Tuning Sequential Monte Carlo Samplers via Greedy Incremental Divergence Minimization
Authors: Kyurae Kim
Zuheng Xu
Jacob R. Gardner
Trevor Campbell
Venue: ICML 2025 poster (2025)
Keywords: sequantial Monte Carlo
annealed importance sampling
Markov chain Monte Carlo
Bayesian inference
Primary area: probabilistic_methods->monte_carlo_and_sampling_methods
TL;DR: We propose a gradient- and tuning-free online scheme for tuning the proposal kernels in sequential Monte Carlo samplers.

Abstract: The performance of sequential Monte Carlo (SMC) samplers heavily depends on the tuning of the Markov kernels used in the path proposal. For SMC samplers with unadjusted Markov kernels, standard tuning objectives, such as the Metropolis-Hastings acceptance rate or the expected-squared jump distance, are no longer applicable. While stochastic gradient-based end-to-end optimization algorithms have been explored for tuning SMC samplers, they often incur excessive training costs, even for tuning just the kernel step sizes. In this work, we propose a general adaptation framework for tuning the Markov kernels in SMC samplers by minimizing the incremental Kullback-Leibler (KL) divergence between the proposal and target paths. For step size tuning, we provide a gradient- and tuning-free algorithm that is generally applicable for kernels such as Langevin Monte Carlo (LMC). We further demonstrate the utility of our approach by providing a tailored scheme for tuning kinetic LMC used in SMC samplers. Our implementations are able to obtain a full schedule of tuned parameters at the cost of a few vanilla SMC runs, which is a fraction of gradient-based approaches.

Tuning Sequential Monte Carlo Samplers via Greedy Incremental Divergence Minimization
Kyurae Kim 1 Zuheng Xu " Jacob R. Gardner'
Trevor Campbell
Abstract
sure. When tuned well, SMC provides state-of-the-ar per- formance in wide range of modern problem settings, from inference in both state-space models and static models (Dai ct al., 2022; Chopin & Papaspiliopoulos, 2020; Doucet & Johansen, 2011; I; Cappé et al., 2007), to training deep gen- erative models (Arbel et al., 2021: Matthews et al., 2022: Doucet al., 2023; Maddison al., 2017), steering large language models (Zhao et al., 2024; Lew et al., 2023), condi- tional generation from diffusion models (Trippe et al., 2023; Wu et al., 2023), and solving inverse problems with diffu- sion model priors (Cardoso et al., 2024; Dou & Song, 2024; Achituve et al., 2025).
The performance of sequential Monte Carlo (SMC) samplers heavily depends on the tuning of the Markov kernels used in the path proposal. For SMC samplers with unadjusted Markov kernels, standard tuning objectives, such as the Metropolis- Hastings acceptance rate or the expected-squart jump distance, are no longer applicable. While stochastic gradient-based end-to-end optimization has been explored for tuning SMC samplers, they often incur excessive training costs, even for tun- ing just the kernel step sizes. In this work, we propose a general adaptation framework for tun- ing the Markov kernels in SMC samplers by min- imizing the incremental Kullback-Leibh divergence between the proposal and target paths. For step size tuning, provide gradient- and tuning-free algorithm that is generally applicable for kernels such as Langevin Monte Carlo (LMC). We further demonstrate the utility of our approach by providing a tailored scheme for tuning kinetic LMC used in SMC samplers. Our implementa- tions are able to obtain a full schedule of tuned parameters at the cost of few vanilla SMC runs, which is a fraction of gradient-based approaches.
In practice, however, tuning SMC samplers is often sig- nificant challenge. For example, for static models (Chopin, 2002; Del Moral et al., 2006), one must tune the number of steps, number of particles, target distribution, and Markov kernel at each step, as well as criteria for triggering par- ticle resampling. Since the asymptotic variance of SMC samplers is additive over the steps (Del Moral et al., 2006; Gerber et al., 2019; Chopin, 2004; Webber, 2019; Bernton et al., 2019), all of the above must be tuned adequately at all times; an SMC run will not be able to recover from single mistuned step. While multiple methods for adapting the path of intermediate targets have been proposed (Zhou et al., 2016; Syed et al., 2024), especially in the AIS context (Ki- waki, 2015; Goshtasbpour et al., 2023; Masrani et al., 2021; Jasra et al., 201 I), methods and criteria for tuning the path proposal kernels are relatively scarce.
(KL)
1. Introduction
Sequential Monte Carlo (SMC; Dai et al., 2022; Del Moral et al., 2006; Chopin & Papaspiliopoulos, 2020) a general methodology for simulating Feynman-Kan mod- els (Del Moral, 2004; 2016), which describe the evolu- tion of distributions through sequential changes of mea-
Markov kernels commonly used in SMC can be divided into two categories: those of the Metropolis- Metsonolis-Hastings (Metropo- lis et al., 1953; Hastings, 1970) type, commonly referred to as adjusted kernels, and unadjusted kernels. For tuning adjusted kernels, one can leverage ideas from the adaptive Markov chain Monte Carlo (MCMC: Robert & Casella, 2004) literature, such controlling the acceptance probabil- ity (Andrieu & Robert, 2001; Atchadé & Rosenthal, 2005) or maximizing the expected-squared jump distance (Pasar- ica & Gelman, 2010). Both have previously been incor- porated into adaptive SMC methods (Fearnhead & Taylor, 2013; Buchholz et al., 2021). On the other hand, tuning unadjusted kernels, which have favorable digensiimnnional convergence properties compared to their adjusted coun-
Equal contribution 'Dept. Computer and Information Science, University of Pennsylvania, Philadelphia, U.S. Dept. Statistics, University of British Columbia, Vancouver, Canada. Cor- respondence to: Kyurae Kim <kyrkim Zuheng Xu Crubhengxuustat.ukee bell <trevor @ @stat.ubc.ca>, Jacob R. Gardner <jaco- geg@scas.upenn.com
edu>> Trevor Camp-
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
terparts (Lee et al., 2021; Chewi al., 2021; Roberts & Rosenthal, 1998; Wu et al., 2022; Biswas et al., 2019) and enable fully differentiable samplers (Geffner & Domke, 2021; Zhang et al., 2021; Doucet et al., 2022), is not as straightforward as most techniques from adaptive MCMC cannot be used.
tion (Duane et al., 1987; Horowitz, 1991; Monmarché, 2021), also known as unadjusted generalized Hamiltonian Monte Carlo (Neal, 2011). I). Our method achieves lower vari- ance in normalizing constant estimates compared to the best fixed step sizes obtained through grid search or SGD-based tuning methods. Additionally, the step size schedules found by our method achieve lower or comparable variance than those found by end-to-end optimization approaches without involving any manual tuning ($ 5).
Instead, the typical approach to tuning unadjusted kernels is to minimize a variational objective via stochastic gradient descent (SGD; Robbins & Monro, 1951; Bottou et al., 2018) in an end-to-end fashion (Doucet et al., 2022; Goshtasbpour & Perez- Cruz- 2023; Salimans et al., 2015; Caterini et al., 2018; Gu et al., 2015; Arbel et al., 2021; Matthews et al.,
2. Background
Notation. Let B (X) be the set of Borel-measurable sub- sets of some set X' C Fd. With some abuse of notation, we use the same symbol to denote both a distribution and its den- sity. Also, log (I) A log max (z, 1), [++ A max (-, O),
2022; Maddison et al., 2017; Geffner & Domke, 2021; Heng et al., 2020; Chehab et al., 2023; Geffner& & Domke, 2023; Naesseth et al., 2018: Le et al., 2018; Zenn & Bamler, 2023). End-to-end optimization approaches are costly: SGD typ- ically requires at least thousands of iterations to converge (e.g, Geffner& & Domke 2021 use 1.5 x 10 SGD steps for tuning AIS), where each iteration itself involves an entire run of SMC/AIS. Moreover, SGD is sensitive to several tun- ing parameters, such as the step size, batch size, and initial- ization (Sivaprasad et al., 2020). But many of the unadjusted kernels, e.g., random walk MH (Metropolis al., 1953; Langevin (Rossky et al., 1978; Besag, 1994), Hamiltonian Monte Carlo (Duane et al., 1987; Neal, 201 have only a few scalar parameters (e.g., step size) subject to tuning. In this setting, the full generality (and cost) of SGD is not required; it is possible to design a simpler and more efficient method for tuning each transition kernel sequentially in single SMC/AIS run.
and [T] _ [L....,T}
2.1. SMC sampler and Feynman Kac Models
Sequential Monte Carlo (SMC; Dai et al., 2022; Del Moral et al., 2006; Chopin & Papaspiliopoulos, 2020) is gen- eral framework for sampling from Feynman-Kat Kac mod- els (Del Moral, 2004; 2016). Consider a space X' with a o-finite base measure. Feynman- Kac models describe a change of measure between the target path distribution
Hastings, 1970), cestropoli--adjustte
POr (dxorr) ) -
1
{ Go (2o)
II
T
G. (x)-1, 21)
} Ql (dzo:))
Z4
I=1
and the proposal path distribution
(dro)))))) a q (dzo) q
I (z--1, dx;) where d=l
In this work, we propose novel strategy for tuning path proposal kernels of SMC samplers. Our approach is based on greedily minimizing the incremental Kullback-Leibler (KL; Kullback & Leibler, 1951) divergence between the target and the proposal path measures at each SMC step (S 3.1). This is reminiscent of annealed flow transport (AFT: Arbel et al., 2021; Matthews et al., 2022), where a normal- izing flow (Papamakarios et al., 2021) proposal is trained at each step by minimizing the incremental KI.. Instead of training a whole normalizing flow, which requires expen- sive gradient- tradient-based optimization, we tune the parameters of off-the-shelf kernels at each step. This simplifies the optimization process, leading to a gradient- and tuning-free step size adaptation algorithm with quantitative convergence guarantees ($ 3.3).
is the reference or iiiiial proposal distribution, (KT) (EET are Markov kernels parameterized with 0, and (G))et)T
are non-negative Q-measurable functions re- ferred to as potentials.
The (intermediate) normalizing constant at time t E [T'] is
z =
==]Jen jxc+1
Go (are
II s=1
Go (I.-1, (dro:))
The goal is often to draw samples from pe 0.r or to estimate the normalizing constant Z7.
At time t = 0, SMC draws N particles xf.N from the initial proposal g, each assigned with equal weights wg = 1 for n E [N]. At each subsequent time / E [7], particles x!=\ are transported via the transition kernel K, reweighted using the potentials G. and optionally resampled to discard particles with low weights. See the textbook by Chopin & Papaspiliopoulos (2020) for more details.
Using our tuning scheme, we provide complete implementa- tions of tuning-free adaptive SMC samplers for static mod- els: (i) SMC-LMC, which is based on Langevin Monte Carlo (LMC; Rossky et al., 1978; Parisi, 1981; Grenander & Miller, 1994), also commonly known as the unadjusted Langevin algorithm, and (ii) SMC-KLMC, which uses ki- netic Langevin Monte Carlo with the "OABAO" discretiza-
At each time t E [T], the SMC sampler outputs a set of weighted particles (W!LN, " xI:N), where w/ A w / / w,",, along with an estimate of the normal- izing constant Žr, Under suitable conditions, SMC sam-
2

