models- the larger the receptive field is, the more accurate the approximation the renormalized Hamiltonian (9). Compared to vanilla generative models (without multilevel sampling), where the receptive field needs to cover the whole correlation range in the original finest-level lattice, our RiGCS approach allows to keep the receptive field of each conditional model small. In particular, we set the number of levels proportional to the lattice size N (per dimension), we can keep the receptive field size constant for different N. This is because RiGCS captures the long-range interactions in the coarser levels- the receptive field size a X a of the coarsest generative model qe, (s") effectively amounts the receptive field size L/2 /2 x aLL 2 in the finest lattice. Although the optimal receptive field size for each level should in principle exist such that the accumulated ap- proximation error is minimized for given computational cost, we use the same architecture for the conditional models for / = L --14 1..... in this work. This choice, with makes the model complexity of RiGCS linear L, enables efficient model transfer in training, as explained below.
3.3 SEQUENTIAL TRAINING WITH MODEL TRANSFER INITIALIZATION
We train our RiGCS by minimizing the reverse Kullback- (KL.) divergence, i.e., min KL (p)))) Eeesv (a(s) log
20(s) p(a) M. log
RJ
Mr.
20(sm) p(*m.
(11)
which is estimated with the generated samples {snm {a)))} trainingg data drawn from the target distribution are not required. However, training all parameters 0 from scratch, e.g., with random initialization, tends to suffer from long initial random walking steps. This is because the randomly initialized RiGCS generates random samples, with which the estimated stochastic gradi- ent of the objective (1T rarely provides useful signal to train the model. We tackle this problem with a specific training procedure with model transfer, again based on RGT. We choose I. be an even
number, and consider set of sequential target Boltzmann distributions pt (s e- (0
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for L' 0,2,4 where ( |= are the approximate renormalized Hamiltonians with NN interactions, defined in 5). and H H creasing order of L', we train a RiGCS
(s). For each target pz (s in the in- ( that shares the same coarsest lattice size Vo as the model, gøs (855) = (s), , for the original target distribution. This allows for initializing the RiGCS parameters for learning pr. (s with the corresponding parameters already trained on 2 - easier (smaller lattice size) system. Figure 2 illustrates this procedure, where the initializations are indicated by the red arrows. Thanks to SIC, the models connected by the red arrows are similar to each other, as detailed in Appendix [..]] Note that all parameters except Oo: 0,,00-100000000 whicc are trained on the three smallest lattice sizes with random initializations-caa be warm-started. The pseudocode for the sampling and training routines of RiGCS is provided in Appendix
NUMERICAL EXPERIMENTS
We evaluate our proposed RiGCS and compare it against several baseline methods. The experimen- tal setup is detailed below. The code is available at http://tww..ootooeeeeee//////e/e
Target Physical Systems We adopt the two-dimensional Ising model, for which the Hamiltonian is given by Eq. 3 with the 2-dimensional binary lattice, i.c., 8 E S = (-1, N and = 1. This commonly used benchmark exhibits second-order phase transition (critical point), and is exactly solvable (Onsager) 1944)-iee. the ground-trune is analytically computed. We set / = 0.44, which corresponds to the critical (inverse) temperature where the phase transition occurs by spontaneous symmetry breaking in the limit of an infinite lattice.
Generative models Since the lattice sites are discrete random variables in a binary domain, use autoregressive neural networks (ARNNs)
go(s) (II" go(su so-1).....81) go(s))
(12)
for unconditional and conditional generative models at each level of RiGCS. More specifically, we adopt PixeICNNs (van den Oord al.) [016c]b), which allow us to control their receptive fields by
adjusting the convolution kernel sizes. Further details on ARNNs are given in Appendix
Baselines We compare our method against three different baselines. MLMC-HB (Schmidt 1983) VAN (plain ARNN without multilevel sampling (Wu et al. 2019)), and HAN (Biatas et al. 2022 We also evaluate the Wolff cluster method (see Appendix BJ. which is widely recognized highly efficient sampler for the Ising model, surpassing the performance of generative modeling ap- proaches. In our experiments, RiGCS achieves comparable (though slightly inferior) performance to the cluster method on large lattices, while outperforming HAN the previous state-of-the-an generative model and other baselines.
Model Architecture: The RiGCS architecture 10 consists of autoregressive models at each level. At the coarsest level (I 0), the generative model qe, (s") employs PixeICNN architecture with three masked convolutional layers, each with 12 channels and a half-kernel size of 6. At all inter- mediate levels except the finest one, the conditional generative models {go, (s' (( utilize convolutional neural networks (CNNs) with two convolutional layers, both featuring with kernel sizes of 5 and 3, respectively. We design the kernel size to ensure that the receptive field adequately captures long-range dependencies at each scale. Consequently, the receptive field of qa, (8") spans the entire coarsest lattice, ensuring comprehensive coverage. For intermediate lev- els (1 = 1,... L 1), the receptive field of qo, (s'|s51- S1-1 extends over an 7 x region, effectively capturing long-range and higher-order interactions up sites that are three steps apart. At the finest level (1 = L)). RiGCS performs exact independent Heat-Bath (HB) sampling using p( s p(*4|1- 5411). For further implementation details, we refer readers to Appendix G
channels
4.1 FREE ENERGY ESTIMATION
We evaluate the sampling methods in terms of the bias and the variance in estimating the free energy thermodynamic observable. For the cluster MCMC method, we combine it with annealed
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MLMC-3B Cluster
1.0
x10-6
vanilla training
VAN
0.8-
sequential training
REGOS
0
0.11
61
0.0000
0.0
3
Final
8
16
32
64 128
N
Figure 3: Left: Relative estimation error for the free energy. Only RiGCS provides estimates for
N 128 that are comparable to those of Cluster-AIS. The vanilla VAN cannot be trained for
N 64 in reasonable time. Right: ESS as a function of training hours for RiGCS with the vanilla training with random initialization (orange) and with the proposed sequential training (purple) for
N 64. The plot of the sequential training starts 2 2.3 hours when the pretraining for smaller lattice systems is finished. The ESS at each training epoch is computed with M 16 samples, and the markers "final" show the ESS computed with M 10 at the end of training.
importance sampling NNall 2001) (Cluster-AIS) for estimating the free energy (Caselle et al. 2016). For generative samplers, i.c. VAN, HAN, and our RiGCS, we use the asymptotically unbiased esti- mator introduced by Nicoli et al. (000)):
F = log 2, where - M Em= -Bff(smm) (g0(m)) 8m ~ g)(
(13)
Here M is the number of generated samples. We use this estimator also for MLMC-HB by com- puting all factors in Eq. (8) including the normalization constants. This is tractable because all
conditionals NN (s'1s - M are products of independent distributions, and the lowest level
marginal (s") the Boltzmann distribution with only 22x2 states.
Figure 3 left shows the relative estimation error (F-F)/(F) where is the ground-truth free energy, computed analytically (Onsager 1944). Each error bar shows one standard deviation of the statistical error. We observe that MLMC-HB already performs poorly for N 2 32, exhibiting strong biases The other four methods achieve compatible (unbiased) estimation up to N < 32, with Cluster-AIS and our RiGCS outperforming VAN and HAN in terms of the variance. Furthermore, for N 2 64, the vanilla VAN cannot be trained because its wall-clock training time exceeds several weeks, and the variance for HAN is much larger compared to RiGCS. For N = 128, only Cluster-AIS and RiGCS achieve results compatible with the ground truth free energy, while HAN gives highly biased estimates incompatible with the ground truth. These results demonstrate the superiority of our RiGCS over existing generative models for estimating thermodynamic observables.
4.2 QUALITY MEASURES FOR GENERAL OBSERVABLE ESTIMATION
We further compare the samplers using recently proposed Effective Mode-Dropping Measure (EMDM) (Nicoli et al., 2023) and the commonly used Effective Sample Size (ESS) as indicators of bias and variance, respectively, for general moontthermodynamii)) observable estimation. The
go(=)* and qo is the renormalized density of qa(s) with effective support i.e., the support excluding the low density areas where no sample appears with high probability. 6 Nicoli et al. (0023) showed that the bias of the importance- weighted estimators for general observables can be bounded with EMDM, and v = indicates that no effective mode-dropping occurs and thus the estimator is unbiased. ESS (per sample), defined as ESS F.p [ E [0, 1], is known to be inversely proportional to the variance of general unbiased estimators, and ESS = implics that qo = p(s).
EMDM defined as Egg [w(s)] E [0, 1], where w(s) p(a)
Figure
4 (left) shows the v (EMDM) for RiGCS, VAN and HAN, We recall that the vanilla VAN can be trained only up to = 32, as explained above. For S 64, the EMDMs of HAN and RiGCS are
'The results for 2 64 are out the range.
TThe threshold for the "low density area" depends on the number M of samples, and (s) go (s) for
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1.2
10a
1.0-.000
10
0.8
110
E5S
10
0.6
HAN
10-3
MLMC-II TIB
0.4
100
Chaster-AIS
10
TIAN
16
32
RUG0S
16
32
64
128
8
16
64
128
Figure 4: EMDM (left) and ESS (right). The vanilla VAN cannot be trained for N 2 64 in reason- able time. The inset in the left plot zooms around the region of Φ ze 1.0, showing that RiGCS does not suffer from mode dropping RiGCS achieves the closest performance to Cluster-AIS for large systems.
for large N 128. Similarly, the ESS plot (right) shows that
compatible with EMDM = 1, indicating no effective mode droe-drop However, for N 128, the EMDM of HAN drops significantly. implying that the model affected by effective mode-dropping. This result is consistent with the biased free energy estimation by HAN observed in Figure 3 (left).
