Bayesian inference via sparse Hamiltonian flows
Authors: Naitong Chen
Zuheng Xu
Trevor Campbell
Venue: NeurIPS 2022 Accept (2022)
Keywords: Bayes
coresets
flows
Hamiltonian

Abstract: A Bayesian coreset is a small, weighted subset of data that replaces the full dataset during Bayesian inference, with the goal of reducing computational cost.  Although past work has shown empirically that there often exists a coreset with low inferential error, efficiently constructing such a coreset remains a challenge.  Current methods tend to be slow, require a secondary inference step after coreset construction, and do not provide bounds on the data marginal evidence.  In this work, we introduce a new method---sparse Hamiltonian flows---that addresses all three of these challenges.  The method involves first subsampling the data uniformly, and then optimizing a Hamiltonian flow parametrized by coreset weights and including periodic momentum quasi-refreshment steps.  Theoretical results show that the method enables an exponential compression of the dataset in a representative model, and that the quasi-refreshment steps reduce the KL divergence to the target.  Real and synthetic experiments demonstrate that sparse Hamiltonian flows provide accurate posterior approximations with significantly reduced runtime compared with competing dynamical-system-based inference methods.

Bayesian Inference via Sparse Hamiltonian Flows
Naitong Chen
Zuheng Xu Department of Statisties University of British Columbia
Trevor Campbell
[naitoong. chen
zuheng. xu
trevor] @stat. ubc. ca
Abstract
A Bayesian coreset is a small, weighted subset of data that replaces the full dataset during Bayesian inference, with the goal of reducing computational cost. Although past work has shown empirically that there often exists a coreset with low inferential enror, efficiently constructing such coreset remains a challenge. Current methods tend to be slow, require a secondary inference step after coreset construction, and do not provide bounds the data marginal evidence. In this work, we introduce a new method-spare these challenges. The method involves first subsampling the data uniformly, and then optimizing Hamiltonian flow parametrized by coreset weights and including periodic momentum meusi-refreshmen steps. Theoretical results show that the method enables an exponential compression of the dataset in representative model, and that the quasi-refreshment steps reduce the KL divergence the target. Real and synthetic experiments demonstrate that sparse Hamiltonian flows provide accurate posterior approximations with significantly reduced runtime compared with competing Soaniccl-system--eee inference methods.
Hamiltonian flows-that that addresses all three
Introduction
Bayesian inference provides a coherent approach learning from data and uncertainty assessment in a wide variety of complex statistical models. Two standard methodologies for performing Bayesian inference in practice are Markov chain Monte Carlo (MCMC) D 3, Ch. 11,12] and variational inference (VI) [4] 5). MCMC simulates Markov chain that targets the posterior distribution. In the increasingly common setting of large-scale data, most exact MCMC methods are intractable. This is essentially because simulating each MCMC step requires (expensive) computation involving each data point, and many steps are required to obtain inferential results a reasonable quality. To reduce cost, typical approach is to perform the computation for a random subsample of the data, rather than the full dataset, at each step [0 (see [T] for recent survey). However, recent work shows that the speed benefits outweighed by the drawbacks; uniformly subsampling each step causes MCMC to either mix slowly or provide poor inferential approximation quality 15 VI, on the other hand, posits family of approximations the posterior and uses optimization to find the closest member, enabling the use of scalable stochastic optimization algorithms []6] [17]. While past work involved simple parametric families, recent work has developed flow families based on Markov and in particular, those based on Langevin and Hamiltonian dynamics 25) However, because these Markov chains are typically designed to target the posterior distribution, each step again requires a computation involving all the data, making KL minimization and sampling slow. Repeated subsampling to reduce cost has the same issues that it does in MCMC.
chains [18 19] and
Although repeated subsampling in each step of a Markov chain (for MCMC or VI) is not generally helpful, recent work on Bayesian coresets [26] has provided empirical evidence that there often exists a fixed small, weighted subset of the data coreset- that one can use to replace the full dataset in standard MCMC or VI inference method [27]. In order for the Bayesian coreset approach to be