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
plers return consistent estimates of the expectation Pt (fp) of measurable function yp X' - R over the marginal P. and
3. Adaptation Methodology
3.1. Adaptation Objective
the normalizing constant Ze (Del Moral, 2004; 2016):
Σ W/"" 40 (x,") N- P. (4p and ZN, No 74
The variance of sequential Monte Carlo is minimized when the target path measure P and the proposal path measure Q are close together (Del Moral et al., 2006; Gerber et al., 2019; Chopin, 2004; Webher, 2019; Bernton et al., 2019). A common practice has been to make them close by solving
nE(N]
Different choices of GO:r, 9, and K0 describe the same target path distribution 0:r but result in vastly dif- ferent SMC algorithm performance. Proper tuning is thus essential for achieving high efficiency and accuracy.
minimize DKL ( Porr).
2.2. Sequential Monte Carlo for Static Models
In this work, we are interested in a scheme enabling efficient online adaptation within SMC samplers. One could appeal to the chain rule of the KI. divergence:
In this work, we focus on SMC samplers for static models where we target "static" distribution R, whose density X' + Rso is known up to normalizing constant Z through the unnormalized density function 7 X Rso:
DKL DKL(Qhr (Qa:T, 1 PO:T)
= (Qu, Po) +
E
A (x) (z)
(E[2
Z where Z = fxx (x)
and attempt to minimize the incremental KI. terms. Unfor- tunately, at each step of SMC, we have access to a parti- cle approximation P=-1 but not the marginal path proposal Q-- due to resampling. Instead, we can consider the for- ward/inclusive KL divergence
This can be embedded into sequential inference targeting "path" of distributions (To..... Wr). where the endpoints are constrained as mo = q and тr = и. is common to choose the geometric annealing path by setting the density of Te, for t E (O....,T), ............
DKL( (Po:r Quer)
Te (1) x 7t (x) _ a (z)) 2in (o)*
= =DLL Qo) + Σ
( Q1j=-))
where the "temperature schedule" (Ai),E(O....))) is mono- tonically increasing as 0 = Ao <... Ar =1
LE[T]
Estimating the incremental forward KL divergence (K Qtj--)). however, is difficult due to the ex- pectation taken over Prie-1. often resulting in high variance. Therefore, we would like to have a proper divergence mea- sure between the joint paths that (i) decomposes into T incremental terms like the chain rule of the KL divergence. (ii) is easy to estimate, just like the exclusive KL divergence.
To implement an SMC sampler that simulates the path (MT) we introduce a sequence of backward Markov kernels (LG-1)rem) (and refer to the (K?))))) as forward kernels). We can then form Feynman-Ka Kac model by set- ting the potential for L 21 1 as
7--1 d (Te 80 I
Notice that naively summing the incremental exclusive KL divergences as
Gl (x)) = Z, d (Te-1 @ k?)
(z))
D path (Pu:T, Qo:))
As long the condition
Te Li < T K
A DKI. (Qo, Po) + Σ E E (DKI. (Qujuc-1, P)
(E[T]
holds for all 2 0 and the Radon-Nikodys derivative can be evaluated pointwise, Eq. (2) is equivalent to
satisfies both requirements and turns out be a valid diver- gence between path measures:
of (Z)-1, = 7/ (ax) L6_1 (Ie, I))
Proposition 1. Consider joint distributions Qui Porr-
(4)
7 (x K. (x)-1,
Then Dpath satisfies the following:
Other than the constraint Eq. (3), the choice of forward and backward kernels is matter of design. Typically, the for- ward kernel K. is selected as a -i-inaariant (i.e., adjusted) MCMC kernel (Del Moral et al., 2006), such that the parti- cles following P are transported to approximately follow Tr. This Feynman-Kac model targets the path measure
(i) Daath (Pozr, Qu:r) 2 0 for any Que Po.r-
(ii) Dpath (Po.r, Qo:r) = 0 if and only if Poer = Qu.. Proof. See the full proof in page 25.
Ideal Adaptation Scheme. Therefore, we propose to adapt SMC samplers by minimizing Dpach
minimize Dpath (Po. Qir))
(dr = n (dar) II 1 (Ire, dr..1)
The key convenience of this objective is that for most cases that we will consider, the tunable parameters 0 decompose into sequence of subsets 0 = (f1...., 0r), where at any
Then, the marginal of Xy is and for / E [T), the intermedi- ate normalizing constant 2 is precisely Z, = fx n (x) dr.
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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
I E (T, K₁ and G2 depend on only O2: while Og dominates their variance contribution. This suggests greedy scheme where we solve for a subset of parameters at time. By fixing -1:11 from previous iterations, we solve for
Algorithm 1: Adaptive Sequential Monte Carlo
28 ~ 4,
w = 1, T do
Zu. +
for t
E ~ v
5 resample (wl-Y)
0, arg min E p {DKL (Q) Pfr
(6) This greedy strategy does not guarantee a solution to the joint optimization in Eq. (5). However, as long as greedily setting O2 does not negatively influence future and past steps, which is reasonable for the kernels we consider, this strategy should yield a good approximate solution.
at
x- = x Wi-1 =1
8,
O, = arg ming, Eo (0s: Rp wi- c! + Freg (0.)
X ~ KZ (x--11))
W. + -wi- G (x==
if resampling is triggered or T then
ŽnN = Z.-1.N 1
N Ene(y w/
a. resamplee (w!
Relation with Annealed Flow Transport. For the static model case, Arbel et al. (2021) noted that the objective in Eq. (6) approximates
x² t xi
wa t- 1
end end
Ex - { (DKL(TT-1 & K, Te & L [x==)]- (7)
3.2. General Adaptation Scheme
Furthermore, Matthews et al. (2022, $3) showed that, when K, is taken to be a normalizing flow Fe (Papamakarios et al., 2021) and Lz-1 = FF,F=, there exists a joint objective associated with Eq. (7),
Estimating the Incremental KL Objective. Now that we have discussed our ideal objective for adaptation in Eq. (9), we turn to estimating this objective in practice. At each iteration & E [T), we have access to a collection of weighted particles
DKL
- I
Σ ME|N]
W 1
p
where. F -e- is the pushforward measure of π pushed through Fr Our derivation of Eq. (6) shows that it is not approximation some joint objective as Eq. (8), but proper divergence between the joint target P and joint path Q. This general principle applies to all Feynman-K Kac models, not just those for static models.
Blu 4-1
up a constant with respect to 0,, 2 where Sep is a Dirac measure centered on x"- Consider the case where sampling from K," can be represented by a map Mfl : X x E - X. where the randomness over the space & following &i B (E) + R20 is captured by eă ~ gh:
just minimizing
Incremental KL Objective for Feynman-Kac Models. For Feynman-Kan Kac models, Eq. (6) takes the form
x = M (x"=;""))
I P1t-1 Pi--1
(DKL(Q (f p)
Then, up a constant, we obtain conditionally unbiased estimate of the expectation in Eq. (9) function of 8:
aQ 0r d
(0:; X-11 w I:N c c!:N)
f|c
4
Σ W log G.T1 (x M, (x (x=))))) (10)
log G.Ll' (I)-1, K (#t-1, de:) aP
ME[N]
Efficiently Optimizing the Objective. Directly optimiz- ing L,, however, is challenging: (i) Evaluating C, takes O(N) evaluations of the potential, which can be expensive. (ii The expectation over the kernel K, or, equivalently, over c ~ 1/, is intractable. We address these issues as follows:
log uQuu f
The normalizing constant ratio forms a telescoping sum such that the path divergence becomes
Zr
Dpath (Pō.r. (br)) = (Qo, Po) - log
Zu
1. Subsampling of Particles. To reduce the O(N) cost of evaluating 2., we apply resampling over the parti- cles according to the weights w/El such that we end up with a smaller subset of particles of size B < N, which remains a valid approximation of P Then, evaluating E. takes O(B) evaluations of the potential.
Σ E (n
2y -ostt--skee SK {-log log Gf"h (x)1, xi)
BE[T]
In practice, Feynman-Kac models are designed such that both Zy and Zo are fixed regardless of 0: Zu is the nor- malizing constant of q, which is usually 1, and ZT is the normalizing constant of the target p por. Therefore, for such Feynman- Kac models, solving Eq. (6) is equivalent to
2. Sample Average Approximation. Properly minimizing the expectation over K, requires stochastic optimiza- tion algorithms, which introduce numerous challenges related to convergence determination, step size tuning,
0, arg min E 10--1"1- p"" oK! {llog Gf (X- 1, X.)}.
(9)
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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Algorithm 2: AdaptStepsize (C, Aguess 8, r,,,)))
Comg Coun-cile regine
Input:
Adaptation objective L: 00) + IR (+00).
Sore Eegine
degenentt rogitnc
SMC iteration t E [2', initial guess Aguuess > D. hacking-off step size 5 exponential search coefficient c. > 0,
E (1)
exponential search exponent r > 1, absolute tolerance >>0 O.
log h
log li 1SS1 h.
Output: Adapted step size h.
_ (E) ê L (exp (f))
f <- log Aguuess
Figure I. Illustration of Assumption 1. The solid line is the empirical objective Le for the LMC kernel computed using the Bones model from PosteriorDB at time 1.
if = 1 then
< FindFeasible (c 2,
end
f < Minimize (c (,,,,,))
The adaptation routine is shown in Alg. 2. First, Line I and 2, we convert the optimiza- tion space to log-space; from (0, oo) to (-o0, 20). At the SMC iteration 1 1, lguuess is provided by the user. Here, it is unsafe to immediately trust Aguues to be non- degenerate (C (Miguess) < oc). Therefore, FindFcasible in Line 4 ensures that L (exp (4)) < oc. At time > 1, we set hguess = h4-1, which should be non non-degenerate as long as adaptation at time t - 1 went successfully. Then we pro- ceed to optimization in Minimize (Alg. 8), which mostly relies on the golden section search algorithm (GSS; Avriel & Wilde, 1968; Kiefer, 1953), a gradient-frre free I-dimensional optimization method. GSS deterministically achieves an absolute tolerance of > O. Since we optimize in log-space, this translates to a natural relative tolerance e with re- spect to the minimizer of C. In our implementation and choice of r, e, e (described in App. B), this procedure termi- nates after around 10 objective evaluations for 1. 1 and few tens of iterations for t 1. For an in-depth discussion on the algorithm, please refer to App. C.
AdaptStepsize.
Return exp (e)
handling instabilities, and such. Instead, we draw sin- gle batch of randomness (c:) and fix it throughout the optimization procedure. This sample average ap- proximation (SAA; Kim et al., 2015) introduces bias in the optimized solution but enables the use of more reliable deterministic techniques.
3. Regularization. Subsampling the particles results in a higher variance for estimating the objective. We coun- teract this by adding weighted regularization term T reg (9t) to the objective. For example, for the case of step sizes at t > 1 such that O, contains heg we will set J reg (h+) = Fllog h log h ?, which has a smoothing effect over the tuned step size schedule. This also makes the objective "more convex," casing opti- mization. For time t = 1, where we don't have h we use guess hg instead. Effective values of 7 depend on the type of kernel in question, but not much the target problem. We thus used a fixed value (App. B) throughout all our experiments.
3.4. Analysis of the Algorithm for Step Size Tuning
The high-level workflow of the proposed adaptive SMC scheme is shown in Alg. 1. The notable change is the addition of the adaptation step in Line 3 (colored region), where the tunable parameters to be used at time I are tuned to perform best at the tth SMC step, which follows the "pre- tuning" principle of Buchholz et al. (2021). In contrast, retrospective tuning (Fearnhead & Taylor, 2013), which uses parameters that performed well in the previous step, forces SMC to run with suboptimal parameters at all times.
We provide quantitative performance guarantees of the pre- sented step size adaptation procedures. To theoretically model various degeneracies that can happen in the large step size regime, we will assume that the objective function & takes the value of +oo beyond some threshold. In practice, whenever a numerical degeneracy is detected when evaluat- ing log y (NaN or -o)), we ensure that the objective value is accordingly set as co. Our algorithm can deal with such cases by design, as reflected in the following assumptions:
3.3. Algorithm for Step Size Tuning
Assumption 1. For the objective L :0, - RU ( (+00), we assume the following:
Recall that for SMC samplers applied static models
($ 2.2), the path proposal kernel is typically chosen to be MCMC kernel. For most popular MCMC kernels such as random walk MH (Metropolis et al., 1953; Hastings, 1970) or Mierropolis-ajjuste Langevin (MALA; Besag, 1994; Rossky et al., 1978), the crucial tunable parameter is a scalar- valued parameter called the step size denoted as ht > for t [T. In this section, we will describe a general procedure for tuning such step sizes.
(a) There exists some h E (0, oo) such that L is finite and continuous on (0, 1x) and too on | co).
(b) There exists some h E (0, h()) such that L is strictly monotonically decreasing on (0, A).
(c) There exists some h E [h. h% such that L is strictly monotonically increasing on [h, h%)
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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Assumption (a) stipulates that degenerate regions are never disconnected and only exist in the direction of large step sizes. Assumptions (b) and (c) represent the intuition that when the step size is too small or too large, the MCMC kernels degenerate predictably. Most of the MCMC kemels used in practice are based on time-discretized diffusions. In these cases, (b) is satisfied as they approach the continuous- time regime, while (c) will be satisfied as the discretization becomes unstable (divergence). Fig. validates this intu- ition on one of the examples.
4. Implementations
Based on the procedure in 3.3, now describe complete implementations of adaptive SMC samplers. Here, we will focus on the static model setting ($ 2.2), where the main objective is tuning of the MCMC kernels (K²) (KT))e(T)-
4.1. SMC with Langevin Monte Carlo
First, we consider SMC with Langevin Monte Carlo (LMC; Grenander & Miller, 1994; Rossky al., 1978; Parisi, 1981), also known the unadjusted Langevin al- gorithm. LMC forms kernel K2 IR x B (R") - Rg=0
Theorem 1. Suppose Assumption holds. Then AdaptStepsize (C, hguues, 8, c,r,c) returns step size E )x) that is 6-close to a local minimum of & in log-scale after Cleas + Com + Cass objective evaluations, where, defining A A log+ (h/ho) + log+ 1o8+(ho/bl (ho/h) and
on the state space X' = RR, which, for 2 0, simulates the Langevin stochastic differential equation (SDE)
dx,- = V log (x.) ds + v2dB,,
(11)
ho & min(hguess,! ))
where (B.).20 is Brownian motion. Under appropriate con- ditions on the target n,, it is well known that the stationary distribution of the process (X+) is TT. where converges exponentially fast in total variation (Roberts & Tweedie, 1996, Thm 2.1). The Buler-Maruyama discretization of Eq. (11) yields a Markov kerne
Cleas = 0 +10++ (Manme/n>m)
Cam Of ) log+ (Arc-)) Cays= O (log_ (13 (1)
Proof. See the full proof in page 28.
K²t (a,, da') N (dx'; + IV log T. (z), 20
This suggests that, ignoring the dependence on r, G, the objective query complexity of our optimization procedure (log (A/s)). Here, A represents the difficulty of the problem, where A 2 |logh - log In essence, loog h - log h| represents how "multimodal" the problem is. In practice, however, many problems result in less pes- simistic objective surfaces such as follows:
where h 0 is the step size, which conveniently has tractable density with respect the Lebesgue measure.
Note that LMC is an approximate MCMC algorithm; for any h > 0, the stationary distribution of K.' is only approx- imately Te. This contrasts with its MH-adjusted counterpart MALA (Besag, 1994; Roberts Tweedie, 1996; Rossky et al., 1978), which stationary on ne.
Assumption 2. L is unimodal on (0, 3)).
Backward Kernel. For the sequence of backward kernels (Li-1):-2....r multiple choices are possible. For instance, in the literature, a typical choice is L K In this work, we instead take the choice of h L (I-) & K (I,,
This is equivalent assuming (b) and (c) in Assumption with h = 1 and implies there is unique global minimum. Then Theorem 1 can be strengthened into the following:
Corollary 1. Suppose Assumption 2 and 3 hold. Then The- orem holds, where AdaptStepsize (C, t., hguess, 5, C, r. returns h E (0, h20) that is e-close to the global optimum h" and A [log h² - hol-
which we call the "time-correct forward kernel." Compared to more popular alternatives, this choice results in signifi- cantly lower variance. (An in-depth discussion can be found in App. E.) The resulting potentials are Yi (#1)
Note that, at / 1, it is sensible to set liguess <- hy-1 since We-1 me by design. Therefore, after t 1, AdaptStepsize will run in "warm start" regime where A R O. For instance, assume the initial guess is warm such that [log Aguess - log h1| S 6 and lgguess E (0, )²). Then Corollary states that the number of objective evaluations will be O (log+ (r2ce-1 + }
G (zo, = K (Fo. 21)
Ghi-ishs (II-1, Yt (xt) T (01,
7 (xt-1) K.t (
where, at each step 1 E [T]. we optimize for hy using the general step size tuning procedure described in $ 3.3 while the backward kernel re-uses the tuned parameter h4-1 from the previous iteration.
The parameters of AdaptStepsize, r and c, must balance the performance of both the warm and cold start cases. For a warm start, -1 = O (e) optimizes performance. For cold start, r needs to be large enough to keep the (log -1 term in Cbm small. Thus, leaning towards making small and r moderately large balances both cases. The values we use in the experiments are shown in App. B.I.
4.2. SMC with Kinetic Langevin Monte Carlo
Next, we consider a variant of the LMC that operates on the augmented state space 2 = X x X, where, for 2 0.
6

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Algorithm 3: AdaptKLMC (L., guess: Pruess, 6, ,,,,),)
kime size h preserving the "Hamiltonian energy" log ". This discretization also coincides with the unadjusted ver- sion of the generalized Hamiltonian Monte Carlo (Duane et al., 1987; Neal, 201 1) with a single leapfrog step.
Input:
Adaptation objective L : x X (0. 1) B (oo),
initial guess (Rguess Pguess) E Ry0 x 1),
backing-oft off step size 5 < 0, grid of refreshment parameters E E (0, exponential search coefficient e > 0,
Backward Kernel. Since the kemel Shh is a deterministic mapping, K.' does not admit a density with respect the Lebesgue measure. Therefore, we restricted to specific backward kernel that satisfies the condition in Eq. (3): Since the leapfrog integrator & is a diffeomorphism, its inverse map (Ø/) exists. Therefore, we can choose
exponential search exponent > 1, absolute tolerance >> O.
Output: Adapted step size and refreshment rate ph.
L (E, p) & C (exp PP)
f 4- log /guuess P + Pquess
ift then
t FindFeasible (t+ ( (()
end
D (z,, .) ((0)))((ar.a ((dz,, det- R (Ue-1/2, (u)-)
while
not converged do
E' +- Minimize (e L (l. cc) d' + arg min Pc= Lb0 (f', p).
This results in the deterministic component of Ke and L being supported on the same pair of points, ensuring ab- solute continuity (Doucet al., 2022; Geffner & Domke, 2023). Then the potential is given by the Radon- NikonnN derivative between the momentum refreshments R as GhaiP (2)
if max (E- -11,1p -p1)) S then
Return (exp (E",,'' end Et E, 5 p
11
12 end
To (xt) (mx; Ta) (6-1; V1 03 02 1/2, P
13
Return (exp (E')
7c-1 (xt-1) (vt-1;0,, (1I-//25 VI - pt vc- pēLo
each state of the Feynman-Kat model is denoted as 24- = (14, 11) E 2, F4, 24 E X, X = TRd, and the target is kmmc (x, v) A Te (x) (u; Oa. La)
with two tunable parameters: (he Pe) E ERyo x 10,
nzz
Adaptation Algorithm. As
Evidently, the z-marginal of the augmented target is m Therefore, a Feynman- Kac model targeting kime is also tar- geting ग by design. Kinetic Langevin Monte Carlo (KLMC; Horowitz, 1991; Duane et al., 1987), also referred to as underdamped Langevin, for 8 2 0, is given by
KLMC has two parameters, cannot immediately apply the tuning procedure offered in 3.3. Thus, we will tailor it to KLMC. At t E [T], we will minimize the incremental KI. objective E. ph) through coordinate descent. That is, we alternate between minimizing over h and P. This is shown in Alg. 3. In particular, ht is updated using the procedure used in $ 3.3, while Pt is directly minimized over a grid 5 E (0, 1)* of k grid points. As shown in Fig, 4, empirically, the minimizers of L. with respect to P tend to concentrate on the boundary, as if the adaptation problem is determining to "fully refresh" or "not refresh at all." Therefore, the grid 5 can be made as coarse as E (0.1, which is what we use in the experiments.
Density
0.2 0.4 0.6 0.8 I Refreshment rate (p)
dx,= v,ds
Figure 4. Distribution of tuned refreshment rates
dv:- = V log π (v.) - 7/yds + 22ddB,,
where ?) > is tunable parameter called the damping coefficient. The stationary distribution of the joint process (xs, v=) is then Himc, This continuous time process cor- responds to the "Nesterov acceleration (Nesterov, 1983; Su et al., 2016)" of Eq. (11) (Ma et al., 2021), meaning that the process should converge faster. We thus expect KLMC to reduce the required number of steps T compared to LMC.
Pt. The results were ob- tained by running adaptive SMC on the Sonar proh- lem with T 256 and N = 1024
To simulate this, we consider the OBABO discretiza- tion (Leimkuhler & Matthews, 2013), which operates in a Gibbs scheme (Geman & Geman, 1984): its kernel K, (zz),,dz))
=R" (0, du)- Sh ((5--1, (dx,, dex)) is a composition of the momentum refreshment kernel
5. Experiments
5.1. Implementation and General Setup
Rº (vt-1, do, _ / (du2-/2i vI -1- v2,,, 02 where p= A 1 1 exp (-nh)= E (0, 1) is the "momentum refreshment rate" for some step size h > O. and the Leapfrog integrator kernel
We implemented our SMC sampler' using the Julia lan- guage (Bezanson et al., 2017). For resampling, we use the Srinivasan sampling process (SSP) by Gerber et al. (2019), which performs similarly to the popular systematic resam-
S: ((5e-1 v/-1/aa ..) A 5gg (1==11),111a) (),
Link GITHUB repository: https://github. Red-Porta (accal//Cooi wow.snt wtt.s......tt.eeee/eeeeeee 0.4.
where B is single step of leapfrog integration with step
7