RiGCS shows no evidence of mode-dropping for N = 128, highlighting its robustness in accurately modeling the target distribution and avoiding mode collapse in high dimensions. Figure | (right) shows that RiGCS outperforms all baselines except the cluster method, a state-of-the-as approach for general observable estimation. Note that for N 128, RiGCS improves the ESS of HAN by several orders magnitude, making it the only generative model with non non-vanishing ESS.
Simulation Costs Generative modeling approaches like VAN, HAN, and RiGCS require training, with wall-clock times for N 64 lattice of approximately 60 days. 2.8 hours, and 3.8 hours, respectively. Sequential training with model transfer (introduced in Section 3.3) offers significant advantage over random initialization, as shown in Figure 3 (right). Sampling costs for generating 100 samples with MLMC-HB, VAN, HAN, and RiGCS are approximately 14, 27, 0.2, and 0.4 seconds, respectively.
Further investigation including additional analysis, ablation study with transformer architectures, and benchmarking for different physical theories are provided in Appendix H
5 CONCLUSIONS
Critical phenomena such as phase transitions are of high relevance in many fields of physies, where Renormalization Group Theory (RGT) plays a central role for theoretical analysis. Insights from RGT were also used for improving tools for numerical analysis, leading to MultiLevel Monte Carlo (MLMC) methods based on the emerging scale invariance at criticality (SIC). In this paper, further enhanced such tools by leveraging machine learning techniques. Specifically, we introduced imnormalizationninfomee informed Generative Critical Sampler (RiGCS), multilevel sampling algorithm where conditional generative models, with appropriate size of receptive fields, are substituted to naive nerghtt heat bath conditional samplers. This modification allows for capturing long- range and higher-order interactions that exist under slight violation of SIC, making RiGCS the new state-of-the-ee generative samplers. Furthermore, we also introduced the sequential training with model transfer, which was again inspired by SIC and significantly reduced the training cost. Al- though many previous works incorporated general domain knowledge, e.g., invariances, equivari- ances, and preservation laws, for machine learning model design, this work is one of the few appli- cations where the knowledge of critical phenomena, i.c., SIC, is incorporated directly in both the architecture design and the training procedure. We see this work specialized machine learning methods for critical regimes, paving the way for more efficient algo- rithms. Future research directions include applying RiGCS to other physical models, such as Potts models and lattice gauge theories, as well as exploring hybrid approaches that combine RiGCS with related methods, such HAN.
first step toward developing
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ACKNOWLEDGMENTS
The authors thank Piotr Bialas, Piotr Korcyl, Tomasz Stebel, and Dawid Zapolski for sharing their code to reproduce the results with their Hierarchical Autoregressive Networks (HAN), and Andrea Bulgarelli for sharing a highly optimized implementation of the cluster algorithm, which was used to obtain the ground truth reference values. The authors also thank Christopher Anders and Pan Kessel for fruitful discussions and inspiration. This work was supported by the German Ministry for Education and Research (BMBF) under the grant BIFOLD24B, the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) as part the CRC 1639 NuMeriQS - project no. 511713970, the SFT Scientific Initiative of INFN, the Simons Foundation grant 994300 (Simons Collaboration on Confinement and QCD Strings), and the European Union's HORIZON MSCA Doctoral Networks program project AQTIVATE (101072344).
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A EXTENDED RELATED WORK
Renormalization Group Theory (RGT) has significantly impacted the study of statistical systems, especially in analyzing critical phenomena and phase transitions. The pioneering works by Wil- son, Kogut, and Kadanoff laid the foundational principles of RG theory (Wilson 1971 Wilson & Kogut 1974 Kadanoff 1966). Subsequent advancements have expanded the application of RGT- inspired methods to disordered systems, random ferromagnetic chains, and Monte Carlo simula- tions (Aharony 1973 Fisher 1973] [wendsen] [979; [errida] 1980; Butera & Comil 2002). These developments have enriched the understanding of critical phenomena and further established the applications of RGT techniques in both theoretical and computational contexts. For further insights
into RG we refer the reader to Appendix C
In computational physics, substantial research has focused on RGT-inspired sampling methods for lattice simulations. For example, early studies on the Ising model (Schmidi 1983, Faas & Hilhorsi 1986) achieved notable success in simulating small systems in one dimension, yet faced limitations as the lattice size increased. More recently,Janseee et al. (2020) introduced a new theoretical frame- work for low variance estimation, showing promising results for one-dimensional quantum systems. In lattice gauge theory, the application of RGT concepts led to the development of several algo-
rithms, including multigrid (Cahill & Kogut 1982. Goodman & Sokal 1986 Hulsebos & Hockney]
1989) multiscale thermalization techniques
Endres et al. 2015), and decimation maps (Matsumoto
al. 2023).
In recent years, RGT-inspired approaches have been combined with machine learning to develop more scalable neural samplers. Notable examples include applications in (1) (Finkenrath 2024)
and (3) gauge theories (Abbott et al., (024)) where a renormalization group (RG) scheme has been combined with normalizing flows. Similarly to our approach, [Biatas et al. (2022) proposed a Hierarchical Autoregressive Network (HAN) for sampling configurations of the 2D Ising model.
This latter leverages a recursive domain decomposition technique (Cè et al. 2016), in which different regions of the configuration are sampled in parallel using the same autoregressive network. thus
replacing the traditional scaling with the system's linear extent L. Biatas et al. (2022) demonstrated the effectiveness of HAN on the two-dimensional Ising model, simulating lattices up to the size of 128 x 128. However, this method has several limitations, which are discussed in Section ! and Appendix in great detail. On side note, we emphasize that our multilevel approach and the domain decomposition proposed in [Balas et al.] (2022) are not mutually exclusive. In fact, these methods could in principle be combined leading to more powerful sampling protocols. We defer this investigation to future work.
Besides the development of enhancing sampling methods, other recent works have leveraged the idea of RG in different ways. Li & Wang (2018) focused on neural network renormalization group, investigating the capability of neural networks to perform hierarchical feature extraction and hierar- chical transformations. To this end, they used bijective transformations to learn hierarchical maps to automatically identify mutually independent collective variables. While inspired by RGT, this ap- proach does not use multilevel sampling, and does not observe favorable scaling- in the paper only results on rather small lattices up to 16 x 16 are shown. Koch-Janusz & Ringel (2018) proposed machine learning approach to identify the relevant degrees of freedom and extract Ising critical exponents in one and two-dimensional systems. |Efthymiou et al. (2019) leveraged the idea of image super-resolution and trained convolutional neural networks that invert real-space renormalization decimations. The authors showed that it is possible to predict thermodynamic quantities for lat- tice sizes larger than those used in training. Lenggenhager al. (2020) drew a connection between real-space RG and real-space mutual information. From an Enformation-thhoree investigated the information loss at arbitrary coarse graining of the lattices through the lens of RG. A Renormalization
standpoint, they
more recent study by Marchand et al. (2023) introduced the Group (WCRG), where fast wavelet transforms are used to build an RG transformation across scales. While similar in spirit, their approach is substantially different from ours. Specifically, they train the model by using a contrastive divergence loss, which requires lot of training samples drawn from the target distributions. Furthermore, Hu et al. (2020) used the neural network renormalization group (Li & Wang 2018) as a universal approach to design generic Exact Holographic Maps (EHMs) for in- teracting field theories. Chung & Kao| (222 used Restricted Boltzmann Machines (RBMs) to learn a valid real-space RG transformation without prior knowledge of the physical system, establishing a solid connection between the RG transformation in physics and the statistical learning theory. Ron
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et al. 1 and Bachtis et al. 20222). instead, used modified block-spin transformations- to im- prove convergence in the Monte Carlo (MC) renormalization group trajectory and inverse RG transformations, respectively, to extract critical exponents of given physical theory.
Recently, a so-called Sechine-Learning Renormalization Group (MLRG) algorithm has been devel- oped to explore and analyze many- bodyy lattice systems in statistical physics (Hou & You 2023). In a recent work by Di Sante et al. (2022), the authors proposed a data-driven dimensionality reduction and used Neural Ordinary Differential Equation (NODE) solver in low-dimensional latent space to efficiently learn the functional RG dynamics. The authors showed promising results in the context of the Hubbard model.
Lin et al. (2117) pointed out that convolutional neural networks in supervised learning tasks can act as a "coarse-graining" procedure, isolating relevant macroscopic features from irrelevant mi- croscopic noise. In recent years, RG-inspired machine learning applications have emerged in the context of variational inference (Ahn al., 2018), regularization techniques (Wang et al. [2024) transfer learning (Redman et al. (022), and multi-scale semantic manipulation of images (Hu et al.) 2022). While all these related works leverage the concept of RG different ways-such as for extracting critical exponents, or interpreting RG as coarse-graining procedures in machine learn- ing-they suffer from a few shortcomings when comes to efficient sampling. First, they often lack the access to tractable probability density, which is required for importance sampling neural MC to obtain unbiased estimates. Second, they do not allow for data-free training of neural sampler as well as rapid and effective sampling at multiple scales RiGCS does.