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
1000
500
-2000
-26000
-500
log Z estimate
-28000
4000
-1000
11000
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-2000
Grid Search
- 1000
-300
Adaptive True
1500
-6000
32000
-1500
4000
2000
-8000
34000
2000
10
10 10"
10
10
5
10
10
10'
10
5
10
10
10'
10
10
10 10'
10
10 10 10'
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
(a) Pines
(b) Capture
(c) Science
(d) Three Men
(e) Rats
Figure SMC-LMC with adaptive tuning v.s. fixed step sizes. The solid lines are the median of the estimates log Z, while the
colored regions are the 80% empirical quantiles computed over 32 replications.
525
25
-2500
26000
log Z esstmaaee
500
20
-1000
-3000-
-88000
475
15
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Grid Search
Adaptive
10
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3500
330000
3000
True
400
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4000
10
10
10"
10
10
10 10
10'
10
5
10a
10"
10'
10 3
10 10
10'
10s
10
10 10'
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
(a) Pines
(b) Capture
(c) Science
(d) Three Men
(e) Rats
Figure 3. SMC-KLMC with adaptive tuning fixed step sizes and refreshment rates. For SMC-KLMC with fixed parameters h, P.
we show the result of the best- perffrrrnnnnng refreshment rate. The solid lines are the median of the estimates of log Z, while the colored
regions the 80% empirical quantiles computed over 32 replications.
pling strategy (Carpenter et al., 1999; Kitagawa, 1996), while having stronger theoretical guarantees. Resampling is triggered adaptively, which has been theoretically shown to work well (Syed et al., 2024) under the typical rule of resampling as soon as the effective sample size (Kong, 1992; Elvira et al., 2022) goes below N/2. In all cases, the ref- erence distribution is standard Gaussian q = .V (0,, Ia), while we use a quadratic annealing schedule A, (t/T)².
Results. A representative subset of the results is shown in Figs. 2 and 3, while the full set of results is shown in App. F.I. First, we can see that SMC with fixed step sizes is strongly affected by tuning. On the other hand, our adaptive sampler obtains estimates that are closer or comparable to the best fixed step size 20 out of 22 benchmark problems. Our method performed poorly the Rats problem, which is shown in the right-most panes in Figs. and 3. Overall, our method results in estimates that are better or comparable to those obtained with the best fixed step size.
Evaluation Metric. We will compare the estimate log Zy.N, where, for unbiased estimates of Z against a ground truth estimate obtained by running a large budget run with N 2 and T = 29. Due adaptivity, our method only yields biased estimates of Z. Therefore, after adaptation,
5.3. Comparison Against End-to-End Optimization
Setup. Now, we compare our adaptive tuning strategy against end-to-end optimization strategies. In particular, we compare against differentiable AIS (Geffner Domke, 2023; 2021; ; Zhang et al., 2021) instead of SMC, as differ- entiating through resampling does not necessarily improve the results (Zenn& & Bamler, 2023). To focus on the tuning capabilities, we do not optimize the reference q. How- ever, results with variational reference tuning can be found in App. F.2. Furthermore, we performed a grid search over the SGD step sizes {10-1, 10-3 10-2) and show the best results. Additional implementation details can be found in App. B.2. Since end-to-end methods need to differenti- ate through the models, we only ran them on the problems with JAX (Bradbury et al., 2018) implementations (Funnel, Sonar, Brownian, and Pine). We use T 32 SMC iterations for all methods. For adaptation, our method uses B = 128 particles out of N = 1024 particles, while end-to-end op- timization uses an SMC sampler with 32 particles during
vanilla SMC with the tuned parameters, which yields unbiased estimates.
Benchmark Problems. For the benchmarks, we ported some problems from the Inference Gym (Sountsov et al., 2020) to Julia, where the rest of the problems are taken from PosteriorDB (Magnusson et al., 2025). Details on the problems considered this work are in App. A, while the configuration of our adaptive method specified in App. B.
5.2. Comparison Against Fixed Step Sizes
First, we evaluate the quality of the parameters tuned through our method. For this, we compare the perfor- mance of SMC-LMC and SMC-KLMC against hand tuning fixed step size h. such that he h, over a grid of step sizes. For KLMC, we also perform grid search of the refreshment rate over 10.1, 0.5, 0.9}. The computational budgets are set as N = 1024, = 128, and T = 64.
Setup.
8

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
20
d=8
d=8
d=8
log Z Estimate
0
Step Size f;
0.8
d= 32
Step Size k:
d=
Step Size h:
d=322
-20
d= 128
d= 128
d= 128
0.6
-40-
SMC-LMC
d= 512
d=512
d=
-60
SMC-MALA-ESID AMC-MALA-ARC
0.4
0.4
-80
True
0.2
0.2
-100
0.1
0.1
0.1
8
32
128
512
0
0.2
0.4
0.6 0.8
1
0.2
0.4 0.6 0.8
0
0.2
0.4
0.6 0.8
Dimension df
Temperature 7,
Temperature 2,
Temperature 2,
(a) log Z Estimates
(b) SMC-LMC
(c) SMC-MALA-ESID
(d) SMC-MALA-ARC
Figure Dimensional scaling of adaptive SMC with Langevin- based kernels with (MALA) and without (LMC) MH adjustment. (a) Comparison of the log 7 estimates under growing dimensionality.. The solid lines are the median, while the shaded regions are the 80% quantiles obtained from 32 replications. (b-d) Tuned step size schedules obtained under each sampler. SMC-MALA-ESID uses ESJD maximization for adaptation, while EMC-MALA-ARC uses acceptance rate control (ARC). Each solid line is step size schedule obtained from a single run (cight examples are shown), while the dotted lines are the average over t.
10
$20
bias in the stationary distribution, which grows as O (d) in squared Wasserstein distance (Dalalyan, 2017: Durmus & Eberle, 2024; Durmus & Moulines, 2019). However, since SMC never operates in the stationary regime (except for the waste-free variant by Dau & Chopin 2022), we expect SMC- LMC to scale better than SMC-MALA with dimensionality d. Here, we will empirically verify this intuition.
2. Estimate
-110-
0-
500
115
480
Z
10-
-120
460
125
Adapcire SMC
End-i-Ead Thue
20
440
-130
-30+
420
0
IM
2M
3M
2M
4M
0
500k
IM
Cost
Cost
Cost
Setup. We set л = N (3 la, Ia) and q = - (Oa, Ia) under varying dimensionality d. The computational bud- gets are set N 1024, 7 4 4[VØ], where the latter is suggested by Syed et al. (2024, $4.7). For MALA, we will consider two common adaptation strategies: control- ling the acceptance rate (Buchholz et al., 2021) such that it is 0.575 (Roberts & Tweedie, 1996) and maximizing the ESJD (Pasarica & Gelman, 2010; Buchholz et al., 2021; Fearnhead & Taylor, 2013). For both, we use the tricks stated in 3.2, such subsampling and SAA.
(a) Sonar
(b) Brownian
(c) Pines
Figure 6. Comparison against end-to-end optimization. The "cost" the cumulative number gradient evaluations of the target. 32 independent runs for end-to-end optimization are shown. The error bars/bands are 80% empirical quantiles of the cost and the estimates of log Z computed from 32 replications.
optimization, and N = 1024 particles when actually esti- mating NT. For both methods, the cost of estimating the unbiased normalizing constant is excluded.
Results. The results shown in Fig. 5. For SMC-LMC, the step size schedule is shown to decrease with (Fig. 5b), Since the smoothness constant of the target density does not change with t, this means our adaptation scheme is automatically performing a trade-off between convergence speed and asymptotic bias. Also, the average step sizes decrease with which is expected since the bias grows with d. However, for d > 128, the step sizes of SMC- LMC tend to be larger than those of SMC-MALA (Figs. 5c and 5d). Consequently, SMC-ULA obtains more accurate estimates in higher dimensions (Fig. 5a).
Results. The results with the KLMC kernel are shown in Fig. 6, while additional results can be found in App. F.2. Our Adaptive SMC sampler achieves more accurate estimates than the best-tuned end-to-end tuning results on Sonar and Brownian, while the estimate on Pines is comparable. This demonstrates that our SMC tuning approach achieves esti- mates that better par with those obtained through end-to-end optimization.
5.4. Dimensional Scaling with and without Metropolis-Hastingg Adjustment
We will now compare the tuned performance of unadjusted versus adjusted kernels, in particular, LMC versus MALA. To maintain a non-zero acceptance rate, MH MH-adjusted meth- ods generally require h to decrease with dimensionality d. Theoretical results suggest that, for MALA, the step size has to decrease as O (d (d31/3) (Chewi et al.. 2021: Roberts & Tweedie, 1996) for Gaussian targets and as O (d-1/2) in general (Chewi et al., 2021; Wu et al., 2022). In contrast, LMC only needs to reduce to counteract the asymptotic
6. Conclusions
In this work, we established a methodology for tuning path proposal kernels in SMC samplers, which involves greedily minimizing incremental KL divergence at each SMC step. We also developed a specific instantiation of the method for tuning scalar-valued step sizes of MCMC kernels used in SMC samplers. A potential future direction would be to investigate the consistency of the proposed scheme, possibly through the framework of Beskos et al. (2016).

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Acknowledgements
Bentley, L. and Yao, A. C.-C. An almost optimal algo- rithm for unbounded searching. Information Processing Letters, 5(3):82-87, 1976. (page 24)
The authors sincerely thank Nicolas Chopin for pointing out relevant theoretical results, Alexandre Nouchard-Cött for discussions throughout the project, and the anonymous reviewers for helpful suggestions.
Bernton, E.. Heng, J., Doucet, A., and Jacob, P. E. Schrödinger bridge samplers. ArXiv Preprint arXiv:1912.13100, arXiv, 2019. (pages 1, 29)
K. Kim and J. R. Gardner were supported through the NSF award [IIS2 145644]; T. Campbell and D. Xu were supported by the NSERC Discovery Grant RGPIN-2025, 2025-04208. We also gratefully acknowledge the use of the ARC Sockeye computing platform at the University British Columbia.
Besag, J. E. Comments on 'Representations of knowledge in complex systems' by U. Grenander and M.I. Miller. Journal of the Royal Statistical Society Series B (Statis- tical Methodology), 56(4):591 1994. (pages 2, 5,
6)
Impact Statement
Beskos, A., Jasra, A., Kantas, N., and Thiery, A. On the convergence of adaptive sequential Monte Carlo meth- ods. The Annals of Applied Probability, 26(2):1111-1165 2016. (page 9)
This paper presents a method for automatically tuning se- quential Monte Carlo (SMC) samplers. Due to the technical nature of the work, we do not expect to have direct soci- etal consequences. Downstream applications of SMC sam- plers, however, include uncertainty quantification (Dai et al., 2022), statistical model comparison (Zhou et al., 2016), and more recently, conditional generation from diffusion mod- els (Wu et al., 2023; Trippe et al., 2023) and steering large language models to ensure their output conforms to ethical constraints (Zhao et al., 2024). Our work may improve the efficiency and efficacy of such tasks, indirectly affecting their societal consequences.
Bezanson, J., Edelman, A., Karpinski, S., and Shah, V. B. Julia: A fresh approach to numerical computing. SIAM Review, 59(1):65-99) 98, 2017. (page 7)
Biswas, N., Jacob, P. E., and Vanetti, P. Estimating con- vergence of Markov chains with L-lag couplings. In Advances in Neural Information Processing Systems, vol- ume 32, pp. 7391-7401. Curran Associates, Inc., 2019. (page 2)
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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Contents
Introduction
2
Background
2.1
SMC sampler and Feynman Kac Models
2.2
Sequential Monte Carlo for Static Models
3
Adaptation Methodology
3.1
Adaptation Objective
3
3.2
General Adaptation Scheme
3.3
Algorithm for Step Size Tuning.
3.4
Analysis of the Algorithm for Step Size Tuning
4
Implementations
6
4.1
SMC with Langevin Monte Carlo
6
4.2
SMC with Kinetic Langevin Monte Carlo
6
5
Experiments
5.1
Implementation and General Setup
5.2
Comparison Against Fixed Step Sizes
8
5.3
Comparison Against End-to-End Optimization
8
5.4
Dimensional Scaling with and without Metropolis- Haatooolss-aastinss Adjustment
Conclusions
Benchmark Problems
18
B Details on the Experimental Setup
21
B.1 Setup of Adaptive SMC Samplers
21
B.2 Setup of End-to-End Optimization
22
Algorithms
23
C.1 PindFeasible (Algorithm 5)
23
C.2 GoldenSectionSearch (Alg. 6)
23
C.3 BracketMinimum (Algorithm 7)
24
C.4 Minimize (Algorithm 8)
24
Theoretical Analysis
25
D.I Definitions and Assumptions
25
D.2 Proof of Proposition 1
25
D.3 Sufficient Condition for an Interval to Contain Local Minimum (Lemma 1). ................................
25
D.4 GoldenSeetiionSearcc (Lemma 2)
26
D.5 BracketMinimun (Lemma 3)
27
D.6 Minimize (Theorem 2)
28
D.7 AdaptStepsize (Proof of Theorem 1)
28
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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Backward Kernels
29
E.1 Some backward kernels are not like the others
29
E.2
Empirical Evaluation
29
E.3
Conclusions
29
F
Additional Experimental Results
30
F1
Comparison Against Fixed Step Sizes
30
F.1.1
SMC-LMC
30
F.1.2
SMC-KLMC
31
F22
Comparison Against End-to-Eng Optimization
32
................................
F.2.1
SMC-LMC
32
F.2.2
SMC-KLMC
33
F3
Adaptation Cost
34
F.3.1
SMC-LMC
34
F33.2
SMC-KLMC
38
E4
Adaptation Results from the Adaptive SMC Samplers
42
F4.1
SMC-LMC
42
F4.2
SMC-KLMC
48
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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Name
Description
Source
Reference
Funnel
Neal's funnel distribution.
10
Inference Gym
Sountsov et al. 2020
Neal 2003
Brownian
Latent Brownian motion with missing observations.
32
Inference Gym
Sountsov et al. 2020
Sonar
Bayesian logistic regression with the sonar classifica- tion dataset.
61
Inference Gym
Sountsov et al. 2020
Gorman & Sejnowski 1988
Pines
Gag-saas Cox process model of the concentration of Scotch pine saplings in Finland over a 40 40 grid.
1600 Inference Gym
Sountsov et al. 2020
Moller et al. 1998
Table 1. Overview of Benchmark Problems
A. Benchmark Problems
Pines. This is a Gag-Gausiin Cox process (LGCP; Møller et al., 1998) model applied to a dataset of Scotch pine saplings in Finland (Møller et al., 1998). A LGCP is a non- parametric model of intensity fields, where the observations are assumed to follow a Poisson point process (PPP). Con- sider 2-dimensional grid of n cells indexed by i [m], each denoted by S, E S and centered on the location E Rd The dataset is the number of points contained in the ith cell, y4 E N2o, for all i E [n]. which is assumed to follow PPP such that
In this section, provide additional details about the bench- mark problems. A full list of the problems is shown in Ta- bles to 3. For the problems we ported from the Inference Gym, we provide additional details for clarity:
Funnel. This is the classic benchmark problem by Neal (2003). We use the formulation:
N (0,
(1 > (cr)
x N (0a-1, 2 Ia-1),
Poisson
(xc)dx)) dx
where d = 10.
with the intensity field
log GP (n,k),
Sonar. This a logistic regression problem with stan- dard normal prior on the coefficients. That is, for d 61, given dataset (X, y). where X nRn and y R", the design matrix is augmented with column containing Is denoted with X to include an intercept. The data mata-generating process is
where GP (H, k) Gaussian process prior (GP; Ras- mussen & Williams, 2005) with mean μ and covariance kernel k R2 R² + Rzo. We use the grid approximation
[ ( do ~ exp (log À (xx))
B ~ N N(a. Ia)
where A, is the area of S. The likelihood then (v., F exp (a(z)) v A, exp (z:)))
y Bernoulli (o
where a (x) e 1/ (1 [1+++= e-") is the logistic function. Here, we use the sonar classification dataset by Gorman & Se- jnowski (1988). The features are pre-processed with z- standardization following Phillips et al. (2024).
Following Møller et al. (1998), the hyperparameters of the GP are set as
o²
= log (126)
(
[_1xr; ||:;;
xz 111
le (xx 2.) = 0² exp
v S
where
0² 1.91 and 8
33
The field [0, 1|2 is discretized into a 40 x 40 grid such that (S] = 40 and A = 1/[S]. Furthermore, to improve the conditioning of the posterior, we whiten the GP prior (Mur- ray & Adams, 2010, $2.1).
18