RG ideas have also been used in the context of Tensor Networks (TNs) to construct an emergent scale invariant description for critical systems. TNs describe the wave function, or the partition function, of system as a contraction of a network of smaller tensors. This approach was shown to be efficient as long as the entanglement in the system is moderate (Bridgeman & Chubb, 2017). Blocking tensors together and coarse graining the system allow for (numerically) obtaining a description of the system a larger length scale. Prominent algorithms for coarse graining the partition function of a critical system are the Tensor Network Renormalization (TNR) group (Evenbly & Vidal 2013) and
loop TNR (Yang et al. (017) for square lattices, as well as grapph-ideependet Local Truncations
(GILT) for arbitrary graphs (Hauru et al., 2018). The Multi-scale Entanglement Renormalization Ansatz (MERA) (Vidal) 2008 [veebbly & Vidal 2014) leverages the hierarchical structure of RG to efficiently represent quantum states for critical systems in 1+1 dimensions that are described by an underlying conformal field theory. Generalizing this idea, and understanding holographic duality as a generalization of the RG flow, Qi (2013) and Lee & Qi (2016) proposed an exact holographic mapping, which is a one-to-one unitary mapping between boundary and bulk degrees of freedom. Unlike MC methods, TN approaches enable the direct computation of the expectation values of observables, as they provide an approximation to the wave function or the partition function of system. However, although the numerical algorithms for TN methods, which allow for recovering exact scale invariance at the critical point, scale polinomially with respect to both system size and tensor size, the computational cost remains challenging due to a large degree in the tensor size X-the leading order costs of TNR and MERA are O(x"), and O(x"), respectively.
Lastly, Cotler and Rezchikov have recently uncovered intriguing connections between RG theory and optimal transport (Cotler & Rezchikov (023a) and diffusion models (Cotler & Rezchikov) 2023b), highlighting promising new directions for further investigation.
B MARKOV CHAIN MONTE CARLO AND CLUSTER METHODS
B.1 MARKOV CHAIN MONTE CARLO (MCMC) SAMPLING
MCMC methods allow for producing sequence of configurations { (()).... through Markov chain, following an target distribution p(s) = 1 ((s)) for which the partition function Z is unknown. To this end, given the current configuration (m)) new configuration s' is proposed, which is either accepted or rejected. If accepted, it becomes the next configuration, i.e., ((m++) = s',
while rejected, the configuration stays, i.c., s (mm-1) = ((m) To ensure that the produced config- urations follow the target distribution, the transition probability T(s - s') from configuration to
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s' has to fulfill
8') EiddyT's' (00666'1))
(14)
One way to ensure the condition above is by making the transition prohability satisfy the detailed balance condition:
(ss) - 8') ;(s) - x)
(15)
Together with the property of ergodiciry-i.e. existence of least one Markov chain connecting any pair of configurations with positive transition probability- the detailed balance condition en- sures that the configurations sampled by the Markov process follow the target distribution p(s) after thermalization.
Once a sufficient number of configurations are sampled, the expectation values of physical observ- ables (ss), e.g., energy, magnetization, and correlation functions, can be estimated by averaging over the configurations
(0)
Σ ()s(m))
(16)
M M7L
where M is the total number of sampled configurations, and mi< M) is the number of burn-in steps for thermalization. Due to the dependence of each sample configuration the previous one in the Markov chain, MCMC samples are inherently correlated. This correlation is quantified by the autocorrelation time т, which characterizes the degree and persistence of correlation between samples. A long autocorrelation time can significantly degrade the accuracy of the Monte Carlo estimator (16), because makes the Effective Sample Size (ESS) much smaller than the generated number of samples.
Crucially, as the system approaches the critical point, the autocorrelation time r of thermodynamic properties diverges, following power law in the correlation length &:
T xx E
(17)
where 2 is known as the dynamical critical exponent. Due to scale invariance at criticality (SIC), the correlation length diverges (g > o0) as system approaches the critical point. This means that the autocorrelation time diverges, as Eq. (17) implies, leading to phenomenon called critical slowing down (Wolff 1990). For finite hypercubic lattices, the correlation length in lattice units bounded by the extent N of the lattice in each dimension, hence
α N
(18)
as one approaches the critical point. From Eq. (8) it is clear that, depending the critical ex- ponent 2, sampling configurations with high ESS at criticality becomes increasingly challenging as the lattice size N increases. For local update protocols, for example Metropolis-Hasting,, one typically obtains values of Rs (Wolff) 1990). This increase in autocorrelation necessitates larger number of samples to achieve accurate statistical estimates, thereby raising the computational cost (Schaefer et al. 2011).
B.2 CLUSTER ALGORITHMS
In contrast to the algorithms relying on local updates, cluster algorithms can yield the dynamical critical exponents close to zero, thus avoiding critical slowing down. In the following, we briefly review two cluster algorithms for efficient sampling close the critical point in the Ising model,
p(s) 1 exp (-B ((ere)) JswSer)
(19)
where V is the spin configuration, and Eg.,) takes the over all nearest neighbor pairs.
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B.2.1 SWENDSEN- WANG ALGORITHM
The basic principle of the Swendsen- Wang algorithm is flip entire clusters of spins instead of single one (Swendsen & Wang 1987p. To this end, given spin configuration is divided into clusters by forming bonds between spins. A cluster then consists of all spins connected directly or indirectly via bonds. Subsequently, all the spins belonging to a cluster are flipped collectively. More specifically, given the current configuration g(m)) E {-1,1} the algorithm proceeds as follows:
1. Inspect all eeearest-neighbe pairs (v. v²) in g(m) If If = (mm) a bond is formed between the pair with probability 1 exp (-)..). Otherwise, no bond is formed.
2. Identify all clusters, i.., all sets of spins connected either directly or indirectly by the bonds.
3. Flip all spins within each cluster collectively with a certain probability pflpp, resulting in a spin configuration s'.
4. Remove all bonds and repeat the steps for the new spin configuration s g(m+1) s'.
In step 3, if pilip is chosen to be close zero, the new configuration s' will, in general, be too similar to In contrast, choosing pilip = 1 will result in full inversion of the configuration g(m), which does not change the energy at all. As both extremal cases do not produce sensible new configurations, pilip is typically to 1/2.
The Swendsen- Wang algorithm can be viewed as a data augmentation method (Higdon 1998 Kaste-
leyn & Fortuin 1969 Fortuin Kasteleyn 1972), where the configuration space the Ising model is extended by introducing auxiliary bond variables u(v,er) E [0, e²a,] for nearest-neighbo pairs. Specifically, we assume that ((v,r") follows the conditional uniform distribution:
p(u 8p)) cxp(-333 (1 + 8p8vr) 1 (M(o)))) E [0, exp(B3((1 + 8o8er))))
(20)
where (.) is the identity function equal to one if the event is true and zero otherwise. This gives the entire joint distribution as
pl Kuutewehoo p p(u(,.r))ssp()))
II (( [0, ex,expp(3 syS )))) (w,e')
(21)
Noting that
0 (exp(28J) > 1
if 8v # 8 if s, = Sy'
exp(3J(1+
Eq. 21
gives the following conditional distribution:
for (sv, sp) E ((-1,-1),(L,11) if u( 1,
p( ((8,sywlus (w(
0 for (p)) E (1-1,1))
if > 1,
(22)
for (v)) E (-1,1)² if ((aa)))) S1.
Since Eq. depends U((eet) only through whether it is larger than 1 or not, and our goal is to sample configurations 8, we can replace the continuous random variable u(,,"") with the binary
random variable by ao E (0, 1} such that the event (lw = corresponds to the event M(n,er) 1.
From Eqs. (20) and (22). we have
p(b(u,o0) = 81))) = p(u > 1/80, (0r) exp(33 + SpSw))) -
cxp(33 + 8. = = (1- (1-exp(-22.)) 1[sp= = =s)),
(23)
and
for 8p) E (1,1,
if b( =1,
p(Su, sv (((p))))))
0 for Sp) E {f (f,,
if b( =1,
(24)
for (8)) E (-1,1) if bf 0.
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respectively. Notably, the spins between the neighboring pairs with bond (o,,e") must be the same, and those without bond (v.,r)) = have no interactions in the conditional (24p. This allows for cluster-wise independent sampling without rejection step. Swendsen- Wang algorithm thus performs Gibbs sampling with Eq. (23) (Step 1) and Eq. 24 (Step 3) iteratively to obtain a
Markov chain spin configurations {s(m)).
The effectiveness of the Swendsen- Wandd algorithm can be understood by the fact that flipping large clusters allows for efficiently destroying the long-range correlations emerging close to the critical point. For D > 2, the Swendsen Wang algorithm becomes less capable. as the majority of the formed clusters tend to be small, with only a few large ones being generated.