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Name
Description
d
Source
References
Bones
Latent trait model for multiple ordered categorical re- sponses for quantifying skeletal maturity from radio- graph maturity ratings with missing entries. (model:
13
PosteriorDB
Magnusson er al. 2025 Spiegelhalter et al. 1996
bones mooel; dataset: bones_data)
Surgical
Binomial regression model for estimating the mor- tality rate of pediatric cardiac surgery. (model: modee: dataset: surgical.data,
PosteriorDB
Magnusson al. 2025
Spiegelhalter al. 1996
HMM
Hidden Markov model with Gaussian emission ap- 14 PosteriorDB
Magnusson et al. 2025 Cappé et al. 2005
plied to a simulated dataset. (model: hmm gaammaaaussin, dataset: hmm gaussian an. wtw.iin....mm..................a simulated)
Loss Curves
Loss model of insurance claims. The model is the sin- gle line-of- single insurer (SISLOB) variant, where the dataset is the "ppauto" line of business, part of the "Schedule P loss data" provided by the Casu- alty Actuarial Society. (model: losscurve.sislok. dataset: loss_curvest
15
PosteriorDB
Magnusson al. 2025
Cooney 2017
Pilots
Linear mixed effects model with varying intercepts for estimating the psychological effect of pilots when per- forming flight simulations various airports. (model: pilots: dataset: pilots)
PosteriorDB
Magnusson al. 2025 Gelman & Hill 2007
Diamonds
Log-log regression model for the price of diamonds with highly correlated predictors. (model: diamonds; dataset: di dimmond)
26
PosteriorDB
Magnusson et al. 2025 Wickham 2016
Seeds
Random effect logistic regression model of the seed germination proportion of seeds from different root extracts. We use the variant with half- Half-Cauchy prior on the scale. (model: seeds seeds_stan's stsassttaiff fiiad medeeele;; dataset: seeds_data,
26
PosteriorDB
Magnusson al. 2025
Crowder 1978
Spiegelhalter et al. 1996
Downloads
Prophet time series model applied to the download count of rstan over time. The model is an additive combination of (i) a trend model, (ii) a model of sea- sonality, and (iii) model for events such holidays. (model: prophet; dataset: rstan downloads)
62
PosteriorDB
Magnusson et al. 2025
Taylor & Letham 2018 Bales al. 2019
Rats
Linear mixed effects model with varying slopes and intercepts for modeling the weight of young rats over five weeks. (model: rats model; data: rats .ata)
65
PosteriorDB
Magnusson et al. 2025
Spiegelhalter et al. 1996
Gelfand et al. 1990
Radon Multilevel mixed effects model with normnllmal likelihood and varying intercepts for modeling the radon level measured in U.S. households. We use the Minnesota state subset. (model: radon www.errrcccca....errrr.....ommm dataset: radon_mm)
90
PosteriorDB
Magnusson al. 2025
Gelman al. 2014
Table 2. Overview of Benchmark Problems
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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Name
Description
d
Source
Reference
Election88
Generalized linear mixed effects model of the vot- ing outcome of individuals at the 1988 U.S. presi- dential election. (model: election88 full; dataset: election88)
90
PosteriorDB
Magnusson et 2025 Gelman & Hill 2007
Butterfly
Multispecies occupancy model with correlation be- tween sites. The dataset contains counts of butterflies from twenty grassland sites in south- Sweden (model: butterfly, dataset: multi occupancy)
106
PosteriorDB
Magnusson et al. 2025 Dorazio et al. 2006
Birds
Mixed effects model with Poisson likelihood and varying intercepts for modeling the occupancy of the Coal tit (Parus ater) bird species during the breed- ing season in Switzerland. (model: ELMM1_model! dataset: GLMM.data)
237
PosteriorDB
Magnusson et al. 2025 Kéry & Schaub 2012
Drivers
Time series model with seasonal effects of 389 driving-related fatalities and serious injuries in the U.K. from Jan. 1969 to Dec. 1984. (model: state_space berochasttcceeelllssccccseeeeeee
PosteriorDB
Magnusson et 2025 Commandeur' & Koopman 2007
dataset: uk dridvires))
Capture
Model of capture-recappe recaptur data for estimating the population size. This is the "heterogeneity model," where the detection probability is assumed to be het- erogeneous across the individuals. The data is simu- lated. (model: Mh model; dataset: Mh data)
388 PostcriorDB
Magnusson et al. 2025 Kéry Schaub 2012
Science
Item response model with generalized rating scale. The dataset was taken from the Consumer Protec- tion and Perceptions of Science and Technology sec- tion of the 1992 Buro-Baroeeter Survey. (model: grsm_latent_regiizz dataset: science.irt)
408
PosteriorDB
Magnusson et al. 2025
Reif & Melich 1993
Furr 2017
Three Men
Latent Dirichlet allocation for topic modeling. The number of topics is set K 2, while the dataset is corpus 3 among pre-processed multilingual corpora of the book "Three Men and a Boat." (model: 1daK2; dataset: three.men3)
505
PosteriorDB
Magnusson et al. 2025
Farkas 2014
Blei et al. 2003
TIMSS
Item response model with generalized partial credit.
530
PosteriorDB
Magnusson et 2025 Muraki 1997
The dataset is from the TIMSS 2011 mathemat- ics assessment of Australian and Taiwanese stu- dents. (model: gpcm latent reg irt; dataset: timssAusTwn.irt,
Mullis et al. 2012
Table 3. Overview of Benchmark Problems (continued)
20

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
B. Details on the Experimental Setup
adaptation scheme estimates the local communication bar- rier and uses it obtain temperature schedule that makes it uniform. Intuitively, LCB Ar) quantifies the "diffi- culty" of approximating क @ L using weighted particles drawn from πe-1 @ K.
B.1. Setup of Adaptive SMC Samplers
Configuration of the Adaptation Procedure. Here, we collected the specifications of the tunable parameters in our adaptive SMC samplers. The parameters of SMC-LMC are set as in Table 4:
In addition, us denote the local communication barrier accumulated up to time step E {O..... .T).
Name
Source
Value
A(A,) Σ LCB(AA-1,A..)
3.2
0.1
Alg.
0.01
Alg. 7
0.1
This serves divergence measure for the "length" of the annealing path from Åo 0 to A Furthermore, the total accumulated local barrier
Alg. 7
2
Alg. 5
-1
igguess
Alg.
exp (-10) x 4.54 x 10 10-5
A A(x))
Table Configuration of SMC-LMC
which is referred the global communication barrier, quantifies the total difficulty of simulating the annealing path (T.). For the normalizing constant to be ac-
The parameters SMC-KLMC are set as in Table 5:
Name
Source
Value
curate, SMC needs to operate in what they call the "stable discretization regime," which occurs at T = O (4) (Syed et al., 2024). Therefore, for tuning the number of SMC steps, good heuristic to set be constant multiple of the estimated global communication barrier.
T
$ 3.2
Alg. 6
0.01
Alg. 7
0.01
r
Alg. 7
3
8
Alg. 5
-1
The corresponding schedule adaptation scheme is fol- lows: From the estimates of the communication barrier (Á (A+)))E(TT obtained from a previous run, the updated schedule for the next run of length T" is set via mapping
5
Alg. 3
{0.1, 0.9)
Pguess
Alg. 3
0.1
hguess
Alg. 5
exp (-7.5) 3 x 5.53 x 10-4
Table 5. Configuration of SMC-KLMC
X = Aun
(x
T 1xx4)
) t' t'
(12)
Schedule Adaptation. In some of the experimental sults in the appendix, evaluate the performance of our step size adaptation procedure when combined with an annealing temperature schedule ((xe) adaptation scheme. In particular, we use the recently proposed method of Syed et al. (2024), which is able to tune both the sched- ule (A:),-o....TT and the number of SMC steps T. Under regularity assumptions, the resulting adaptation schedule asymptotically (N and T oo) approximates the op- timal geometric annealing path that minimizes the variance of the normalizing constant estimator. For a detailed de- scription, see Syed et al. (2024, Sec. 5). Below, we provide a concise description of the schedule adaptation process.
where the inverse mapping Amy is approximated using a monotonic spline with knots ((A(),), )=)) In our case, the length of the new schedule is set as T = 24.
Below, we summarize the general steps for adaptive SMC with round- schedule adaptation:
Algorithm 4: Round-Based Annealing Schedule Adap-
tation
Input:
Number of rounds Tmax- Initial number of SMC iterations Tí.
Initial schedule (X, (
for = 1l.... ... Fmax do
Under suitable regularity assumptions, the asymptotically optimal schedule is the one that makes the "local communi- cation barrier"
2
Run adaptive SMC with the schedule (X))r=o....r.
3 Estimate A and compute Any-
Set Ty41 241.
LCB(A-11, Ad) R( & KPll @ I
Obtain (2g (X.+1) Tr. ) using Eq. (12).
end
uniform across all adjacent steps de. A-11 for all t E (7) (Syed al., 2024, $4.3). As such, the corresponding
21

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
B.2. Setup of End-to-End Optimization
We provide additional implementation details for end-to- end optimization". We implemented two differentiable AIS methods: one based on LMC (Thin et al., 1021) and another based on KLMC (Geffner & Domke, 2021; Doucet et al., 2022). Both methods are implemented in JAX (Bradbury et al., 2018), modified from the code provided by Geffner & Domke (2023).
For optimization, we used the Adam optimizer (Kingma & Ba, 2015) with three different learning rates (10-4,10-) 1,,00,100 2 for 5,000 iterations, with batch size of 32. We evaluated two different annealing step sizes (32 and 64), keeping the number of steps fixed during training while optimizing the annealing schedule (detailed App. F.2). Each setting was repeated 32 times, and we report results from the best-performing configurations.
Following the setup of Doucet al. (2022), the vector- valued step sizes are amortized through function el()) [0, 1] + Rd. This function is parametrized through a 2- layer fully connected neural network with 32 hidden units and ReLU activation, followed by a scaled sigmoid func- tion which enforces (t
for the ULA variant and (t) < 0.25 for the KLMC variant. Enforcing these step size constraints is necessary prevent numerical issues during training, which was acknowledged in prior works (Doucet et al., 2022; Geffner & Domke, 2021).
For schedule adaptation, Doucet et al. (2022); Geffner & Domke (2021), parametrize the temperature schedule as
A Eesoo a (bv)
Σ o (br))
where the sigmoid function, Ag fixed be 0, and bo:T-1 is subject optimization. Following Doucet al. (2022), additionally learn the momentum refreshment rate p (shared across & E [T] for SMC-KLMC. That is, we parametrize p with a parameter u as = _98 + ,01, which ensures p E (0.01, 0.99).
PLink to GITHUB repository: zuhengxu/dais--/it weepp//eeeess/taaaeeeeeeeeeeee
https://github..om
22

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Algorithm 5: FindFeasiibe (f, 4)
Input: Objective f': IR IR, initial guess To E IR, backing off step size E R {0}.
Output: Feasible initial point 20
< 20
while [xx do do
- 6
end
Algorithm 6: GoldenSeettionSearch (f, b,,,e)
Return r
Input: Objective / + R (+xx),
initial triplet (a, b, R2 satisfying Eq. (13),
C. Algorithms
absolute tolerance >> 0.
の (v5-1)/
In this section, we will provide detailed description of our proposed adaptation algorithms and their components.
+ a
3 t c
4 if cc- bl > - al then
C.1. Fi FindFeasible (Algorithm 5)
#1 b
Our adaptation schemes receive a guess from the user. For robustness, it is safe to assume that this guess may not be non-degenerate. As such, we must first check that it is non- degenerate, and if it is not, move This is done by FindFeasible (f, zo, 8) shown in Alg. 5. If tgg already non-degenerate, immediately returns the initial point zo- Otherwise, if is degenerate, it increases or decreases with step size of 5 until the objective function becomes finite.
22 t b + (1 - -6-1) (c-1)
else
x b + b (1- (b- (b-a)
somewhere that is.
end
fi / (x1)
f2 +- f (x2)
while
[] - T2l > e/2
14
if f2 < f1 then
is
Iq + 21
C.2. GoldenSectionSeardd (Alg. 6)
I x2
The workhorse of our step size adaptation scheme is the golden section search (GSS) algorithm (Avriel & Wilde, 1968), which is a variation of the Fibonacci search algo- rithm (Kiefer, 1953). In particular, we are using the imple- mentation of Press al. (1992, $10.1), shown in Alg. 6, which uses triplet, (a, b, e) R&, for initialization. This triplet requires the condition
32 + o 001x2
T2 + (1-6) (1-0-1) 23
fi t f2
f2 f (x2)
else
I3 s x2
F2 E
F1 x- & 1r1 + (1-617 (1- 00
f2 fi
a<dsc, (b) < (a), and (b) < (c)
(13)
fi + f (x1)
end
hold. Then, by Lemma 1, this implies that the open interval (a,c) contains local minimum. Then GSS is guar-
27 end
28 if fi S f2 then
anteed to find the contained local minimum at "linear rate" of (1-v5)/2 RJ 1.62, the golden ratio. For finding point e-close to a local minimum, this translates into an objective query complexity of (log /c al/e). Furthermore, unimodal (Assumption 2), then the solution will be c-close
Return #
else
Return #22
is
32 end the global minimum (Luenberger & Ye, 2008, $7.1). The key is to find triplet (a, b, c) satisfying the condition in Eq. (13), which is done in Alg. 7 in the next section.
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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
Algorithm 7: BracketMinimum (f, To. Fo c.r))
Algorithm 8: Minimize (f. c,r,c
Input: Objective f: IR U (+00), initial point E (-00, exponential search coefficient c > 0,
Input: objective f R IR (+00),
initial point æ E R such that (xx) < <0,
exponential search coefficient > 0,
exponential search base r > 1.
exponential search exponent r > 1,
Output: Triplet (x7 , miid: x*)
absolute tolerance >>
x < 26
Tint; + BracketMinimum (o, C,
y + f(x) k 5
2 x= GoldensectionSeardl (f, x Tint :6)
while true do
x' f- To + cr
efficiency and quality of the output of this algorithm are not analyzed. Furthermore, the presence of discontinuities in our objective function (Assumption 3) warrants a sim- pler algorithm that is provably robust. Therefore, we use a specialized routine, BracketMinimun. shown in Alg. 7.
y' 4 f (x')
y < y
then
x+ 4- x¹
Zo +
break
Bracket Bocckttiinimur works in two stages: Given an initial point Zg. it expands the search interval to the right (towards +00; Line 4-15) and then to the left (towards - oo; Line 17-28). During this, it generates a sequence of exponentially increasing intervals (Lines 5 and 18) and, in the second stage, stops when it detects points that satisfy the condition in Equation (13). This algorithm was inspired by a Stack Exchange post by Lavrov (2017), which was in turn inspired by the exponential search algorithm (Bentley & Yao, 1976).
11
end
12
y y'
14
l t- k: +
end
k 6-00 0
17
while true do
18
x' +- To- c
y' + (x')
19
20
y < y then
Most tunable parameters of the adaptation method in $ 3.3 come from BracketMinimun. In fact, the parameters of BracketMinimum most crucially affect the overall com- putational performance our schemes. Recall that the convergence speed of GSS depends on the width of the pro- vided triplet, Ja cl. Given, this and affect performance as follows: (a) The width of the resulting triplet, |a cl. increases with and c. (b) Smaller and requires more time to find valid triplet. For discussion on how to set these parameters, see $ 3.4.
21
x + x'
22
Tmid + x break
24
end
25
+ x!
26
y y'
27
k 1 k +
end
29
Return (x F'iid x²))
C.4. Minimize (Algorithm 8)
C.3. BracketMinimum (Algorithm 7)
We finally discuss our complete optimization routine, which is shown in Alg. 8. Given an initial point vo and suit- able assumptions, Minimize (f, zo, C, r,)) finds a point that e-close to a local minimum. This is done by first finding an interval that contains the minimum (Line 1) by calling BracketMinimum. which is then used by DeldenSectionSeare
The main difficulty of applying GSS in practice is setting the initial bracketing interval. If the bracketing interval does not contain local minimum, nothing can be said about what GSS is converging towards. In our case, we require a triplet (a, b,c) that satisfies the sufficient conditions in Lemma 2. Therefore, an algorithm for finding such an interval is necessary. Naturally, this algorithm should have computational cost that better or at least comparable to GSS. Otherwise, more naive way of setting the intervals would make more sense. Furthermore, the width of the interval found by the algorithm should be as narrow as possible so that GSS can be run more efficiently.
for proper optimization (Line 2). As such, the computation cost of the routine is the sum of the two stages.
There are four parameters: xo, C, e, Admissible values of will depend on the requirements of the downstream task. On the other hand, and r can be optimized. The effect of these parameters on the execution time is analyzed in Theorem 2, while discussion on how to interpret the thcoretical analysis is in 3.4.
While Press et al. (1992, $10.1) presents an algorithm for finding such a bracket using parabolic interpolation, the
24

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
D. Theoretical Analysis
D.2. Proof of Proposition
In this section, we will provide a formal theoretical analysis of the algorithms presented in App. C as well as the omitted proof of the theorems in the main text.
Proposition 1. Consider joint distributions QU.T, Port-
Then D Paath satisfies the following:
(i) Patth (Por,,Qor)) 2 0 for any Qu:,, Po-r-
(i) Patth (Po.r, Qo:r) = =0 if and only if Porr = Qo.r-
D.1. Definitions and Assumptions
Proof. ①i is trivial. (ii) follows from the fact that if Po.r = Qo:r, the incremental KL divergences are all O, while if Porr # Qo:r- Dpath (Qu.r: Por)) 2 D path (Q,jo=-1, P1 > 0 for any t E [7] by the fact that the conditional KL divergence if and only if Q1jo:t11 Pro--11.
Formally, when we say "local minimum," we follow the following definition:
Definition 1 (Definition 7: Rudin, 1976). Consider some continuous function f: X - R on a metric space X' C R. We say/ f has a local minimum at x² E X if there exists some E > such that f (x*) S f (x)) for all æ X' with |x- 2| < c.
D.3. Sufficient Condition for an Interval to Contain a Local Minimum (Lemma 1)
Also, unimodal functions are defined as follows:
Our adaptation algorithms primarily rely on GSS (Algo- rithm 6) to identify local optimum. To guarantee this, however, GSS needs to be initialized contains local minimum. For this, the following lemma establishes sufficient condition for identifying such inter- vals. As such, we will use these conditions as invariants during the execution of GSS, such that it finds narrower and narrower intervals that continue to contain local minimum.
Definition 2. We say f: [a. -> IR is unimodal if there exists some point x" such that f is monotonically strictly decreasing on [a, ] and strictly increasing on [x",b]. b).
an interval that
Now, recall that our adaptation objective in 3.2 operates on Rz0. During adaptation, however, the objectives are log-tre -egtrransformed so that optimization is performed on R. Therefore, it is convenient to assume everything happens on R. That is, instead of Assumption and 2, we will work with the following assumptions that are equivalent up to log transformation:
Lemma 1. Suppose f satisfies Assumption 3 and there exist some a < b < c. such that
(b) < S (a) < and (bb)
Assumption 3. For the objective /R - IR U (+00), we assume the following:
Then (a, c) contains a local minimum of f on (-00, x))
(a) There exists some x E (-20, oo] such that æ is finite and continuous on (-oc, and oc on
Proof. Consider any triplet (a', c') consisting of three points < b < d' with S(b') < f(a') < oo and f(H) S f(c)) < 00. Then dd' contains a local min- imum of f: f attains its minimum on [a', e) by the ex- treme value theorem, which is either on (al, c')-in which case the result holds immediately or on (a', (M,c'--i which case the result holds because b' is a local minimum since S(b') I
(b) There exists some x E (-00, such that of is strictly monotonically decreasing on (-o0, z]-
(c) There exists some I E [x, 20) such that f is strictly monotonically increasing on [ 2)
Assumption 4. f is unimodal on 2-),
Evidently, these assumptions are equivalent to Assumption and by setting
We now apply this result to triplets contained in [a, c]. First, if c < x use the triplet (a', B' d') = (a, b).. For the remaining cases, assume e 2 x 00 If 2 x, set the triplet dd,b',c))
f (x) (exp (I))
b, for any d E (b,x If (a, b, z). and f(z) 2 f(b), set the triplet (a',b'c)
Σ = log h
Otherwise, if b < I and f(T) < f(b), set the triplet (a' (', e') = (b, æ. d) for any d E (I
g log h
log m0
25