B.2.2 WOLFF ALGORITHM
The Wolff algorithm Wolfi 1989a is a single-cluster variant of the Swendsen wrrendse--aangg algorithm. Instead of dividing the entire configuration into clusters and flipping each of them, the Wolff algo- rithm only forms single cluster and collectively flips the spins inside this cluster. Given the current configuration (mm) E {-1, the Wolff algorithm proceeds with the following steps:
1. Randomly choose site E [I....,V}
2. Form bonds, analogous to the Swendsen- Wang algorithm, with probability 1 1-exp (-23J) with all nearest neighbors {v'} such that s, = Sy'
3. For each of the bonded sites {u'}, form bonds with its respective neighbors that have not been bonded, according to Step
4. Repeat Step 3 iteratively until more bonds can be formed.
5. Flip all spins within the cluster and obtain spin configuration s'.
6. Remove all bonds and repeat the steps for the new spin configuration g(m+1) s'.
Note that compared to the Swendsen- Swendsen-Wang algorithm, the cluster is flipped with probability pflpp
1. the cluster formed by the Wolff algorithm is large, the long-range correlations are broken up essentially as effectively as in the Swendsen- Wvendsen-Wang algorithm with smaller computational cost- because the Wolff algorithm focuses only on single cluster. If the cluster formed by the Wolff algorithm is small, the configuration does not change significantly, however, at the same time, the computational cost is also small. Thus, the Wolff algorithm turns out to be even more efficient in decreasing the dynamical critical exponent z, compared to the Swendsen- Wang approach (Swendsen) Wolff 1989b).
& Wang 1987
In our experiments Section 4 and Appendix we used the Wolff algorithm as the state-of-the-art Cluster method, which is more efficient than the Swendsen-Wand algorithm.
C RENORMALIZATION GROUP
The Renormalization Group (RG) (Wilson 1971 Cardy 1996) is powerful framework in theoret- ical physics for studying the behavior systems as they are progressively coarse- -rarine to larger length scales. During this process, microscopic degrees of freedom are systematically marginalized, generating flow in parameter space known as the RG flow. More formally, given a Hamiltonian H describing the system given length-seale, can define RG transformation RR
H' RR [H]
Namely, the operator RA changes the scale of the system and yields renormalized Hamiltonian H' describing the system at larger length scale, indexed by X. with less degrees of freedom. For Rx to be proper RG transformation, it has to fulfiil the semi-group property, i.e., there exists neutral element Ro that does not change the scale and the composition of two transformations to different length scales À and X' has to fulfill Rx RR Rv+*- This transformation generates flow on the space of Hamiltonians that can yield crucial insights into the macroscopic properties of physical systems. In particular, the critical points correspond the fixed points H* satisfying
H' = Rr [H*]
(25)
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in the RG flow, as the system exhibits scale invariance at criticality (SIC).
Close to the critical point, one can approximate the Hamiltonian the system as = H* + öH, where 8H is small perturbation. Expanding the RG transformation around the fixed point, one finds
RA [H' + +8H H + + 8 + O (8ff") R H" + 8H',
(26)
where 5H' = L [H*] 8H. Applying the transformation R. for 7-times, we find in leading order that RxXa [H* + +8H] H + + [H*]* 8H.
(27)
Expanding &H in the eigenoperators [M₃} of C [H*), one finds that the leading order correction can be expressed as
C [ 8H - Me.
(28)
where {ej} are the expansion coefficients, and {e₂} are the eigenvalues corresponding the eigen- operators {Mz}-
For large T, equivalently at large length scales, one observes that the eigenvalues {e, } determine the behavior of the system: the renormalized Hamiltonian tends to be dominated by the eigenop- erators with larger eigenvalues. We say that the eigenoperator Mj is relevant, marginally relevant, or irrelevant if the corresponding eigenvalues are é; 1, = 1, or < 1, respectively. Then, Eq. (28) implies that the relevant and marginally relevant operators determine the macroscopic be- havior of the system. Thus, close to the critical point, the systems sharing the same (marginally) rel- evant operators will exhibit the same behavior at macroscopic scales, regardless of the microscopic degrees of freedom. This gives rise to the notion of universality classes, i.e., physical systems show- ing the same scaling behavior criticality described by typically a few critical exponents, despite being microscopically different (Cardy) 1996).. Thus, information about system's 's behavior close to criticality can be obtained by studying another model within the same universality class. For ex- ample, the Ising model, originally developed to describe phase transitions in ferromagnetic systems, can also used to study the liquid-gas transition, superfluids, and the Higgs mechanism (Wilson 1971 Wilson & Kogut 1974).
simple example an RG transformation is the Kadanoff block spin transformation, which will be illustrated below. The partition function of the Ising Hamiltonian is given by
Z = Σ exp exp((3H(s)) = Σ exp (-3J 81 NN )
(29)
where I NN is the nearest neighbor matrix (of the appropriate size depending the context) defined
as
{
if v') are 2 D nearest neighbor pairs,
)o
(30)
otherwise.
For D (see the top row in Figure
5 this can rewritten Maris & Kadanoff 1978) as
Z
Σ
(
(31)
$2.81.0a...
Σ
((st+to)) + +(()+ (((s++s)) + K(sg+
(32)
where K -BJ. Here, we have separated the sum over the even and odd spins in Eq. 3 31 and
explicitly performed the sum over the even spins in Eq. (32). Since s, (-1, it holds that
k(s + k(n
f(K) $1
(33)
where
f(K) 2/cohh(2K)
K² In(cosh(2K)))
(34)
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Markov Blanket: RiGCS
Markov Blanket: MLMC-HB
Markov Blanket: True
sampled sites
sampled sites condition sifes
sampled sites
condition sites
Site partitionings based on the block-spin transformation (adanofff 1966). This figure is similar to, yet more accurate than, Figure [T] Specifically, the only difference is in the true Markov blanket (yellow shadow) of the second finest level in the 2D case (bottom row in the middle). While Figure Tis for explaining that the true Markov blankets of the intermediate levels-which in general unknown- possibly covers the entire lattice, this figure faithfully shows the Markov blanket for the true renormalized Hamiltonian (36 at the second finest level.
Figure 5:
Substituting Eq.
33
into Eq. 32 one finds
Z f(K)
Σ
exp(K's
~ Σ exp (-8HL-1(8 S1. -K' -k'18))
(35)
where
-J)
is the nearest-neighbor renormalized Hamiltonian. This demonstrates that the partition function of the system on the fine lattice is related to the one coarser lattice described by the same type of Hamiltonian with K" different from K = -BJ. Moreover, looking at Eq. (34), the only fixed points K in the recursion relation for the renormalized couplings
the trivial ones, i.e., K"* 0,
corresponding to the temperature 7' = B-1 + oo, where the system is in the paramagnetic phase, and K 0o, corresponding to T' 0, where the system is in the ferromagnetic phase.
For D = 2 (see the bottom row in Figure 5). one can follow similar approach by marginalizing at each iteration over the even (or odd) degrees of freedom in "checker board" pattern (see also Fig- ure 6). After a single step of the procedure, one obtains the following partition function (Maris & Kadanofi 1978):
Z = N(K N/2
Σ exp L-6-1
K
Σ 818j + K2 (v)
Σ 848, + K3
Σ 0
81858p8t
(36)
((0))
where (ij) corresponds the nearest-neighhors on the lattice after summing over one sublattice, ((i))) to the spins on next nearest-neighhor sites, O indicates the spins plaquette of the coarser lattice, and
1 In (cosh (2K)). Ki In (cosh (4K)), K2 - In (cosh (4K)), K3 In (cosh (4K)) -
1
Note that in this case, the partition function is not given by the exponential of the same type of Hamiltonian as the original model just with different parameters on a coarser lattice. This is illus- trated in Figure as as the "true Markov blanket" (yellow shadow) of the second finest level lattice in
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Figure 6: Illustration of the Kadanoff block spin method on square lattice, the spheres indicate the spins and the solid black lines the original lattice. After summing over the configurations of the orange spins, one obtains renormalized Hamiltonian for the blue spins on square lattice that is tilted by 45 relative to the original lattice (indicated by the grey dashed lines). The resulting Hamiltonian on the dashed lattice contains nearest-neighbor interactions, interactions of all four spins along square as well as next nearest-neighbor interactions along the diagonal of the squares.
the 2D case (the middle figure in the bottom row). Continuing this procedure, one would generate various long-range and multi-body interactions in the renormalized Hamiltonian, which is captured in the Hamiltonian in Eq. (9).
A practical example of RG flow in machine learning is the application of CNNs to a classification problem (Lin et al.) 2017). CNNs perform form of coarse- maars--raiiing, where successive convolu- tional layers progressively filter out microscopic noise (irrelevant operators) and isolate high-level features (relevant operators) essential for distinguishing the target classes. The latter example can be tested by training CNN to classify the phase (ferromagnetic or paramagnetic) of the Ising model. As shown in Carrasquilla & Melko (017), the output such a CNN is strongly correlated with the magnetization, indicating that both neural networks and the RG flow capture the same key param- eter mage-naanetizttonn as a relevant feature to characterize the phase transition in the Ising model. Interestingly, a similar behavior was observed in [Alexandrou al. (2020), where the latent repre- sentation of an autoencoder trained on Ising configurations was studied.
D MULTILEVEL MONTE CARLO WITH HEAT BATH (MLMC-HB) ALGORITHM
With the site partitioning (see Figure T based the block-spin transformations MLMC-HB performs ancestral sampling, according to Eq. (8) i.e.
[aaannofi 1966).
NN (8) ( (TI/= N (8)1 g,))
(s"),
which approximates the target distribution
p(0) = p(=l p(00).
Here,
q NN (o = (II g NN -15") uN (80)
for lorl= = o...., L approximates the true marginal distribution (4) i.e.
p(s = p(s)D[s>>]] = 1/2 -BHi(
with nearest-neighhor (NN) interaction Hamiltonians, i.e..