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
D.4. GoldenSectionSearch (Lemma 2)
First, let's establish that k- cc. GSS terminates as soon as | =a S for some 0 2 k co. We will establish this by showing that |a = satisfies a contraction. For this, however, we first have to show that xf, u satisfy
We first establish that, under suitable initialization, GSS is able to locate local minimum. Most existing results assume that f is unimodal (Luenberger & Ye, 2008, $7.1) and show that GSS converges to the unique global mini- mum. Here, we prove a more general result that holds under weaker conditions: GSS can also find a local minimum even when unimodality doesn't hold. For this, we establish that our assumptions in Assumption 3 and initializing at a triplet (a, b, e) satisfying the condition in Lemma 1 are sufficient. Furthermore, while it is well known that GSS achieves linear convergence rate with coefficient & ف +1++0) / we could not find a proof that exactly applied to the GSS variant by Press et al. (1992), which is the one we use. There- fore, we also prove linear convergence rate with a proof that precisely applies to Algorithm
F - + (1-0- 2
(15)
x (1 - 1) + 2
(16)
at all k: 2 O. We will show this via induction. Before we proceed, notice that the name "golden" section search comes from the fact that ø, the golden ratio, is the solution to the equation
Φ²
= - +1 + 1- --
(17)
Now, for some k > O, suppose Equations (15) and (16) hold. Then, if f2 < fi.
Lemma 2. Suppose Assumption 3 holds. Then, for any triplet (a, b, e) satisfying a < <b <, f(b) S f(a)= < co, and (bb S fc). GoldenSectionSeacch (f,a,b,e,e) re- turns a point x² E (-00,200) that is c-close to local minimum after
of
E+1 = 1k
= (1- - x z + -
0 -220 + (1 (1- zz
(Eq. (17))
20 + +1+ 1) (1 - の x
+ log |c - (10g)/c-a)
a12)
= o (0 2 + (1- x14)
objective evaluations, where Φb = (1+ + v5) /2
+
= o - + + k
(Eq. (15))
Proof. For clarity, let us denote the value of the variables To. I1, 22, 23 set at iteration & 2 1 of the while loop in Line 13-27 as at; of, at, wz.. Before the while loop at k 0, they are initialized as follows: If |c- 6l 2 16- al.
= x - 2
x+1 + (1 (1- 4 z3
k+
(Eq. (14))
This establishes Equation (15) for k + 1. Similarly,
122 A+1 = -6 + (1-11-111) ot.
(a8. 29. 222 x²) = (a, b. b ++( (1 x 1) (c b). c)
= = ((1 2 Φ 23)
and
+11-
(Eq. (16)
(x), 21,
(b 12. z.)) = (a, b+ (1- a), b, c)
= 1 (1-0-)) d
+ +(1 + 0- 2-2) 23
otherwise. For all k 2 0, the following set of variables is
set as follows: If f (05) < (1f), the next set of variables
= (1 - 0 4 (0- 1/ + (1 - x -) x4$)
is set as
2/
(1 1+1 1 , x22 33g
1+1 k+1 k+1)
(1 ) 23
(Eq. (15)) (Eq. (14))
A (ut, 22, Φ x2 (1- (1- 2gz x5), , ;(14)
= (1 - (1---11 )- 2 x+1
+
+ 0 23
k+1
and
This establishes Equation (16) for k + 1. The proof for the remaining case of få 2 ft is identical due to symmetry. Furthermore, the base case for k = 0 automatically holds due to the condition on (a, b, e). Therefore, Equations (15) and (16) hold for all h: 2 0.
(z) k+1
k+1 k+1 k+1
Izz
A (1
-- (1 - ---
x0 21 25/)
otherwise. We also denote f 2 (a and A 2 f (at).
From Equations (15) and (16), we now have a precise rate of decrease for the interval | - 2t That is, for f < ft.
Assuming the algorithm terminates at some K- < 00, the algorithm outputs either =x² or 2th"" Therefore, it suffice to show that A* < co, |a - =f < e/2, and that the interval (ar)" xgg ) contains local minimum.
|
kl x
2r x
k-1
= | k-1 (ob x
+ - (1- 2 -11
26

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
25- D3
0
D.5. BracketMinimun (Lemma 3)
and for f2 2 f
We now prove that BracketMinimun returns a triplet "F=iimid:2" 2+) satisfying the condition in Lemma 1. Fur- thermore, under Assumption 3, we analyze the width of the initial search interval represented by the triplet, la+ - |_|. Note that, while BracketMinimum is designed to be valid even if zo 2 x², accommodating this complicates the anal- ysis. Therefore, in the analysis that will follow, we will assume Z0 xxx0
/6 = | 1-1 4-11
= 1((11 - x x1 20
k-1 +
23 1)
k-1
0
x 1 23
k-1 0
Furthermore, This implies, for all k 2 1, the interval =b] shrinks a geometrical rate
[ = = & $10 =88
Lemma 3. Suppose Assumption 3 holds. Then BracketMinimum (f, r,) for xo E (-00,x returns triplet (x", ,Tmid, ('mid,x*)) where x- < Imid < xxt.
Then
-1 = (20 - 1) - - uلي S e/2
22
I (Imia) S f (x < <00 I (Imii) S (f(()) (21)
can be guaranteed by iterating until the smallest iteration count l 25 1 that satisfies
and
xxt after
x S 22 ((r 1)(z - sol+ + m00 - x]+) + 3r
ke 13
2 1 - 2
of (logr)) ((rf - mol+ + mm0 x]+//}}
1
2 - |c al
k=
log
log Φ
objective evaluations.
which yields the execution time complexity statement.
Proof. BracketMinimus has two stages; exponential search to the right (Stage I) and exponential search to the left (Stage IT). In the worst case, Stage I must pass 7 moving to the right starting from Ty. which takes at most O(k,) iterations, where
We now prove that the interval (zb". "=0",""")) contains a lo- cal minima. For this, we will prove a stronger result that (ar; 2²t) contains a local minimum for all l 2 0 by induc-
tion. Suppose, for some k 124 I,
min ft) S f(xb) oc min (J+, f2) S ((x))
(18)
k [logr) log, ((T - xo)//))]
(19)
Similarly, in the worst case Stage II must pass x moving to the left starting from xo + crti,, which takes at most O(ks)
hold. If 13 < f the next set of variables is set as
iterations, where
(oot), k+1 k+1
,_1 2 ) (of, a =3)),
(20)
ke = (logr) (oogr))loo+++ ((x0+
+ to (oto)t)r))))) < (log (ogr))>loo++ (ro + (r(x= - xol+ -2) -2///) /c)
- x0) -x) - /c)l
which guarantees that the inequalities
min (81+1 1(t1) < fi+1 = 5 < f(zb+2) <
min (ft+) .22 pk+1) S fi+ f S f(altl)
Adding these two costs yields the stated result. At the end of Stage I, by inspection, we know that f(x) & f(x+), and that f(x) < co. Also, Stage II continues until the first increase in objective value, which guarantees that co > 1(x 2 (T and [(5miaa S f(x) S f(x+). Finally,
hold. Otherwise, if f$ 2 fh;
(n k+1 k+1
.22 ; . +++1)= = (x) at. 2b)),
guarantee
|x+ -x x 1 (20 + crh.tt) (xo + cr - co crfett)
+++ 1 = 1 < s( < <0 min (Jl+1 s S
rr + rke
min (fl+1, 112 ,h+11 S - f
< f(a)t)).
S rc
r| sol+ + rc rc r + cr - =/++ + rc)
The base case le 0 trivially holds by assumption f(b)- < If (ag) < co, f(b) < f(2)). and the fact that either z or 22 is set as b. Therefore, Equations (18) and (19) hold for all k 2 O. Now, Equations (18) and (19) imply that either (xt, .f. xz) or (a.) .r. 2g} satisfy the condition in Lemma 1. Therefore, local minimum is contained in (xt, for all k 2 0.
5 r
<r²
[ xol+ + [To - - + cr
5, + +20)
S 1² ([T - =0o + 0z0 - -]+ + r[ā - zo]+ + 3c)
= r² ((r + 1)[z - zol+ + + - ==++)=++++ + 3r
27

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
D.6. Minimize (Theorem 2)
D.7. AdaptStepaize (Proof of Theorem
We prove that combining BracketMinimum and GoldenSectionSearchh which call Minimize, results in optimization algorithm that finds point c-close to local minimum in O (log (A/c) time.
We now present the proof for the theoretical guarantees of Alg. 2 in the main text, Theorem 1. Since most of the heavy lifting in Alg. 2 is done by Alg. 8, Theorem is almost corollary of Theorem 2. The main difference is that Alg. invokes Alg. 5 at = 1 and operates in log-space. Therefore, the proof incorporates these two modifications into the results of Theorem 2.
holds. Then Minimize (f, no C, r, c) returns a point that is c-close to a local minimum after Com + Cass objective evaluations, where
Theorem 2. Suppose Assumption
Theorem Suppose Assumption 1 holds. Then AdaptStepsize (C, Nguess: ò, c, r, returns a step size h (0, hov) that is c-close
Cbm = o 1 - log+ (4
local minimum of C in log-scale after Creas + Com + Cgss objective evaluations, where, defining A - log log+ (T//ho) + llog. (ho/h) and ho - min(hguess: h)²)
log
Cgss = 0 { log (n + r2c) 1 c
Creas = 0f5-1 10-1log+ (Mxuem/h>>)}
where A A [20 + + Fo
Com = Of (logr)~((og) log, (Arc-))
Proof. Com immediately follows from Lemma 3, while Cass. on the other hand, follows from Lemma
Cgs = O {log
c c-
00108. Cass 0 log+ (x (x- 1 } x
Proof. Since Assumption implies that the function Clog (h) satisfies Assumption 3 with
= = -0f188+ log+ (r (+*++ro))
f = clog (h). I = log h x = log h, x = log h
Then the result is simple application of the lemmas in the previous sections.
where we plugged in the bound on |x+ - x: from Lemma 3. This yields the stated result. Furthermore, since BracketMin imum returns a triplet (x=, Ximid, that satis- fies the requirement Lemma 2, the output x" E (-00, x 20) is c-close to a local minimum.
First, under Assumption 1, Alg. 5 can find a point l' (-oo, log ho) that guarantees Llog (e) oo within
foldenSectionSearcl as stated in
Creas S 0 10 log+ (Aiguess/ ))))
Remark In Theorem the "difficulty" of the problem is represented by A O, where the magnitude of [Iro [ =]+ and [x: - xo represent the quality of the initialization 2o (how much zo undershoots or overshoots T and z). Further- more, we have 4 2 I z where [ æl can be thought as the quantitative multimodality of the problem. There- fore, the exccution time of Minimize becomes longer as the problem becomes more multimodal and the initialization far from [F,a]. x).
log hguess if hgueess h and e' < log h otherwise. Then, Theorem 2 states that Line 6 of Alg. 2 is guaranteed to find local minimum of L after Com + Cass iterations, while
steps. Furthermore,
A t zol4 + [xt - -21+ [log h {) + [e _f log h]+
log, (h/ho) + loog, (ha/h) hh)
Remark 2. The execution time of Minimize(J, C, r,e) depends on r and c. In general, the best-case performance (4 = 0) can only become worse as c increases. On the other hand, in the worst-case when
large, increasing r reduces Cam- while slowly making Cgss worse. Therefore, large improves the worst-case performance.
28

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
E. Backward Kernels
E.1. Some backward kernels are not like the others
Here, we will discuss some options for the "backward kernel" used in SMC samplers the static model setting (Del Moral al., 2006; Neal, 2001).
log Z.
-20
Detailed Balance Formula. In the literature, the choice
Detailed Balance Formula Forward
-40
Time-Coree Forward True
I.cbf (12, A H/ (Ir) Kr (Z1-1, 24)
(22)
7/ (11-1)
60
128
256
512
1024 2048 4096
which we will refer as the "detailed balance formula backward kernel," has been the widely used (Dai et al., 2022; Bernton et al., 2019; Heng al., 2020). Conveniently, Eq. (22) results in a simple expression for the potential
Number of Particles (N)
Figure 7. Comparison of backward kernels for SMC-LMC. The solid lines are the median, while the colored bands mark the 80% empirical quantile over replications.
G1 (2)-1, = Ytt (x1-1)
Time-Correct Forward Kernel. In $ 4.1, we used the forward kernel at time t 6 1,
N-- (21-1))
I. 1e (t-) A K,- (t-1)
which does not involve the densities of K, The origin of this backward kernel is that many 7e-invariant MCMC kernels used in practice satisfy the detailed balance formula (Robert & Casella, 2004, Def. 6.45) with Te.
which we will refer to the time-correct forward kernel. Unlike I.fed the stationary distribution of this transport map is properly -T- Informally, the reasoning is that
(TT @ 1) (t-,
(Te @ TT-1) (Iz, 2x-1) = 1.
Te (me-1) K2 (2)- = ne. (1+) Ke (Iit 1e-1)
(Tr-1 @ Kτ) (xt-1,
(Te-1 me) (z)
which, given h 24) = K2 (#1-1, yields Eq. (22) after re-aranging.
Therefore, this should result in lower variance.
E.2. Empirical Evaluation
The detailed balance formula backward kernel biased. Now, let's focus on the fact that K, Lt only holds under the detailed balance condition. Said differently, L is properly normalized kernel only when K, satisfies de- tailed balance. This implies that, for non-reversible kernels like LMC, using the detailed balance formula kernel with > may result in biased normalized constant estimates. This bias can be substantial, as we will see in App. E.2. Fortunately, this bias does diminish as h, -> and T - oo since the continuous Langevin dynamics reversible un- der stationarity (Heng et al., 2020). However, the need for smaller step sizes means that larger number of SMC steps T has to be taken for the Markov process to converge.
Setup. We compare the three backward kernels on a toy problem with d = 10 dimensional Gaussians: π = N (30 La. Ia), qu = / (0a, I2). Since the scale of the tar- get distribution constant under geometric annealing, a fixed step size he 0.5 should work well. We use linear schedule with T = 64.
Results. The results are shown in Fig. 7. The backward kernel from the detailed balance formula severely overesti- mates the normalizing constant due to bias, while the for- ward kernel exhibits significantly higher variance than the time-correct forward kernel.
Forward Kernel. The properly normalized analog of dbf at time t > 1 is the "forward" kernel
E.3. Conclusions
We have demonstrated that caution must be taken when us- ing the popular backward kernel based on the detail-balance formula. Instead, we have proposed the "time-correct for- ward kernel," which is not only valid but also results in sub- stantially lower variance. Unfortunately. the time-correct forward kernel is only available for MCMC kernels that have a tractable density, which may not be the case; for instance, the KLMC kernel used in $ 4.2 does not have this option. However, whenever it is available, it should be preferred.
I fwdd 1--1 11-1 K2 (I1,
This has been used, for example, by Thin et al. (2021). Recall that the "optimal" L,-1 is kernel that transports the particles following P. to follow P.-1. The fact that we are using K, to do this implies that we are assuming Pa R Pt-1, which only true if T' is large. We propose a different option, which should work even when T is moderate or small.
29