NN
+ (a with in(s5)) -8 N
Based on RGT (Appendix C. the interaction coefficients {Ji} (J in Eq. 2 for each NN are com- puted by the following recursive equation in the two-dimensional Ising model (Maris & Kadanoff #1_1
1 for lattice spacing a; /2 i.e., 45 degree rotated lattice,
3 log log(cosh(44K))
K
(37)
8
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where K; = 3J for L, with Jn J.
Thanks to the site partitioning and the approximation with NN interactions, the conditional sampling probability can be fully decomposed into independent distributions
q NN ( ا = -I q NÑ (s
(38)
Sampling from the independent distribution NN (5-1) can be easily performed by the heat bath (HB) algorithm: for the Ising models, each site o 1- follows
(see the Markov blankets shown Figure D
exp Sy Eo (J Je S1-1)
q NN ( exp EE (J S. () + exp
Je,ors st
and can therefore be sampled by using probability versus state space table. This is efficient because each site has only 2D nearest neighbors. When sampling coarser levels, where the nearest-neight neighhr interactions are governed by the recursion relation
(37. the sampling distribution can be tuned by
adjusting Jj to improve the performance.
E ALGORITHMIC DETAILS OF RIGCS
In this section, we provide a detailed description of the specialized sequential training with model transfer for RiGCS. Furthermore, also provide pseudocode for both training and sampling.
E.1 DETAILS OF SEQUENTIAL TRAINING WITH MODEL TRANSFER INITIALIZATION
As mentioned in Section 3.3 we train our RiGCS by minimizing the reverse Kullback-Leein (KL) divergence (TT), which can suffer from long initial random walking steps if the training parameters 0 are not well initialized, c.g., by randomly initialization. This is because randomly initialized RiGCS, (s), generates random samples, for which the stochastic gradient of the objective (11) rarely provides useful signal to train the model for large lattice system. We tackle this problem with a specialized training procedure for RiGCS using model transfer, again based on RGT.
We choose L be even number, and consider tions (s x c-A L' = 2, L). where (H are the approximate Hamiltonians with NN interactions, defined in (5. and (s H(s). Let consider the corresponding set of RiGCS models {go, (s²), (9 (s (1 L' 2, 2....... L}} that share the same coarsest lattice size Vo. We train each RiGCS model to the sequential targets in the increasing order L', namely,
of sequential target Boltzmann distribu-
At level the O-layered RiGCS (plain VAN) qa, (so) is trained on Po(s") x e -B.((())
At level 2, the 2-layered RiGCS qe (s52) = p < )oe, (s*1s") Seggee(e (00)
trained on P2(sS²) oc ee (Ha(ss5)
At level 4, the 4-layered RiGCS (0 (854) = p50
II go (8 ((|1s5-11)
9o, (80)
trained on (1 xx e -8i
1--1
At level L', the L'-layered RiGCS 40 90 ($<1) (
p(s (
II
20, (s)1))
qo, (sO)
is trained on pr:(eSh)
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h-1
At level L, the L-layered RiGCS (0=z_1 (sgL) = (8--1)))11)
II
2o (s (1115-11))
100 (80)
is trained on (r
S & (BB8)
Figure 2 iillsttrae this procedure in the case of = 6 for the 16 x 16 lattice, where the parameters {0t} be trained are explicitly shown.
Now, assume that scale invariance at criticality (SIC) holds approximately. Then, the renormal- ized Hamiltonian should be well approximated with NN interactions, i.e., Hi(s 2 Hi (see Eq. 5). This means that the sequential target distributions (s 0,2.4.... ................................ L} have similar marginal distributions on the sites 8S (for 1 I I.). Therefore, once the parameters (0:) of the (L' 22-llyeredd RiGCS have been trained on the corresponding target PL' P1 5²'), they represent suitable initializations for the corresponding components of the L'-layered RiGCS to be ( (s This justifies the model transfer initializations depicted as the vertical red arrows in Figure 2 Furthermore, SIC-stating that the interaction terms in the renormalized Hamiltonians ( }} quickly converge to fixed point for
trained on the next level, i.e., for the target distribution Pr
1 < T. with some T. < I-iiplies that the renormalized Hamiltonians for different scales, e.g. (i-2(sS1-2) and Hi(sS should consist of similar sets of interaction terms. Therefore, thanks to our choice of using the same architecture for all conditional models over different levels, we can also apply model transfer initializations from 0 to 0 as depicted as the diagonal red arrows in Figure
In summary, our sequential training with model transfer follows the following procedure:
1. Train the (unconditional) generative model (8,()) to approximate ((s0) x e -a with JNN. Set On Oo-
2. Refine Ogi from its initial value ش=1 = (01, 0o). where 0 is set randomly, by training yoo (s* (a))),,("" to approximate p(s = Set
90.i (8g2)
0-1 + 0g1.
3. Refine Oga from its initial value 0g3 (01, 01, 0n). where O2 is randomly, by train-
ing 90 (854) p(8 [M 20, (s" 18"1"" 111 go, (8") to approximate ((sSt) x
e- -Hale 5t) Set 0g3 - gg
4. For L' 6, 8, .... refine 0z1 from its initial value 0 10 0. -a)
by training 90 90g ($21. = 11 SL' -1) (T go, (s" (511) 90, (s") to approxi-
=t) mate "(855') x - (" Set 0 + -0
Note that all parameters except Og. 0, which are trained on the three smallest lattice sizes with random initializations can be initialized to the parameters trained on casier problems (smaller lattice), which significantly accelerates the training process, as shown in Figure (right). For large L and & L. the approximate renormalized Hamiltonian (s!) with NN interactions might be significantly different from the true renormalized Hamiltonian (sS)) that may have long-range and higher-order interaction terms. Crucially, our training procedure helps reducing this gap step by step by fine-tuning the generative models with receptive fields extending beyond nearest neighbors.
E.2 PSEUDOCODE FOR RIGCS
The pseudocodes provided in Algorithm I and Algorithm 2 describe the practical steps for training RiGCS and for sampling from a trained RIGCS, respectively.
F AUTOREGRESSIVE NEURAL NETWORKS
Autoregressive neural networks are a class of generative models, where the elements of random configurations ordered, and each element is sampled, depending only on the previous elements.
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Algorithm RiGCS training
1: Input:
Coarsest lattice size No
PixcICNN qo,
numbers of levels L
Conditional networks {go, (s'1 ((=1s))-1))
HB algorithm p(s'|s" |s
2: Output:
Trained RiGCS (PixeICNN-based generative model): list of conditional models for sampling sh E SD with D No x No
3: Train the PixeICNN qe, No No target lattices.
4: Add to the RiGCS's list of models the conditional VAN qo, (s* (randomly initialized) and the HB p( )
5: Train the RiGCS on 2Ng x 2No lattices.
6: Replace the HB p(s² s with the conditional VAN qo, (s² S randomly initialized.
7: Add to the RiGCS's list of models the conditional VAN yog (s 222) initialized with the weights of the trained model qø and the HB p(s ($))
8: Train the RiGCS on 4Ng x 4N2 lattices.
9: for 5, L. + do
10: Replace the HB
with the conditional VAN qo (8' (//-118)-22 initialized with the trained model qø weights.
11: Add to the RiGCS's list of models the conditional VAN qo, (s1|)5111)) initialized with the trained model ج weights, and HB ((s/+1185))
12: Train the RiGCS on lattices of size 2 x 2
13: end for
return List (trained) conditional models {go, (s"|85(-1))
14:
These models are widely used in time series forecasting (Triebe et al.) 2019). natural language processing (yan den Oord et al. 2016a), large language models (Brown et al.) 2020), and generative modeling (van den Oord al. 2016b) they explicitly capture the dependencies between elements in sequence.
In the last decades, autoregressive neural networks have been extensively deployed in different sci-
entific domains including statistical physics (Wu et al.) 2019 Nicoli et al., 2020 Wang et al. 2022
Biazzo 2023 Biazzo et al., 2024). quantum chemistry (Gebauer al. 2019 Joshi al.
2021
Gebauer et al., 2022). learning wave functions of many body systems Hibat- et al.
2020
tensor newtorks (Chen al. 2023) and quantum computing (Liu et al. 2021).
Relying on the factorizability of arbitrary distributions as
V II
p(s)=
p(sn sp-1)..., p(si),
(39)
autoregressive models approximate each factor in the right-hand side with neural network models ge with the parameters 0 to be optimized so that (s(s) x p(s). The ancestral sampling in the order of $1...., sy allows sampling and density evaluation at the same time. State-of the-hhe-tttt art architectures often use convolutional neural networks, leveraging masked filters ensure that the conditional de- pendencies are restricted to previous elements in the sequence (van den Oord et al. 2016b Salimans
al. 2017
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Algorithm 2 RiGCS sampling
1: Input:
Coarsest lattice size No
PixeICNN qo,
List of conditional models {go, (s'|sSt-1)]
Heatbath algorithm for sampling the finest level La (81-11)
2:
Output:
Samples: sf S with D 2d No x 2dN Log-prob: Exact sampling probability qo (s*).
3: Sample 8 qe, and compute qo, (80).
4: for - 1. [ L, - 1 do
5: Embed the sample s/ into 2N x 2N--1 with zeros in the lattice sites of the levels 1,
6:
Sample 81 ~ yo, (8 (1-1) and compute In qo, (81).
7:
i 1 * L then
Sample 8141 ~ qo._ (s' ( and compute In q (8 11) 9: else
8:
10: Sample sh ~ 1(14|11-11) with HB and compute go (o(
11: end if
12:
end for
return 81, In (s+)
G IMPLEMENTATION DETAILS FOR VAN, HAN AND RIGCS
G.1 VAN
Two possible architectures often used for implementing VAN-like networks (Wu et al. 2019 are Masked Autoencoder for Distribution Estimation (MADE) (Germain et al. 2015) and Pixel- CNN (van den Oord al.] 2016c|b) which rely on fully-connected and convolutional layers, re- spectively. In order ensure the autoregressive properties, the weights of these architectures are masked (see Figure 1 in van den Oord et al. (2016b)) so that v-th entry of the output 8, = go of the network go depends only on the previous values <<v (Su-1...., i.c.
3, go (.c)).
With the sigmoid function used in the last layer of the network go (8<x),, the output bounded as S. (0, 1), which expresses the probability that the entry has the positive spin S. +1. Namely, the corresponding entry is drawn from the following Bernoulli distribution:
go (s.18ccc)) S.0. (1 (1-35)) 8.).
(40)
In our experiments, we used the PixelCNN-based JAN-implemented by Wu et al. (2019 which leverages masked convolutional kernels of the size of K x K for odd K. Let M {0, K be the mask for the kernel. Noting that (k, k') ((K 1)/2 + 1, (K - 1)/2 + corresponds to the center the kernel, setting the mask such that
{1 MR to
if k (K 1)/2 + 1 [[k (K - 1)/2 + 1] A I < (K - 1 I]], otherwise
ensures the autoregressive properties. When more than two layers used, PixeICNN uses residual connections (He et al. (016)) (i.c., the original input to the layer summed to the output) for each layer, except for the first and the last layers. Within the residual connections before each masked convolutional layer, as well as at the end of the network before the sigmoid function, a standard convolutional layer with the kernel size of 1 x is added.
The PixeICNN, used in our experiments as the plain VAN, has masked convolutional layers with 32 channels and the kernel size of K 13.
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G.2 HAN
The HAN (Biatas et al. 2022) model leverages recursive domain decomposition (Cè et al. 2016) in order to sample in parallel different regions of the lattice configurations. The crucial aspect of the domain decomposition is that the domains must be connected through a common boundary, and, once it is given, each domain can be sampled independently by using the same model. In the HAN implementation, a boundary By that divides the lattice into four domains is first sampled by using a standard MADE architecture. Then, each domain further split into four by sampling in parallel four boundaries }B;} using another MADE model conditioned on the boundary Bo. This procedure repeated until the remaining entries have all the neighbors fixed and therefore can be sampled by HB.
In our experiments, where HAN is evaluated as baseline, we used the code and the hyperparameters provided by the authors of Biakas et al. (2022).
G.3 RIGCS
In our implementation of RiGCS, the (unconditional) generative model (80) at the coarsest level I 0, consists of VAN with 3 masked convolutional layers with 12 kernels of the size of K 13.
For the conditional models (go, (s')s (st which correspond to upsampling from lattice size N; x N2 to Ni42 x N142 = 2N, x 2N, for L- 2). Each block takes input coarser configuration 8 of size Nr N₁ and embeds it into a 2N, x 2N, configuration where all unsampled entries for the levels + 1 and + 2 are set to O. Then, the spins s and s are sampled sequentially according to the output of a standard CNN that takes as input the embedded configuration. Each conditional CNN (conditional VAN) of RiGCS has one hidden layer with 12 kernels of the sizes of K and K = 3, respectively, for the hidden and output layers, making its receptive field size 7 x 7. Within each block, we use the same CNN architecture for both levels. The same architecture applies all blocks except the last. In the last level, i.e., I = L, RiGCS performs the HB sampling because the target Hamiltonian has only nearest-neighbor interactions by assumption.
define one "block" nwo consecutive levels,
I {0, 2, 4,
Similarly to the VAN (see Appendix G. b. the conditional VAN uses a sigmoid activation in the final layer, such that the output is bounded as ā, E (0, 1). With this output, the spin is drawn from the Bernoulli distribution:
q0 |s (o.
3 +1 + (1 30)5
During the training procedure described Section 3.3 the weights in each block are initialized with those of the conditional VANs in the coarser block.
G.4 TRAINING
All the generative neural samplers (RiGCS, VAN, HAN) used in our experiments are trained by minimizing the reverse Kullback-Leible (KL) divergence
min (L
(41)
with the gradient estimator
(oKK(g()))lp(ss) Egr-go [(HH()) log go (s)) Vo log [s)]
In order to make the variance of the estimator more stable, we leverage control variates method (Mnih & Gregor 2014) as suggested in Wu et al. (019)) We use the ADAM opti- mizer (Kingma & Bal 2015) with learning rate 0.001 and standard 3s for training all models.
We trained VANs for 50000 gradient updates (steps) with batch size 100, and HANs for 100000 gradient updates with batch size 1000. For RiGCS, training is performed for a total of 3000 steps for each sequential (upscaled) target lattice. When training on target lattice N₂ = N, the pretraining phase involves training at coarser levels as follows: 2000 steps for level L - 2. 1500 steps for level 4, and 1000 steps for all previous levels, except for the coarsest one which is always trained for 500 steps.
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0.3
MEMG-R
VAN
(6' -Gu)
0055
[mn] (mnos///mm mmmss
RAN
RIOOS
-0.04
-0.06
-0.08
VAN
MIMMC-BB
12
0.1
-CANCE
-0.10-
16
32
64
Figure 7: Relative estimation error for the internal energy (left) and absolute magnetization (right). The vanilla VAN cannot be trained for N 2 64 in reasonable time. Note that, unlike in Figure (left), we used the MC estimators by the Cluster-AIS Uets and mels the reference values, as the ground truth cannot be computed analytically.
MIMC-HB
D
Traniing ttme hour
10
VAN
10
Sompling time
10/
TEGCS
10"
10-1
10
HAN
10
8
16
32
64
128
16
32
64
128
Figure 8: Total training time (left) and sampling time (right) for different lattice sizes.
G.5 SAMPLING
We computed the MC estimates with million samples for the Cluster(-AIS), HAN and RiGCS.
For VAN and MLMC-HB we used 100k samples. For the largest lattice considered in this work, i... 128 x 128, Cluster-AIS sampled only 500k configurations due its high computational costs. Errors are estimated using an automatic differentiation method introduced in Ramos (2019) and implemented Joswig et al. (2023).
H ADDITIONAL NUMERICAL RESULTS
NON-THERMODYNAMER OBSERVABLE ESTIMATION
Figure 7 shows additional numerical results of estimating the internal energy and the magnetization, where the estimators by the cluster method are used as the reference values. Consistently with EMDM and ESS shown in Figure 4 our RiGCS provides unbiased estimates with lowest variances compared to other generative neural samplers.
H.2 COMPUTATION TIME
Figure shows empirical training (left) and sampling (right) time. For all models (RiGCS and the baselines), we used a single NVIDIA A100 GPU with 80 GB of memory.
H.3 ABLATION STUDY: TRANSFORMER ARCHITECTURE FOR CONDITIONAL VAN
In order to investigate the impact different neural network architectures for the conditional VAN used in RiGCS, compared the performances between CNN- and transformer-basee neural net- works. For the latter, we replaced the CNN of our proposed implementation with transformer (encoder only) with 5 multi-head attention block (8 heads) and 1024 neurons in the feed-forward
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1.0
0.8
0.6
BSS
0.4-
0.2-
Tronsfommr
CNN
10000
20000
30000
steps
Figure 9: Comparison between CNN-based RiGCS (purple) and transformer- based RiGCS (or- ange). Both CNN and transformer refer to the type of neural network architectures used for the conditional models in RiGCS, Both models use our proposed warm-start technique for training for the target (finest) lattice size of 16 x 16.
networks. However, for the unconditional VAN the coarsest level O, we kept using the Pix- cICNN. Figure Setrre@diipin displays the results of sequential training with model transfer up to lattices of size 16 x 16. Note that, unlike in Figure 3 (right), the plot shows all training procedure including the pretraining phase. Specifically, the initial phase (steps: to 1001) corresponds to training of the un- conditional VAN at / = 0, the second (steps: 1002 to 1 11002) and the third (steps: 11003 to 21003) phases involve training of the conditional VANs at 1,2, respectively, and the last phase (steps: 21004 to the end of training) corresponds to training of the full RiGCS for the target lattice of size 16 16.
We observe that, except the initial phase, where both RiGCS models use the same unconditional VAN with PixeICNN, RiGCS with CNN (purple) clearly outperforms RiGCS with Transformer (orange). This result is consistent with recent work (Abbott et al. 2023 Liu et al.) 2024). where transformer- based architectures do not appear be well-suited for lattice simulations of physical systems, despite their great success across various domains. The reasons for this remain to be theoretically understood.
H.4 PRELIMINARY EXPERIMENT FOR LATTICE QUANTUM FIELD THEORIES
To further validate our approach, we tested RiGCS on another popular benchmark in computational physics: the o² scalar field theory. This field theory, when discretized on lattice, serves as an important benchmark for validating sampling algorithms. In fact, it has been extensively studied the literature (Albergo et al.) 2019; Nicoli et al.) 202T) for benchmarking the performance of generative models (e.g., normalizing flows) with continuous degrees of freedom. The gi field theory belongs to the same universality class as the Ising model (i.e., as discussed in Appendix C they share toy model in the 2-dimensional lattice,
the same critical exponents). Despite being recognized the o² theory is of high-relevance to physicists for the following reasons. In the standard model of particle physics, the field of the Higgs boson possesses the same type of interaction described by the ø² lattice field theory. Moreover, the o² scalar field theory exhibits spontaneous symmetry breaking of the Za invariance, which property central the Higgs mechanism well.