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
F. Additional Experimental Results
F.I Comparison Against Fixed Step Sizes
F.1.1. SMC-LMC
5
50
1000
250
log Z estimate
0
0
0
0
-5-
200
1000
250
50
-10
Grid Search
400
2000
500
Adaptive
100
3000
...
Trie
-2
-550
600
4000
1000
10"
10a
10
10'
10
10
10
10'
10
10 10
10'
10
10
10
10'
10
10
10"
10'
Step Size (h)
Ste Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
(a) Funnel
(b) Brownian
(c) Sonar
(d) Pines
(c) Bones
6600
1500
-300
100
10000
log Z estimate
800
2000
400
1000
10000
-1000
2500
500
-200
20000
1200
3000
600
-300
30000
10 < 10"
10" 10'
10
10"
10" 10'
10
10" 10"
10'
10
10"
10
10'
10
10"
10"
10'
Step Size (h)
Step Size (h)
Step Size (4)
Step Size (h)
Step Size (h)
(f) Surgical
(g) HMM
(h) Loss Curve
(i) Pilots
Diamonds
5000
4000
500
500
7000
log estimate
5500
2000
1000
8000
-600
1500
1500
9000
-550
2000
4000
2000
1x10
10
10
10" 10'
10
10 10
10
10
10
10
10
10th
10th 10
10'
10
10
10
10'
Step Size (h)
Step Size (f)
Step Size (h)
Step Size (h)
Step Size (h)
(k) Seeds
()) Downloads
(m) Rats
(n) Radon
(0) Election88
-1000
8x10
1000
500
2000
log Z estimate
6x10*
-1500
-400
-500
4x
-1000
-1000
-2000-
-6000
2x10
2000
1500
-2500
3000
2000
8000
1
10 10" 10'
10"
1 10"1
10
10
10t
10"1
10'
10
10 10
10'
10
5
10
10 10'
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
(p) Butterfly
(q) Birds
(r) Drivers
(s) Capture
(0) Science
26600
2000
log estimate
-28000
4400
-30000
60000
-30000
-34000
8000
10 3 10
10'
10
10 10
10'
Step Size (h)
Step Size (f)
(u) Three Men
(v) TIMSS
Figure 8 SMC-LMC with adaptive tuning v.s. fixed step sizes. The solid lines are the median estimate of log Z, while the colored regions the 80% empirical quantiles computed over replications.
30

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
F. 1.2 SMC-KLMC
10
525
250
log Z estimate
500
-200
-10
475
250
-1-
Grid Search
20
450
500
Adaptive
400
30
425
750
True
40
600
400
1000
10
10 10"
10'
10"
10
10" 10'
10
10 10" 10'
10
10
10
10'
10 3
10 10" 10'
Step Size (h)
Step Size (h)
Step Size (/)
Step Size (4)
Step Size (h)
(a) Funnel
(b) Brownian
(c) Sonar
(d) Pines
(c) Bones
600
1500
350
50
10000
log Z estimatee
0
8000
2000
400
-10000
-1000-
2500
450
-50
20000
-1200
3000
500
30000
10 1 10 10'
10 10 10 10
10 10t 10
10'
10
10ta
10
10'
100
10
10
10'
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
(I) Surgical
(g) HMM
(h) Loss Curve
(i) Pilots
i
Diamonds
500
4000
500
7000
log Z. estimate
520
2000
1000
1000
8000
5440
2000
1500
9000
5660
-2000
3000
-580
4000
4000
2000
=x000
10
1
10" 10'
10
10
10" 10
10
10t
10"
10
10
10
10
10'
10
10
10
10'
Step Size (h)
Step Size
Step Size (h)
Step Size (h)
Step Size (h)
(k) Seeds
()
Downloads
(m) Rats
(n) Radon
(0) Election88
-1000
80000
400
25
2500
log Z estimate
1200
60000
200
20
3000
15
1400
40000
10
3500
1600
20000
200
-8800
400
4000
10
10"
10 3
10
10
10
10
3
10
10"
10'
10
10t 10'
5
10
10 10
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
Step Size (h)
(p) Butterfly
(q) Birds
(r) Drivers
(s) Capture
(1) Science
-26000
2500
log estimate
3000
-28000
3500
4000
30000
4500
-32000-
-5000
1 10 10
10'
10Th
10th 10
10'
Step Size (h)
Step Size (h)
(u) Three Men
(v) TIMSS
Figure SMC-KLMC with adaptive tuning v.s. fixed step sizes and refreshment rates. SMC-KLMC fixed parameters h we show the result of the best-perfominng refreshment rate. The solid lines are the median estimate of log Z, while the colored regions are the 80% empirical quantiles computed over 32 replications.
31

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
F.2. Comparison Against End-to-End Optimization
For all results, the "cost" is calculated as the cumulative number of gradients and Hessian evaluations used by each method. Also, end-to-end optimization with variational tuning of the reference is referred as "End-to-End + VI." (End-to-end optimization methods require Hessians.) For all figures, the error bars/bands are 80% empirical quantiles computed from 32 replications, while y is the Adam step size used for end-to-end optimization.
F.2.1. SMC-LMC
2
log Z Estimate
2
Adaptive SMC
End-to-End
End-to-End VI
True
-8
IM 2M 3M
4M 5M
0
1M
2M
3M
4M
5M
0
500k
1M
1.5M
2M
Cost
Cost
Cost
(a) Y 10
(b)
10
(c)
10
Figure 10. Comparison against end-to-end optimization on Funnel.
10
10
0
0-
log Z Esttmate
10
10
-20
+
Adaptive SMC
20
20
-30
End-to-End
30
30
End-to-End VI
40
40
True
500
50
50
-60
IM 2M 3M
4M
SM
0
2.5M
5M
7.5M 10M
0
IM
2M
3M
4M
5M
Cost
Cost
Cost
(a) 10
(b)
10
(c)
10
Figure 11. Comparison against end-to-end optimization Brownian. The result for variational reference tuning (End-to-Bae End + VI)
for 7 = 10 2 is omitted as most rums diverged.
log Estimate
Adaptive SMC
200
End-to-End
4400
300
150
End-to-End VI
True
400
600
200
500
-800
600
-250
IM 2M 3M
4M
5M
0
1M 2M
3M
4M
SM
0
1M
2M
3M
4M
5M
Cost
Cost
Cost
(a) 7 10-4
(b) 7
10
7
10
Figure 12. Comparison against end-to-end optimization on Sonar.
1000
1000
1000
log Z Estimate
500
500
0
Adaptive SMC
-500
End-to-Ena End
500
5000
1000
End-to-End
1000
1000
1500
1500
1500
0
IM 2M
3M 4M 5M
IM
2M
3M
4M
5M
IM
2M
3M
4M
5M
Cost
Cost
Cost
(a)
10-4
(b) 7 = 10-3
(c) = 10-2
Figure 13. Comparison against end-to-end optimization on Pines.
32

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
F.2.2. SMC-KLMC
0.25
0.25
0.25
log Z Estimate
0.25
0.25
0.25
Adaptive SMC
-0.5
-0.5
0..
End-to-End all
-0.75
End-to-End + VI True
0.75
0.75
2M 3M Cost
5M
2.5M
5M
7.5M
10M
3M
5M
Cost
Cost
(a) 10
(b)
10-3
(c) 7 10-2
Figure 14. Comparison against end-to-end optimization Funnel.
10
10
10
log Z Estimate
0
0
-10
10
Adaptive SMC
-20
20
20
End-to-End
-30-
End-to-End VI
-30-
-30
True
440
-40
-40
0
IM 2M 3M
4M 5M
0
2.5M
5M
7.5M 10M
0
IM
2M 3M
4M
SM
Cost
Cost
Cost
(a) 7
10 10m4
(b) 7 10
(c) 7 10
Figure 15. Comparison against end-to-end optimization on Brownian.
-100
105
105
log Z Estimate
110
-110
-10
120
Adaptive SMC
-130
End-to-End
140
End + VI
-120
-120-
True
-150
125
125
0
IM 2M 3M 4M 5M
0
1M
2M
3M
4M
5M
0
1M
2M
3M
4M
5M
Cost
Cost
Cost
(a)
10-4
(b) 7 = 10-3
(c) 7 = 10-2
Figure 16. Comparison against end-to-end optimization on Sonar.
520
520
520
log EEtimatte
500
500
500
480
480
480
Adaptive SMC
460
460
460
End-to-End
440
End-to-End + VI True
440
440
420
420
420
IM
2M 3M
4M 5M
0
500k
1M
1.5M
2M
0
250k
500k
750k
1M
Cost
Cost
Cost
7 10"4
(b)
10
(c) 10
Figure 17. Comparison against end-to-end optimization Pines.
33

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
F3. Adaptation Cost
In this section, we will visualize the cost of adaptation of our algorithm. In particular, we show the number of objective evaluations used at each SMC iteration during adaptation. Recall that the cost of evaluating our objective is in the order of (B) unnormalized log-density evaluations (log 7) and its gradients (V log ?), where B is the number of subsampled particles ($ 3.2). Therefore, the cost of N/ B adaptation objective evaluations every SMC step roughly amounts the cost of single vanilla SMC rum with N particles. That is, for N 1024 and B 128, the cost of running our adaptive SMC sampler is comparable two three times that of vanilla SMC sampler. The exact number of objective evaluations spent at each SMC iteration is shown in the figures that will follow. All experiments used N 1024, B 128, and T 64
F.3.1. SMC-LMC
# obj. evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (1)
(a) Funnel
# of obj. evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (1
(b) Brownian
# of evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (1)
(c) Sonar
# of obj. evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (1)
(d) Pines
# obj. evals.
30
20
10-
10
20
30
40
50
60
SMC Iteration (/)
(e) Bones
Figure Number of objective evaluations spent during adaptation each SMC iteration. The size of the markers represents the proportion of runs that spent each respective number of evaluations among 32 independent runs.
34

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
# of of evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (f)
(a) Surgical
# of obj. evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (f)
(b) HMM
# of evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (1)
(c) Loss Curves
# obj. evals.
30
20
10
10
20
30
50
60
SMC Iteration (0)
(d) Pilots
# obj. evals.
30
20
10-
0
10
20
30
40
50
60
SMC Itcration (f)
Diamonds
# ofj evals.
30
20
10
0
10
20
50
SMC Iteration (/)
(f) Seeds
Figure 19. Number of objective evaluations spent during adaptation at each SMC iteration. The size of the markers represents the proportion runs spent each respective number evaluations among 32 independent runs.
35

---

Tuning SMC Samplers via Greedy Incremental Divergence Minimization
# of of evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (f)
(a) Downloads
# of obj evals.
30
20
10-
10
20
30
40
50
60
SMC Iteration (/)
(b) Rats
# of evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (/)
(c) Radon
# obj. evals.
30
20
10
10
20
50
60
SMC Iteration (1)
(d) Election88
# of obj. evals.
30
20
10
10
20
30
40
50
60
SMC Itcration (f)
(e) Butterfly
# ofj evals.
30
20
10-
10
20
50
SMC Iteration (/)
(i) Birds
Figure 20. Number of objective evaluations spent during adaptation at each SMC iteration (continued). The size of the markers represents the proportion runs that spent respective number of evaluations among 32 independent runs.
36

---

Tuning SMC Samplers via Greedy Incremental Divergence Minimization
# of evals.
30
20
10
10
20
30
40
50
60
SMC Iteration (7)
(a) Drivers
# of evals.
30
20
10-
10
20
50
SMC Iteration (f)
(b) Capture
of obj. evals.
30
20
10-
20
50
SMC Iteration (/)
(c) Science
# ofj. evals.
30
20
10
20
50
SMC Iteration (/)
(d) Three Men
obj. evals.
30
20
10
10
20
30
50
60
SMC Iteration (/)
(c) TIMSS
Figure 21. Number of objective evaluations spent during adaptation at each SMC iteration (continued). The size of the markers represents proportion runs that spent respective number of evaluations among 32 independent runs.
37

---

Tuning SMC Samplers via Greedy Incremental Divergence Minimization
F.3.2. SMC-KLMC
Atfob.. evals.
60
40
# of
20
0
10
20
30
40
50
60
SMC Iteration (1)
(a) Funnel
# #ofobj. evals.
60
20
0
10
20
30
40
50
60
SMC Tteration (t)
(b) Brownian
# of obj. evals.
60
40
20
0
20
30
40
60
SMC Iteration (f)
(c) Sonar
ofj.. evals.
40
0
20
30
40
60
SMC Iteration (f)
(d) Pines
off... evals.
60
40
# of obj.
20
0
10
20
30
40
50
60
SMC Iteration (/)
(e) Bones
obj. evals.
60
40
20
0
10
20
30
40
50
60
SMC Iteration ())
(f) Surgical
Figure 22. Number of objective evaluations spent during adaptation at each SMC iteration. The size of the markers represents the proportion of runs that spent each respective number evaluations among 32 independent runs.
38

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
# of evals.
60
40
20
10
20
30
40
50
60
SMC Iteration (f)
(a) HMM
# of obj. evals.
60
40
20
10
20
30
40
50
60
SMC Iteration (f)
(b) Loss Curves
# of evals.
60
40
20
10
20
30
40
50
60
SMC Iteration (/)
(c) Pilots
# of obj. evals.
60
40
20
10
20
30
50
60
SMC Iteration (1)
(d) Diamonds
# of of obj. evals.
60
40
20
10
20
30
50
60
SMC Itcration (1)
(e) Seeds
# of evals.
80
60
40
20
10
20
50
SMC Iteration (/)
(f) Downloads
Figure 23. Number of objective evaluations spent during adaptation at each SMC iteration (continued). The size of the markers represents the proportion runs that spent respective number of evaluations among 32 independent runs.
39

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
# of evals.
60
40
20
10
20
30
40
50
60
SMC Iteration (f)
(a) Rats
# oo obj. evals.
60
40
20
10
20
30
40
50
60
SMC Iteration (/)
(b) Radon
# of evals.
60
40
20
10
20
30
40
50
60
SMC Iteration (n)
(c) Election88
# of evals.
60
40
20
10
20
30
50
60
SMC Iteration (1)
(d) Butterfly
# of obj. evals.
60
40
20
0
10
20
30
50
60
SMC Itcration (1)
(e) Birds
# of evals.
60
40
20
10
20
50
SMC Iteration (/)
(f) Drivers
Figure 24. Number of objective evaluations spent during adaptation at each SMC iteration (continued). The size of the markers represents the proportion runs that spent respective number of evaluations among 32 independent runs.
40

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
# of evals.
60
40
20
10
20
30
40
50
60
SMC Iteration (f)
(a) Capture
# of objj. evals.
60
40
20
10
20
30
40
50
60
SMC Iteration (/)
(b) Science
# of evals.
60
40
20
10
20
30
40
50
60
SMC Iteration (/)
(c) Three Men
# obj. evals.
60
40
20
0
10
20
30
50
60
SMC Iteration (0)
(d) TIMSS
Figure 25. Number of objective evaluations spent during adaptation each SMC iteration (continued). The size of the markers represents the proportion runs that spent each respective number of evaluations among 32 independent runs.

---

Tuning SMC Samplers via Greedy Incremental Divergence Minimization
F.4. Adaptation Results from the Adaptive SMC Samplers
Finally, we will present additional results generated from our adaptive SMC samplers, including the adapted temperature schedule, step size schedule, and normalizing constant estimates. The computational budgets set Tj 64 and N 1024 with B 256.
F.4.1. SMC-LMC
0.5
10'
10*
Temperature (2.)
10a
Local Barrier
Step Size (ht)
0
10'
Z |oo
10
0.1
0.5
10
10a
-1
10
10
0.01
0
500k
IM
1.5M
0
25
50
75
100
10
10
10
10"
10"
10"
10
10
10" 10
of Gradient Evaluations
SMC Iteration (1)
Temperature (2.)
Temperature (2.
(a) Funnel
5
10
10
Temperature (2)
0-
Local Barrier
10*
Step Size (h))
10"
Z
0.01
log
-5-
10a
10
10a
-10-
10
10a
10'
10
10
0
1M
2M
3M
0
100
200
300
10
$10
"10
'10 '10"'''0''00 10
10
$10
$10
'10
'10" 10
of Gradient Evaluations
SMC Iteration (1)
Temperature (2.
Temperature (2.)
(b) Brownian
106
10"
10
10
108
Temperature (2.)
10"1
Local Barrier
10a-
Step Size (h)
10a
Z log
10
-110
10a-
10
10"
-112
10
10²
-114-
10"
10
10ah
0
1M
2M
3M
4M
0
100
200
300 400
10
10th
10
10"
10
10
10
10
10 10º
of Gradient Evaluations
SMC Iteration (1)
Temperature (2.
Temperature (2.)
(c) Sonar
550
10
10a
500
Temperature (2)
10a1
Local Barrier
Step Size (h.)
10"
log Z
10
450
0.1
10
10²
400
10
350
10
10'
0.01
0
2M
4M
6M
8M
0
250
500 750
10 $
$10110 '10'100' 10"
10 *10 '10
'100"100"" '10'110 '1'" 10
of Gradient Evaluations
SMC Iteration (f)
Temperature (2.)
Temperature (2.)
(d) Pines
Figure 26. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-LMC. The dotted line is the ground truth value obtained from large budget run. For the normalizing constant estimate, the confidence intervals in the vertical and horizontal dircctions the 80% quantiles obtained from replications. The temperature schedule, local communication barriers, and the step sizes from subset of runs are shown.
42