Configurations of the scalar field theory are sampled from the Boltzmann distribution
p(0) e -S
7
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1.00
VAN RIGCS
0.75
+
ESS
0.50
0.25
0.00
4
8 16
32
Figure 10: Effective Sample Size (ESS) for different lattice sizes for the Ø² theory, after 5k steps of training for N (4, 8, 16) and 10k for N 32. Blue squares and magenta circles refer to the plain VAN (baseline) and RiGCS (ours), respectively. The plain VAN can be trained in reasonable time only up 16 16 lattices, and RiGCS consistently achieves higher ESS than VAN.
where Z is the unknown partition function, and S(Gb is the action-a functional describing the physics of the system in the Lagrangian formalism. For the ot theory, the action reads
Sløl ||ø|=> Σ
2k Σ
uu 2X)Ø + AO!
(42)
where k is the hopping parameter controlling the nearest-neighbor interactions, and À is the bare coupling parameter determining the strength of the quartic self-interactions. The index μ labels the basis vectors in the D D-dimensional space.
In the limit of À oc, the self-interaction terms enforce de > =1, effectively reducing the field to discrete spin values 8, = =1 (Vierhaus) 2010; Wolff 2009). The resulting effective action in this limit corresponds to the ferromagnetic Ising Hamiltonian
Sgls] (s) -3 Σ SuSa' for 8 = 2k.
(43)
(o.e*)
In the limit of - 0, the quartic interaction vanishes, and the theory reduces to free (Gaussian) field theory. We refer to Vierhaus (2010); Wolff (2099) for further details on the connection between the Ising model and the o² theory.
The architecture of RiGCS used in our preliminary experiment for simulating the di theory closely follows that for the Ising model used in the main paper, with few key differences outlined below. As in the Ising case, we use PixeICNN (plain VAN) at the coarsest level, and conditional CNNs the intermediate levels. However, unlike for the Ising model, we also use the conditional CNN, instead of the HB sampler, at the finest level. For sampling continuous variables, we use Gaussian Mixture Models (GMMs), following the approach by Faraz et al. (2025). Specifically, at level GMM represents the conditional probability density for new site o,, in the configuration as
qo, (o. cv:
Σ h=1
The N.Holehh of
Here, T/,,, T &e 01), whe (n (0 Pes 01), and oh_-- on (Φ) <. (51; 01) neural networks, which parameterized with 0 and represent the mixing coefficient, the mean, and the standard deviation of the h-th Gaussian component, respectively. We set H =
We perform sequential training similar to the Ising model case with minor modification: since we use conditional CNN at the finest level. all HB blocks in Figure [ are replaced with conditional CNNs parameterized by 0z. More specifically, the HB block level is replaced with Og which
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initialized randomly. The trained 02 is transferred to 02 and 0₂ (substituted for HB) at Level 4. We continue this model transfer to Level and higher.
We conducted preliminary experiments for comparing the performance of VAN and RiGCS in sam- pling configurations for the o$ theory at the critical point. We evaluated the Effective Sample Size (ESS) for lattices up 32 32, and plotted the results in Figure [O We observe that RiGCS again consistently outperforms the plain VAN (Wu et al. 2019), always converging to higher ESS. Furthermore, for lattices larger than 32 32, the plain VAN could not be trained due to excessive computational costs, while RiGCS can be trained with reasonable computational costs. This further highlights the superior performance and efficiency of RiGCS. We note that the VAN architecture which is particularly suited for sampling discrete variables- is not necessarily an optimal choice for sampling continuous variables. Exploring RiGCS with more suitable hase generative samplers, e.g., normalizing flows, for continuous systems is promising direction future research.
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Reviews:
[{"review_id": "N7tvplKnsw", "reviewer": "Reviewer_DmX4", "summary": "The paper presents a novel approach, the Renormalization-informed Generative Critical Sampler (RiGCS), designed for efficient sampling in near-critical regimes, leveraging the concept of scale invariance at criticality (SIC). By integrating MultiLevel Monte Carlo (MLMC) with Heat Bath (HB) algorithms, the authors aim to improve sampling efficiency while addressing challenges associated with critical slowing down in Markov chain Monte Carlo (MCMC) methods.", "questions": "(1) Provide a detailed algorithm or flowchart that outlines the key steps of RiGCS, including how training proceeds at each resolution level. This would help clarify the simultaneous sampling and training processes.\n\n(2) Address whether RiGCS is susceptible to the mode-seeking, if so, how they might mitigate them.\n\n(3) Compare the proposed approach to methods like the Neural Network Renormalization Group, highlighting potential advantages or trade-offs of RiGCS in relation to these continuous relaxation based techniques.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The proposed method utilizes SIC effectively, enabling ancestral sampling from low-resolution to high-resolution lattice configurations with site-wise-independent conditional HB sampling. This innovation enhances both scalability and efficiency, making it a significant contribution to the field of generative models.", "weaknesses": "(1) It appears that the training of the algorithm at each resolution follows a simultaneous sampling and training paradigm. However, this is not clearly articulated in the manuscript. I recommend providing complete pseudocode for the algorithm, which would greatly aid readers in understanding the implementation and workflow.\n\n(2) The loss function presented in equation (11) is known to have a mode-seeking tendency when training generative models. This could lead to the neglect of certain peaks in multimodal distributions. It would be beneficial for the authors to address whether RiGCS is susceptible to similar issues and, if so, how they might mitigate them.\n\n(3) The paper does not mention an important class of methods for the Ising model that rely on relaxation to continuous variable representations. Approaches such as the Neural Network Renormalization Group by Li and Wang use this relaxation technique to enable generative sampling in a continuous variable space. Including a discussion of these relaxation-based methods would provide a more comprehensive comparison and allow readers to better appreciate where RiGCS stands in relation to existing methods.", "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": "wyWBgSUq97", "reviewer": "Reviewer_cr5x", "summary": "This paper deals with the problem of sampling systems at criticality, i.e. systems typically displaying some form of self-similarity w.r.t change of scale, and estimating their associated thermodynamical quantities like the free energy. This is a notoriously difficult problem of interest in physics, because it requires to deal with arbitrary long range correlations and for which standard MC samplers based on local moves undergo-critical slowing down, meaning that the auto-correlation time become arbitrarily large with system size. As explained in the introduction, there exists some specific algorithms in the physics literature, like cluster methods specifically designed for lattice models,\ntypically working for discrete or continuous variables (Ising, Potts, O(n) ...). In this work, the authors propose to leverage machine learning methods to generalized some existing multi-scale methods following closely the spatial RG procedures, which basically exploits statistical dependencies between coarse grained variables at different scales. These use rather drastic approximations on the conditional models between coarse grained variables at different scales, leading sometime to very low acceptance rate, and ML methods offer the possibility to used more complex conditional models between scales.", "questions": "If the author could comment on the points raised in the previous paragraph:\nin particular\n- How does the algorithm scale  system size (even empirically) ?\n- Is there a key difference of RIGCS wrt   the model developed by Marchand et al. (Wavelet conditional renormalization group)?\n\n\nMinor points:\n\nInstead of what is written on p4, the Wolff algorithm seems also to work for continuous variables like O(n) variables as can be seen  for instance in Kent-Dobia, Sethna PRE (2018)\n\nI had difficulties to understand the notations in the begining, concerning in particular the definition of the hierarchy of variables.  I am not sure how I should understand the last sentence of p4 for instance, whether (i) there are coarse grained variables which are defined hierarchically or whether (ii) the original variables are organized hierarchically depending on their position on the lattice as in the Schmidt paper (I believe it is (ii) after reading it)\n\nbibliography could be ordered in alphabetic order (to help old school readers using printed version:) )", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "the paper is well written and rather clear with an impressive bibliography on the subject which is indeed quite vast and old. A synthetic and pedagogical section in the supplementary material which should be useful to non-physicists is devoted to the RG theory, and should make the whole paper readable for non-physicists. The proposed algorithm is quite generic and and is competitive with the Wolff algorithm on 2d lattices up to L=128, where it seems to actually yield better estimates of the free energy. and shows clear advantage w.r.t MLMC-HB (used in it orginal version maybe outdated, as it use the original  Kadanov block-spin  transformation, subsequent refined algorithms cited in the text are not considered for  baselines) and other baselines considered like HAN which uses a similar idea, in the form of an auto-regressive\nNN based on a hierarchical representation of the lattice.", "weaknesses": "I found some aspects of the experimental not completely satisfactory. The claim that RiGCS performs better than Wolff is a strong claim that should be better sustained in my opinion. Authors concentrate in the experiments on the computation of the free energy, but characterization of physical properties involve many other possible indicators, which the  Wolff algorithm is known to directly compute very efficiently,\nlike the magnetization, susceptibility, Binder parameters which could be easily displayed without further work it seems ...\nsome exponents could also be extracted by finite size scaling and compared to exact ones. The combination of Wolff with AIS even though reasonable seems a bit like of an ad-hoc procedure to yield a baseline comparison for the computation of the free energy, I would expect that integrating the mean energy with respect to beta or simply using directly the empirical distribution of energy delivered by the Wolff algorithm would yield more robust estimation but I maybe wrong? The size itself (128x128) seems rather limited compared to what can commonly be achieved with the Wolff algorithm. It seems to be a limitation in term of computation time but this is not discussed. Actually the scaling of the the computational time  of RiGCS w.r.t to L is actually not given. Another possible weak  point is that the proposal of the paper is very similar in spirit with another recent one which also uses a top-down regressive model  in scales (also explicitly using  RG concepts), in combination with wavelets (Marchand et al. 2022). While this work is mentioned in the paper, it is not really discussed, no element of comparison or even qualitative elements describing advantages of  RiGCS over this one are being given.  