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
-60
10
10"
Temperature (2)
-80
10
Local Barrier
Step Size (h)
10'
log Z
10
-100
0.1
10
10a
-120
10
140
10"
10'
0.01
IM
2M
3M
50
100
150
10
10
10
10"
10"
10
10
10
10" 10
# of Gradient Evaluations
SMC Iteration (1)
Temperature (2.)
Temperature (2.)
(a) Bones
-725
10'
10
10
Temperature (2)
10"1
Local Barrier
10*
Step Size (h.)
log Z
-726
10
10a1
10a
10
-727-
10
10
10
-728
10
10
10
0
500k
1M
1.5M 2M
0
50
100
150 200
10
10
10a
10
10
10
10
10
10 10
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Temperature
(2)
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(b) Surgical
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10
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SMC Itcration (f)
Temperature (2))
Temperature (2.)
(d) Loss Curves
Figure 27. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-LMC (continued). The dotted line is the ground truth value obtained from large budget run. For the normalizing constant estimate, the confidence intervals in the vertical and horizontal directions are the 80% quantiles obtained from 32 replications. The temperature schedule, local communication barriers, and the step sizes from a subset of runs are shown.
43

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
25
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10
Temperature (2)
20
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IM
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50
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# of Gradient Evaluations
SMC Iteration (0)
Temperature (2.)
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(a) Pilots
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10
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Temperature (2)
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0-
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10*
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SMC
Iteration (f)
Temperature (2)
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(b) Diamonds
-22
10
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Temperature (2)
10
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10*
Step Size (h.)
10a1
log Z
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100 200
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Temperature (2))
Temperature (2.
(c) Seeds
3000
10
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10a1
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Local Barrier
Step Size (h.)
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10
10ah
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0
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10
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10a
10
IM 2M
4M SM
100 200 300 400500
10 '10 $10'10 "100"00""00000000 "1" 10
$1010 $10 $10'00000
"10"10" 10
of Gradient Evaluations
SMC Itcration (f)
Temperature (2))
Temperature (2.)
Downloads
Figure 28. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-LMC (continued). The dotted line is the ground truth value obtained from large budget run. For the normalizing constant estimate, the confidence intervals in the vertical and horizontal directions are the 80% quantiles obtained from 32 replications. The temperature schedule, local communication barriers, and the step sizes from a subset of runs are shown.
44

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
800
10"
10"
10
Temperature (2)
-900
Local Barrier
Step Size (h)
log
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10a
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10
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# of Gradient Evaluations
SMC Iteration (7)
Temperature (2.)
Temperature (2.)
(a) Rats
-900
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Temperature (2)
Local Barrier
10*
Step Size (h.)
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-1000
le
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4M
0
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10 $10 '10 $10 "10 10
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10
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SMC Iteration (f)
Temperature (2)
(b) Radon
-7400
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Temperature (2)
10
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10
Step Size (h.)
10a²
log Z
-7600-
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4M 5M
200
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Temperature (2)
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10
Step Size (h))
10a1
z
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log
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200
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10 $10 $10
10 Temperature (2.)
'10 $10 $10'100 $10"10 10
of Gradient Evaluations
SMC Itcration (f)
Temperature (2)
(d) Butterfly
Figure 29. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-LMC (continued). The dotted line is the ground truth value obtained from large budget run. For the normalizing constant estimate, the confidence intervals in the vertical and horizontal directions are the 80% quantiles obtained from 32 replications. The temperature schedule, local communication barriers, and the step sizes from a subset of runs are shown.
45

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
60050
10
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10
60000
Temperature (2)
10
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Step Size (h)
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# of Gradient Evaluations
SMC Iteration (1)
Temperature (2.)
Temperature (2.)
(a) Birds
400
10'
10
10
Temperature (2)
10"
Local Barrier
Step Size (h))
10a1
10*
log Z
200
10
10"²
100
10
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10
0-
10
-100
10
10a
10
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2M
4M
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0 200
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10 Temperature (2)
$10 $10 $10011000"''''' 10
10 Temperature (2.)
10 $100110"00 '10 "10" 10
of Gradient Evaluations
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(b) Drivers
40
10
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Temperature (2.)
20
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Local Barrier
Step Size (h.)
10
log Z
10a
0.1
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10"5
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0.01
6M
200
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10
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SMC Iteration (!)
Temperature (2)
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(c) Capture
-2900
10
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Temperature (2.)
-3000
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Local Barrier
Step Size (h.)
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10
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10
-3200
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-3300
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of Gradient Evaluations
SMC Itcration (f)
Temperature (2))
Temperature (2.)
Science
Figure 30. Normalizing constant estimate, temperature schedule, local communication barrier, and stepsize schedules obtained by running SMC-LMC (continued). The dotted line is the ground truth value obtained from large budget run. For the normalizing constant estimate, the confidence intervals in the vertical and horizontal directions are the 80% quantiles obtained from 32 replications. The temperature schedule, local communication barriers, and the step sizes from a subset of runs are shown.
46

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
-27200
10
10"
10"
-27300
Temperature (2)
10
Local Barrier
Step Size (h)
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# of Gradient Evaluations
SMC Iteration
Temperature (2.)
Temperature (2.)
(a) Three Men
-8800
10
10
10
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Temperature (2)
10"1
Local Barrier
Step Size (h.)
10*
10a1
-3200
10
log
-3400-
10
10a
10
-3600-
10
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10
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2M
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0
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of Gradient Evaluations
SMC Iteration (f)
Temperature (2)
Temperature (2.)
(b) TIMSS
Figure 31. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-LMC (continued). The dotted line is the ground truth value obtained from a large budget run. For the normalizing constant estimate, the confidence intervals in the vertical and horizontal directions the 80% quantiles obtained from 32 replications. The temperature schedule, local communication barriers, and the step sizes from a subset of runs are shown.
47

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
F.4.2. SMC-KLMC
10'
10"
Temperature (2)
10
Local Barrier
10
Step Size (h.)
-.2
log
10
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0.1
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IM
1.5M
30
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# Gradient Evaluations
SMC Iteration (f)
Temperature (4.)
Temperature (2.)
(a) Funnel
10"
10
Temperature (2.)
10"
Local Barrier
Step Size (h.)
10a
log Z
10"3
0.1
10
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10a
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0.01
500k
IM 1.5M 2M
20
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Gradient Evaluations
SMC Iteration (t)
Temperature (2))
Temperature (2.)
(b) Brownian
-107
10
10
-088
Temperature (2)
10"
Local Barrier
10a
Step Size (h:)
log z
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IM
1.5M 2M
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# Gradient Evaluations
SMC Iteration
Temperature (1.)
Temperature (2.)
(c) Sonar
504
10"
10a
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Temperature (2.)
10"1
Local Barrier
10a
Step Size (h.)
log Z
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494
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0.1
500k
IM 1.5M 2M
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30
40
10
110 100011000000 10" 10
10 100 10 10
# of Gradient Evaluations
SMC Iteration
Temperature (2.)
Temperature (2.)
(d) Pines
Figure Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-KLMC. The dotted line the ground truth value obtained from large budget run. For the normalizing constant estimate, the contidence intervals in the vertical and horizontal directions are the 80% quantiles obtained from 32 replications. The temperature schedule, local communication barriers, and the step sizes from subset of runs shown.
48

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
-70
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Temperature (2)
10
Local Barrier
Step Size (h)
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# of Gradient Evaluations
SMC Iteration (0)
Temperature (2,)
(a) Bones
-725.6
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10
Temperature (2)
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10a
Local Barrier
Step Size (h.)
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log Z
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SMC Iteration (f)
Temperature (2)
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(b) Surgical
-1660
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Temperature (2)
10a
Local Barrier
Step Size (h.)
-1680
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log Z
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10
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25
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of Gradient Evaluations
SMC Iteration (!)
Temperature (2,)
Temperature (2.
(c) HMM
-370
10
10
Temperature (2.)
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Local Barrier
Step Size (h.)
10²
log Z
1
-390
0.1
10
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of Gradient Evaluations
SMC Itcration (f)
Temperature (2))
Temperature (2.)
(d) Loss Curves
Figure 33. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-KLMC (continued). The dotted line is the ground truth value obtained from large budget For the normalizing constant estimate, the confidence intervals in the vertical and horizontal directions are the 80% quantiles obtained from 32 replications. The temperature schedule, local communication barriers, and the step sizes from a subset of runs are shown.
49

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
25
10
10
20
Temperature (2)
10
Local Barrier
Step Size (h.
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10
log
15-
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3M
0
25
50
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SMC Iteration (/)
Temperature (2))
Temperature (2.)
(a) Pilots
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10
10*
0
10
Local Barrier
Step Size (h)
Temperature
10"
log 2
-1 10000
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10
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10
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0
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of Gradient Evaluations
SMC Iteration (f)
Temperature (2)
Temperature (2.
(b) Diamonds
-527
10
10
10a
Temperature (2)
10a
Local Barrier
10'
Step Size (h)
log Z
-528
10
10a
10"
10a
-529
10
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10
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IM
1.5M 2M
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SMC Iteration (1)
Temperature (2)
Temperature (2.)
(c) Sceds
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10
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Temperature (2.)
10
Local Barrier
Step Size (h)
2000
10a
10a1
log 7
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10
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10
4
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# of Gradient Evaluations
SMC Iteration (1)
Temperature (2.)
Temperature (2.)
(d) Downloads
Figure 34. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-KLMC (continued). The dotted line is the ground truth value obtained from large budget run. For the normalizing constant estimate, the confidence intervals the vertical and horizontal directions are the 80% quantiles obtained from replications. The temperature schedule, local communication barriers, and the step sizes from a subset of 8 runs are shown.
50

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
10"
109
0.1
-1000
Temperature (2)
Local Barrier
10"
Step Size (h)
log Z
-2000
10a
10
10a
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0.01
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10
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40
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# of Gradient Evaluations
SMC Iteration (1)
Temperature (2.)
Temperature (7.)
(a) Rats
-900
10
10
10
Temperature (2)
1000
10"
Local Barrier
10*
Step Size (h.)
log Z
10
-1000
10²
10
10
-1200
10
10
-1300
10
10
10a
0
500k
IM
1.5M 2M
0
20
40
60
10
10
10a
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10 4
10
10
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of Gradient Evaluations
SMC Iteration (f)
Temperature (2)
Temperature (2.
(b) Radon
-7400
10
10
10a
-7500-
Temperature (2)
10
Local Barrier
Step Size (h.)
10'
log Z
10
-6600
10"1_
10ah
10a
7700
10
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10
10
10
3M
20
40
4
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10
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of Gradient Evaluations
SMC Iteration (!)
Temperature (2))
Temperature (2.
(c) Election88
-175
10
10*
Temperature (2.)
1180
10"1
Local Barrier
Step Size (h.)
10'
log Z
10
-188
0.1
10
10a
-1190
10
-1195
10
10
0.01
3M
60
10
10
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10
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of Gradient Evaluations
SMC Itcration (f)
Temperature (2))
Temperature (2.)
(d) Butterfly
Figure 35. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-KLMC (continued). The dotted line is the ground truth value obtained from large budget For the normalizing constant estimate, the confidence intervals in the vertical and horizontal directions are the 80% quantiles obtained from 32 replications. The temperature schedule, local communication barriers, and the step sizes from a subset of runs are shown.
51

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
60050
10
10*
60000
Temperature (2)
10
Local Barrier
Step Size (h.)
10'
log 2g
59950
10
0.1
59900
10
10a
59850
10
59800
10
10
0.01
0
IM
2M
3M
0
50
100
150
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10S
10
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10 1
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10
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10
# of Gradient Evaluations
SMC Iteration (/)
Temperature (2))
Temperature (2.
(a) Birds
400
10
10
Camperaiai (2.)
10
Local Barrier
Step Size (h))
300
10
log Z
10
0.1
10
200
10"
10
100
10
10'
0.01
0
1M
2M
3M
0
20
40
60
80
10
10Th
10
10
10"
10
10th
10
10 10"
of Gradient Evaluations
SMC Iteration (1)
Temperature (2)
Temperature (2.
(b) Drivers
22
10
10
Temperature (2.)
10a
Local Barrier
Step Size (h.)
20
10'
10
log
10
18
10a
10
16
10
10
0.1
0
IM
2M
3M
0
10
20
30
40
10
10
10
10 10
10 100
"10 10
of Gradient Evaluations
SMC Iteration (1)
Temperature (2)
Temperature (2.)
(c) Capture
-2950
10
10
3000
Temperature (2)
10
Local Barrier
Step Size (h))
10a
log Z
10
-3050
0.1
10
10a
3100
10
-3150-
10
10
0.01
0
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3M
0
20
40
60
10
4
10
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10"
10º
10
4
10
10
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# of Gradient Evaluations
SMC Iteration (1)
Temperature (2.
Temperature (2.)
(d) Science
Figure 36. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-KLMC (continued). The dotted line is the ground truth value obtained from large budget run. For the normalizing constant estimate, the confidence intervals the vertical and horizontal directions are the 80% quantiles obtained from replications. The temperature schedule, local communication barriers, and the step sizes from a subset of 8 runs are shown.
52

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Tuning SMC Samplers via Greedy Incremental Divergence Minimization
-27100
10
10"
Temperature (2)
-27200
10
Local Barrier
Step Size (h.)
10'
10
log
27300
10
10a
-27400-
10
27500
10"
10'
0.1
IM
2M
3M
25
50
75
100125
10
1
10
10
10
10"
10
10
10S
10" 10
# of Gradient Evaluations
SMC Iteration (1)
Temperature (2.)
Temperature (2.)
(a) Three Men
-8800
10
10*
10a
3000
Temperature (2)
10"
Local Barrier
Step Size (h.)
10'
10
log
-3200
10a1
10
10
-3400
10
-3600
10
10
10a
0
1M
2M
3M
0
20
40
60
80
10
4
10
10a
10
10
10
10
10
10
of Gradient Evaluations
SMC Iteration (f)
Temperature (2)
Temperature (2.)
(b) TIMSS
Figure 37. Normalizing constant estimate, temperature schedule, local communication barrier, and step size schedules obtained by running SMC-KLMC (continued). The dotted line is the ground truth value obtained from large budget rum. For the normalizing constant estimate, the confidence intervals in the vertical and horizontal directions the 80% quantiles obtained from 32 replications. The temperature schedule, local communication barriers, and the step sizes from a subset of runs are shown.
53