Finally the experiments concentrate on 2D Ising paradigm, a larger spectrum of experiments, on models with continuous variables in particular, would be beneficial to convince the reader of the flexibility of the method.\n\nDespite these reserves, I find the method quite appealing due to its simplicity and the performances which are displayed.", "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": "vHBMlr8rDM", "reviewer": "Reviewer_1WHN", "summary": "In this paper, the authors present RiGCS, a method to sample from Boltzmann distributions in systems with critical phenomena such as phase transitions. This is challenging due to the problem of scale invariance at criticality, in which the correlation length between sites on a lattice diverges. This leads to a high rejection rate in standard Markov Chain Monte Carlo (MCMC) algorithms which mainly focus on local correlations to propose new configurations. The authors also state that this poses a challenge for generative models due to the need for large receptive fields. The proposed approach breaks down the system lattice into a hierarchy of resolutions, essentially factoring the probability distribution at the finest level into a product of distributions each conditioned on a coarser level. Each distribution is parameterized by autoregressive neural networks (PixelCNN) which are learned sequentially, and whose parameters are used to warm-start the next level in the hierarchy. The authors demonstrate that their method is able to generate samples which estimate free energies more accurately than the baselines, as well as have a higher effective sample size due to lower correlation between samples.", "questions": "1. Presenting the proposed method as an alternative to MCMC sampling feels confusing to me. Doesn't the proposed method generate i.i.d samples from the Boltzmann distribution? As opposed to MCMC approaches which produce correlated samples via a Markov Chain? Unless I am misunderstanding something, I think it would help to clarify this point.\n\n2. How is the training data obtained for optimizing the KL divergence loss? Is it from long MCMC simulations? I think this is a critical point that was not focused upon enough in the paper, unless I missed something.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The paper tackles an important and challenging problem in physics (critical phenomena/phase transitions), and one that has received comparitively less attention from the AI for Science community, which is largely still focused on equilibrium phenomena. Thus, I think the topic/paper would be of interest to many in the community. Additionally, the proposed approach (hierarchical sampling at multiple resolutions) is interesting and solidly grounded in well-known techniques from scientific computing such as multigrid algorithms.", "weaknesses": "My primary concerns center around the \"receptive field\" argument, and the quantity and quality of the experimental results. I will elaborate further. \n\nReceptive field:\n\nThere is a large and growing body of literature on long-context models in the fields of language and computer vision [1, 2, 3, 4, 5]. The goal of these works is to provide models with more global reasoning capabilities over, e.g. extremely long textual or video contexts, and/or to reason over extremely high-resolution images, while remaining memory and compute-efficient. The central problem tackled by this paper - diverging correlation lengths in critical systems require large receptive fields - is related to this line of work. The proposed solution - a hierarchical decomposition of the lattice - is also related to the scale hierarchies proposed in these works.\n\nMy first criticism is that the authors do not mention these works or position their contribution in the backdrop of this existing work. Can the authors comment on how their problem setting is substantively different from what is tackled in these existing works, and/or why the approaches from this line of work cannot be applied \"zero-shot\" to this application domain? My next and primary criticism is that given the current state-of-the-art in long context generative modeling, the proposed method is behind the curve, and the size of systems considered in this work (at most $128^2$) is not a sufficiently challenging setting to warrant the proposed approach. First of all, the authors use convolutional neural networks, which have the inherent limitation of a local receptive field. Can the authors justify this choice? Transformers have a global receptive field at the expense of an increased computational/memory footprint, but for 128x128 systems this shouldn't pose an issue. As far as I could tell, there was no mention of the Transformer architecture in the paper. Even if you consider much larger lattices (~$100^4$) used in e.g. lattice gauge theories and quantum chromodynamics, this is still comparable to the number of pixels in, e.g. satellite imagery data, for which solutions to reason over long context have been proposed [5]. I understand that there is not a one-to-one mapping between modeling critical phenomena and long context image/video models, but I think this disparity is telling and worth pointing out. \n\nThe choice to use an autoregressive model (PixelCNN) for sampling individual states also seems dubious to me. There is nothing inherently causal about the lattice sites, and so it seems it would make more sense to employ a non-autoregressive generative paradigm, such as diffusion. By parameterizing a diffusion model with e.g. a Transformer with bidirectional/unmasked attention, a global receptive field is naturally obtained at every stage of the generation process. I understand the binary nature of spin systems poses a slight hurdle, but one that is seemingly surmountable given the growing body of work on discrete diffusion [6] (not to mention more naive solutions like quantizing continuous predictions, clipping, binning, etc.). Plus, the authors mention that their approach is valid for continuous systems, so this seems like a natural choice (non autoregressive approaches are the method of choice for continuous data modalities).\n\nQuantity/quality of experimental results:\n\nGiven my previous comments, I feel that the amount/nature of the experimental validation of the proposed method is too narrow. The evaluation is restricted to 2D Ising systems with size at most $128^2$. Can the authors include results on larger systems where, e.g. using a global receptive field is actually out of the question due to the memory constraints?\n\nAdditionally, the improvements in free energy estimation and EMDM shown in Figure 2 and 3 appear to be very marginal. The cluster method baseline significantly outperforms RiGCS on effective sample size - the authors claim that it is not applicable to continuous settings, but they do not demonstrate results in these settings either.\n\nIn summary, in order to reconsider my score, I would like to see considerable evidence that the proposed approach is superior to the many existing ways to achieve global receptive fields in generative models, including empirically on larger and continuous systems.\n \n[1] Child, Rewon, et al. \"Generating long sequences with sparse transformers.\" arXiv preprint arXiv:1904.10509 (2019).\n\n[2] Liu, Ze, et al. \"Swin transformer: Hierarchical vision transformer using shifted windows.\" Proceedings of the IEEE/CVF international conference on computer vision. 2021.\n\n[3] Dao, Tri, et al. \"Flashattention: Fast and memory-efficient exact attention with io-awareness.\" Advances in Neural Information Processing Systems 35 (2022): 16344-16359.\n\n[4] Liu, Hao, Matei Zaharia, and Pieter Abbeel. \"Ring attention with blockwise transformers for near-infinite context.\" arXiv preprint arXiv:2310.01889 (2023).\n\n[5] Gupta, Ritwik, et al. \"xT: Nested Tokenization for Larger Context in Large Images.\" arXiv preprint arXiv:2403.01915 (2024).\n\n[6] Austin, Jacob, et al. \"Structured denoising diffusion models in discrete state-spaces.\" Advances in Neural Information Processing Systems 34 (2021): 17981-17993.", "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": "4hN4ab1OQh", "reviewer": "Reviewer_pv8Q", "summary": "This paper introduces the Renormalization-informed Generative Critical Sampler (RiGCS), a novel generative sampling algorithm designed for near-critical systems where scale-invariant criticality (SIC) can hinder the efficiency of standard Monte Carlo simulations. The proposed RiGCS builds on Multilevel Monte Carlo algorithms with Heat Bath sampling and incorporates generative models to address the shortcomings of HB under slight SIC violations.", "questions": "1. Why is addressing SIC important in this context? While the authors claim that SIC reduces sampling efficiency, they do not clearly explain how RiGCS specifically addresses this issue. A more detailed justification is required.\n2. Although the warm start is listed as a key contribution, the implementation details are unclear. How is the warm start applied across different resolution levels, and why is it considered a significant innovation?\n3. If the authors only employ standard metrics (e.g., effective sample size, free energy bias), it is unclear why performance evaluation is listed as a major contribution. Were any new metrics introduced? If not, the framing of performance evaluation as a contribution should be reconsidered.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The proposed method integrates renormalization group theory with generative models in a novel attempt to address critical slowing down in Monte Carlo simulations.\n2. Employing generative models to enhance MLMC-HB sampling seems to be a promising approach with potential applications in statistical physics and lattice systems.", "weaknesses": "1. The core contributions (lines 90–99) are not sufficiently clear. See Questions 1-3.\n2. The mathematical formulations are not clear. For example, Equation (9) is not well defined and difficult to interpret. The algorithm design between lines 355–369 is unclear, and the lack of theoretical guarantees on convergence or sample complexity weakens the technical rigor.\n3. The experiments are limited in scope. To demonstrate generalizability, the authors should evaluate RiGCS on more complex benchmarks beyond the 2D Ising model.\n4. The appendix introduces unrelated terminologies (e.g., Swendsen-Wang algorithm, Wolff algorithm) without meaningful connections to the core work or further experimental analysis. Moreover, the lack of theoretical support or detailed experimental setups in the appendix limits its usefulness.\n5. The manuscript cites more than 100 references, many of which seem only loosely related to the topic.", "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)