Reviews:
[{"review_id": "762sLLWShb", "reviewer": "Reviewer_3HR9", "summary": "Main problem and approach: The performance of sequential Monte Carlo (SMC) samplers heavily depends on the tuning of the Markov kernels used in the path proposal. The paper proposes a framework for tuning the Markov kernels in SMC samplers by minimizing the incremental Kullback-Leibler (KL) divergence between the proposal and target paths. \nMain Result: the paper show that the approach and implementation are able to obtain a full schedule of tuned parameters at the cost of a few vanilla SMC runs, which is a fraction of gradient-based approaches.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review.\n\n> The experiment set up is too simple on a collection of toy benchmark datasets. It is not clear how it is applicable to the application domains of SMC, such as steering large language models and conditional generation from diffusion models.\n\nWe agree that more realistic experiments are always a good thing. As such, we added new benchmarks from both PosteriorDB and Inference Gym, including a high-dimensional benchmark problem (Pines) from Inference Gym, which is 1600 dimensional and multiple problems from PosteriorDB, including TIMSS and Three Men, both of which have more than 500 dimensions. Here is the full list of problems with their corresponding dimensionality:\n\n| Name | dims. | Source |\n| ----- | --- | ---- |\n| Funnel | 10 | Inference Gym |\n| Brownian | 32 | Inference Gym |\n| Sonar | 61 | Inference Gym | \n| Pines | 1600 | Inference Gym |\n| Bones | 13 | PosteriorDB |\n| Surgical | 14 | PosteriorDB |\n| HMM | 14 | PosteriorDB |\n| Loss Curves | 15 | PosteriorDB |\n| Pilots | 18 | PosteriorDB |\n| Diamonds | 26 | PosteriorDB |\n| Seeds | 26 | PosteriorDB |\n| Rats | 65 | PosteriorDB |\n| Radon | 90 | PosteriorDB |\n| Election88 | 90 | PosteriorDB |\n| Butterfly | 106 | PosteriorDB |\n| Birds | 237 | PosteriorDB |\n| Drivers | 389 | PosteriorDB |\n| Capture | 388 | PosteriorDB |\n| Science | 408 | PosteriorDB |\n| Three Men | 505 | PosteriorDB |\n| TIMSS | 530 | PosteriorDB |\n\n\n\nThe new experimental results can be found in the anonymized links below.\n\nComparison against end-to-end optimization:\n\nhttps://imgur.com/pdk3Bza\n\n(red dot: normalizing constant estimate obtained by our adaptive SMC sampler; blue line: normalizing constant estimate obtained by end-to-end optimization; dotted line: ground truth)\n\nHere, we can see the comparison against end-to-end optimization results with the new Pines benchmark problem. Furthermore, unlike the experiments included in the original submission, we use the official code provided by Geffner and Domke [1], which we find to be better tuned. In particular, on Pines, we observe that end-to-end optimization performs slightly better than our adaptive SMC procedure, while on other problems, we obtain more accurate results.\n\nComparison against grid search for SMC with Langevin Monte Carlo kernels:\n\nhttps://imgur.com/9JjJEiK\n\nComparison against grid search for SMC with kinetic Langevin Monte Carlo kernels:\n\nhttps://imgur.com/mUeWbuz\n\n(red line: the normalizing constant obtained by our adaptive SMC sampler; purple line: the normalizing constant obtained by vanilla SMC with a fixed stepsize; dotted line: ground truth estimate.)\n\nOverall, for the comparisons against grid search, we observe a similar trend where our tuning procedure finds better or comparable results to the best-fixed step size. The only exception appears to be Rat, as observed in the original submission.\n\nWe would also like to point out that PosteriorDB is a state-of-the-art benchmark (with the corresponding paper published this year at AISTATS’25 [2]) that contains problems that have actually been used for practical applications or closely resemble such problems. As such, we believe our experiments adequately represent problems to which SMC is expected to be applied in practice. While evaluating our method on applications such as LLM steering and conditional generation from diffusion models would definitely be interesting, they will probably need their own investigation with more specialized solutions.\n\n1. Geffner, Tomas, and Justin Domke. \"Langevin diffusion variational inference.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2023.\n2. Magnusson, Måns, et al. \"posteriordb: Testing, benchmarking and developing Bayesian inference algorithms.\" AISTATS’25, to be presented.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "The method is sound. The evaluation and benchmark datasets are too simple.", "theoretical_claims": "No fully check the correctness of  theoretical claims", "experimental_designs_or_analyses": "The experiment set up is too simple on a collection of toy benchmark datsets. It is not clear how it is applicable to the application domains of SMC, such as steering large language models and conditional generation from diffusion models.", "relation_to_broader_scientific_literature": "It address an open questions in the broader scientific literature.  Tuning SMC is often a significant challenge.  methods and criteria for tuning the path proposal kernels are relatively scarce. This paper focuses on the setting where a few scalar parameters (e.g., step size) subject to tuning. In this setting, the full generality (and cost) of SGD is not required; it is possible to design a simpler and more efficient method for tuning each\ntransition kernel sequentially in a single SMC/AIS run.", "essential_references_not_discussed": "I am not familiar with the domain and did not check thoroughly.", "other_strengths_and_weaknesses": "I am not familiar with the domain. The paper is well written and the claims seem solid. The experiment setup is too simple, it is hard to assess the practical value of this work.", "other_comments_or_suggestions": "Provide a practical example to demonstrate the value of this work.", "questions_for_authors": "How the proposed tuned SMC samplers can bring benefit to its applications such as steering large language models and conditional generation from diffusion models.", "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": "9MHKZD0nUd", "reviewer": "Reviewer_NHUm", "summary": "This paper proposes a novel method for tuning sequential Monte Carlo (SMC) samplers by greedily minimizing the incremental KL divergence between the target and proposal path measures. The authors develop efficient, gradient-free algorithms for tuning key parameters—such as step sizes in unadjusted Langevin Monte Carlo kernels. Experimental results demonstrate that their approach reduces variance in estimates and outperforms both fixed-parameter and gradient-based tuning methods on various benchmark problems.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review.\n\n> One potential caveat is that some theoretical guarantees rely on assumptions (e.g., unimodality of the tuning objective) that may limit generalizability,\n\nWe agree that those theoretical assumptions are restrictive. As such, we are happy to mention that we were able to relax those assumptions during the review period. Now, even without unimodality, we are able to guarantee that our algorithm finds a point $\\epsilon$-close to a local minimum with similar computational cost. Additionally assuming unimodality strengthens the guarantee so that the returned point is $\\epsilon$-close to the global optimum.\n\nThe only major change we had to introduce is that we now use a different variant of the golden section search algorithm. In particular, we use the version described in “Numerical Recipes” (Section 10.1 in [1]). We found that this variant is able to guarantee finding a local minimum as long as the initialization satisfies a certain condition implying that the initial search interval contains a local minimum. \n\n> How sensitive are the experimental results to changes in key parameters, and does this sensitivity mirror the challenges encountered in practical scenarios?\n\nThank you for the great question. Overall, we didn’t spend much effort tuning the parameters, and all of our experiments were run with a single fixed set of parameters for each type of kernel. In more detail though, the only parameters that primarily affect the results are the regularization strength $\\tau$ and the optimization accuracy budget $\\epsilon$. (The rest of the parameters only change the amount of computation spent obtaining similar solutions.) $\\epsilon$, for instance, does not affect the result much as long as it is small enough. On the other hand, the effect of the regularization strength $\\tau$ seems to depend on the kernel but not so much on the problem. For instance, the performance of LMC isn’t very sensitive to $\\tau$, and we thus use a small amount. On the other hand, KLMC seems to require stronger regularization to obtain good performance. While we suspect this has something with the persistence of the momentum, it is not entirely clear why. Overall, we do not expect the parameters to require much tuning, except when swapping the MCMC kernel.\n\n> Can the proposed method scale efficiently with the increased dimensionality and sample sizes often found in real-world problems?\n\nWe extended our experiments with higher dimensional problems since the initial submission. (Please refer to the response to 3HR9 for the new results.) In principle, dimensionality shouldn’t be a problem for our method. As shown in the experiment, our method should perform better or as well as the best-tuned vanilla SMC sampler. Therefore, as long as vanilla SMC can scale, our method should be able to scale as well. In terms of scaling with respect to the sample size $N$, as long as the subsampling size $B$ is smaller than $N$, the added overhead of our method should be negligible. More concretely, if at most $C$ objective evaluations are spent during each adaptation step, the total cost of running SMC with our adaptation is $O(B C +  N)$. Thus, if $BC = o(N)$, our adaptive SMC scheme is just as scalable as vanilla SMC.\n\n1. Press, William H. Numerical recipes 3rd edition: The art of scientific computing. Cambridge University Press, 2007.", "claims_and_evidence": "The submission's claims are largely supported by comprehensive empirical results. The authors verified their method through experiments on multiple benchmarks that their adaptive tuning yields lower variance estimates and comparable or improved performance compared to fixed or gradient-based methods. One potential caveat is that some theoretical guarantees rely on assumptions (e.g., unimodality of the tuning objective) that may limit generalizability, but within the presented contexts the evidence is clear and convincing.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are well-aligned with the problem at hand. The paper’s approach—minimizing incremental KL divergence to tune SMC samplers—is both innovative and practical, addressing the challenges of tuning unadjusted kernels without incurring high computational costs. The evaluation criteria, such as the variance and accuracy of normalizing constant estimates, are standard in the SMC literature. Additionally, the benchmark datasets are used.", "theoretical_claims": "I only checked the main body of the theoretical claims and did not find any issues.", "experimental_designs_or_analyses": "I briefly reviewed the experiments; however, I have some concerns about whether they accurately reflect real-world scenarios.", "relation_to_broader_scientific_literature": "The paper’s contributions build directly on a rich body of work in sequential Monte Carlo (SMC) methods and adaptive tuning techniques.", "essential_references_not_discussed": "No, all the essential related works appear to be appropriately cited and discussed.", "other_strengths_and_weaknesses": "Strengths: \n\n- Novel and efficient approach that combines divergence minimization with adaptive SMC tuning.\n- Provides effective empirical results.", "other_comments_or_suggestions": "The document is well written and the technical terminology is used correctly.", "questions_for_authors": "- How sensitive are the experimental results to changes in key parameters, and does this sensitivity mirror the challenges encountered in practical scenarios?\n\n- Can the proposed method scale efficiently with the increased dimensionality and sample sizes often found in real-world problems?", "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": "kGmHqXueR9", "reviewer": "Reviewer_7uSZ", "summary": "The paper provides a gradient-free, hyperparameter-free tuning algorithm for proposal step sizes and particle refresh/resampling rates in Sequential Monte Carlo pipelines.  Experiments on example graphical models from PosteriorDB demonstrate that the new algorithm not only provides a great advantage in log-evidence estimation relative to a fixed step size, but in fact quickly approaches the true log-evidence after adaptation. In experiments comparing to end-to-end adaptive SMC approaches, the new method performs no worse than end-to-end optimization and, in one experiment, significantly better.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review.\n\n> Do the authors want to minimize incremental divergences in order to target the path measure of the annealed importance sampling procedure, or fine-tune an annealed importance sampling procedure in order to later perform SMC on the target density?\n\nWe kindly request more details on this question as our tuning procedure does not involve nor is specialized to annealed importance sampling (AIS).\n\nPerhaps the question arose from the fact that the Feynman-Kac model formalism does not explicitly involve resampling, which might have led to the impression that optimizing the target path of the Feynman-Kac model is tuning an AIS procedure, not an SMC procedure. If this is the case, we would like to clarify that while our procedure can absolutely be applied to AIS (after all, SMC and AIS are a few resampling steps away from each other), our focus is purely on SMC samplers. That is, both AIS and SMC are algorithms for simulating the same Feynman-Kac model (it’s only the procedure for simulation that differs.) To be clear, we would like to bring attention to Algorithm 1, which shows the adaptive SMC implementation with our tuning scheme embedded in it. All we do is run Algorithm 1.\n\nIf this does not fully address the Reviewer’s question, we would be very happy to further clarify any point of ambiguity.", "claims_and_evidence": "This paper poses a bit of a clarity problem, at least for this reviewer.  The paper takes its starting notation from \"Introduction to Sequential Monte Carlo\" by Chopin and Papaspiliopoulos, and as a result, is somewhat notationally overloaded.", "methods_and_evaluation_criteria": "Yes, in fact PosteriorDB is just the benchmark for Bayesian inference problems that I would recommend.", "theoretical_claims": "Proposition 1 more-or-less follows from the definition of a divergence, though a proof summary in the main text would be preferable.", "experimental_designs_or_analyses": "PosteriorDB is the gold-standard for benchmarking Bayesian inference and the numerical metrics used here are the valid ones.", "relation_to_broader_scientific_literature": "SMC is a back-end workhorse for many scientific computations, and so the paper fits well into the broader literature.  I do not know of a previous paper doing what this paper does.", "essential_references_not_discussed": "I do not have any missing references to report.", "other_strengths_and_weaknesses": "The problem of approximating an idealized annealing path distribution is, to my knowledge, a relatively original one to pose.  I would somewhat like the authors to motivate it more.", "other_comments_or_suggestions": "There are a few grammar and usage typos, such as \"distribution distribution\" and \"the goal is often to infer... or estimating\" (either the infinitive or gerund should be used consistently).\n\nScore has been revised in light of the authors' response to both my review and to others asking for additional experiments.", "questions_for_authors": "Solved in author response.", "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": "hy14OBg4yq", "reviewer": "Reviewer_beQ4", "summary": "The authors propose a method for tuning kernels in SMC by minimising a KL-divergence between proposal paths and target paths. This optimisation is done using a gradient free method. They show that by adapting the step size with their method they get better normalising constant estimates than using fixed step sizes, and that their adaptation scheme has a smaller computation cost compared to setting the step size using gradients.\n\n## update after rebuttal\n\nI maintain my recommendation for acceptance.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your review.\n\n> The datasets used seem quite standard for evaluating SMC methods. One thing I am not sure of is whether just reporting normalising constant is sufficient to show the superiority of this method, it would have been nice to see something like ESS or some assessment of posterior quality like moment estimates.\n\nThank you for the suggestion! Theoretically speaking, the accuracy of the normalizing constant and expectations taken over the particles are closely related. That is, the variance of the normalizing constant estimate is the cumulation of the variance of the normalizing constant ratios $Z_t/Z_{t-1}$, which corresponds to setting the integrand of expectation taken over the particles to be the indicator function as $\\varphi = \\mathbb{1}$. Therefore, if the normalizing constant estimate is accurate, the particle expectations will also be reasonably accurate. Indeed, we ran some experiments on two benchmarks: seeds and radon, where we computed the Wasserstein-2 distance of $N = 1024$ particles from SMC-LMC against ground truth samples from $10^5$ samples from the no-u-turn sampler. The Wasserstein-2 distance is closely related to the accuracy of the first and second moments. Also, it bounds the Wasserstein-1 distance, an integral probability metric, which quantifies the worst-case absolute error for 1-Lipschitz integrands. Therefore, it is a good metric for quantifying the error of particle expectations.\n\nThe results are shown in the following link:\n\nhttps://imgur.com/a/80l62vS\n\n(red line: Wasserstein-2 distance between the samples from vanilla SMC with the corresponding fixed stepsize and NUTS; blue line: Wasserstein-2 distance between the samples from our adaptive SMC sampler and NUTS.)\n\nWe can see that our adaptive SMC samplers achieve Wasserstein-2 distances close to the best fixed step size. However, it does not outperform fixed stepsizes as much as for estimating normalizing constants. This suggests that our adaptation scheme could be tailored to expectations taken over the particles for better performance.\n\n> Some interpretation as to why, eg, the method under-performed on the rats dataset would be welcome.\n\nThank you for pointing this out. With full honesty, it is unclear why this is the case. Upon close inspection, the optimization algorithm seems to correctly find the minimizer of the objective. Therefore, two explanations are possible: Either the sampler is underperforming for the given budget such that the estimate of the incremental KL is inaccurate, or the greediness of the scheme results in a suboptimal solution.\n\n> The experiments seem sound if a bit limited (as I previously mentioned) from what is relayed in the paper. I am curious why 32 replications were used to get the confidence intervals in the adaptive tuning vs fixed experiments but only 5 in the end-to-end optimisation experiments.\n\nWe agree with the reviewer that more than 5 evaluations for the end-to-end results would have been better. Therefore, since the submission, we re-ran the experiments using 32 replications for evaluation, which is now the same number as adaptive SMC, and included a more challenging problem (Pines). We observe that on Pines, our method does not outperform end-to-end optimization, which is unsurprising: end-to-end optimization should outperform our greedy scheme on some problems where the gradient noise is negligible. For the grid search experiments, we ran more problems from PosteriorDB, which now totals 21 benchmark problems. Please refer to the response reviewer 3HR9 above!\n\n> The idea of using the KL divergence to tune parameters of the proposal distribution in SMC also appears in (Shixiang et. al. Neural Adaptive Sequential Monte Carlo).\n\nThank you for pointing this out! We will add this to the list of works performing end-to-end optimization.\n\n> Is there any indication that this method under-explored the posterior or suffered more from particle degeneracy than the other compared methods?\n\nGenerally, our method should perform as well as a well-tuned vanilla SMC sampler, which may or may not fully explore the posterior depending on the problem and other configurations. In terms of particle degeneracy, our method should be the least susceptible since we are essentially maximizing the incremental weights. In a sense, our scheme is actively preventing degeneracy. The side effect would be that the biased normalizing constant estimates obtained during adaptive runs will be overestimated.", "claims_and_evidence": "From the paper the two key empirical claims are:\n\n1) Our methods achieve lower variance in normalizing\nconstant estimates compared to the best fixed step sizes\nobtained through grid search or SGD-based tuning methods\n\nFigures 2,3,8 and 9 evidence this, with the exception of one or two examples. Some interpretation as to why, eg, the method under-performed on the rats dataset would be welcome. \n\n2) Additionally, the computational cost of our adaptation\nscheme is several orders of magnitude lower than that of\nSGD-based approaches (Section 5).\n\nFigures 5, 10, 11, 12 and 13 show this. The choices made in the experimental setups for these didn't seem unreasonable so I think these claims were well evidenced.", "methods_and_evaluation_criteria": "The datasets used seem quite standard for evaluating SMC methods. One thing I am not sure of is whether just reporting normalising constant is sufficient to show the superiority of this method, it would have been nice to see something like ESS or some assessment of posterior quality like moment estimates. \n\nFor comparing against end to end optimisation it wasn't clear to me why a neural network as opposed to any other method would be chosen for learning the step size.", "theoretical_claims": "I didn't check these very closely, but appreciated that some empirical study was done to see how realistic the assumptions were (figure 1.)", "experimental_designs_or_analyses": "The experiments seem sound if a bit limited (as I previously mentioned) from what is relayed in the paper. I am curious why 32 replications were used to get the confidence intervals in the adaptive tuning vs fixed experiments but only 5 in the end-to-end optimisation experiments.", "relation_to_broader_scientific_literature": "This paper aims to tune the hyperparameters of SMC kernels by minimising the divergence between the proposal path and the target path. This idea has similarities to annealed flow transport Monte Carlo and normalising flow. \n\nThe issue of adapting the hyperparamters of SMC kernels is a long standing one (P Fearnhead and Benjamin M. Taylor, An Adaptive Sequential Monte Carlo Sampler).\n\nThe idea of using the KL divergence to tune parameters of the proposal distribution in SMC also appears in (Shixiang et. al. Neural Adaptive Sequential Monte Carlo).", "essential_references_not_discussed": "Not that I know of.", "other_strengths_and_weaknesses": "I found this paper quite clear and this method seems much simpler than previous methods for tuning SMC samplers.", "other_comments_or_suggestions": "On line 210 in the second column you refer to h_t before it has been introduced which is potentially confusing. Also I wasn't sure what \\theta_t contains h_t meant.\n\n\"distribution\" is repeated on line 85 in the second column.", "questions_for_authors": "Is there any indication that this method under-explored the posterior or suffered more from particle degeneracy than the other compared methods?", "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:

Meta-review: The paper introduces a novel approach for tuning SMC proposal kernels; all reviewers agree that this is a clear method, praise the fact it does not require expensive or difficult end-to-end gradient-based training, and the paper itself is well-written. Additional experiments were added during the rebuttal period to strengthen the experimental validation of the approach. All reviewers agreed the paper should be accepted.
Decision: Accept (poster)