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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practically useful, one must (1) find a suitable coreset that provides a good posterior approximation; and (2) do so quickly enough that the speed-up of inference is worth the time takes to find the coreset. There is currently no option that satisfies these two desiderata. Importance weighting methods [26) are fast, but do not provide adequate approximations in practice. Sparse linear regression methods (28430] are fast and sometimes provide high-quality approximations, but are very difficult to tune well. And sparse variational methods [27 31 find very high quality coreset approximations without undue tuning effort, but are too slow to be practical.
This work introduces three key insights. First, we can uniformly subsample the dataset once to pick the points in the coreset (the weights still need to be optimized). This selection is not only significantly simpler than past algorithms; we show that it enables constructing an exact coreset-with KL divergence 0 to the posterior- of size ( for N data points in representative model (Proposition 3.1 Second, we can then construct normalizing flow family based on Hamiltonian dynamics 22 32 that targets the coreset posterior (parametrized by coreset weights) rather than the expensive full posterior. This method address all of the current challenges with coresets: enables tractable i..d. sampling, provides known density and normalization constant, and is tuned using straightforward KL minimization with stochastic gradients. also addresses the inefficiency of Markov-chain-basee VI families, as the Markov chain steps are computed using the inexpensive coreset posterior density rather than the full posterior density. The final insight is that past momentum tempering methods [2]] do not provide sufficient flexibility for arbitrary approximation to the posterior, even in a simple setting (Proposition (.2). Thus, we introduce novel periodic momentum muasi-refreshment steps that provably reduce the KL objective (Propositions 3.3 and A.2). The paper concludes with real and synthetic experiments, demonstrating that sparse Hamiltonian flows compare favourably to both current coreset compression methods and variational flow-based families. Proofs of all theoretical results may be found in the appendix.
It is worth noting that Hamiltonian flow posterior approximations based weighted data subsample were also developed concurrent work in the context of variational annealed importance sampling 33 and subsampling prior to weight optimization was developed in concurrent work on MCMC
34 In this work, we focus on incorporating Bayesian coresets into Hamiltonian based normalizing flows to obtain fast and accurate posterior approximations.
2
Background
2.1 Bayesian coresets
We are given a target probability density r0) for variables A E IRd that takes the following form:
N
(0) 1
Σ (n) (0))
exp
In a Bayesian inference problem with i.i.d. data, To is the prior density, the fn are the log-likeli terms for N data points, and the normalization constant is in general not known. The goal is to take samples from the distribution corresponding to density (0)).
In order to avoid the O(N) cost of evaluating log T (0) or V log (0) (at least one of which must be
conducted numerous times in most standard inference algorithms), Bayesian coresets (26) involve replacing the target with surrogate density of the form
N
Tee (0) = ((w)
cxp Σ w. (w) (o)))
where w N w 2 are a set of weights. If w has at most M < N nonzeros, the O(M) cost of (0) evaluating log T. (0) V log Tw (0) is a significant improvement upon the original ((N) cost.
The baseline method construct coreset is draw a uniformly random subsample of M data points, and give each weight of N/M: although this method is fast in practice, it typically generates poor posterior approximations. More advanced techniques generally involve significant user tuning effort [26 28 30]. The current state-of-the art black box approach formulates the problem as variational inference 22 and provides a stochastic gradient scheme using samples from Tup
w² = arg min DRI. (Twll s.t. || Ilo S M.
weR
2

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Empirically, this method tends to produce very high-quality coresets [27]. However, to estimate the gradient at each iteration of the optimization, we require MCMC samples from the weighted coreset posterior at that iteration. While generating MCMC samples from sparse coreset posterior is not expensive, it difficult to tune the algorithm to ensure the quality of these MCMC samples across iterations (as the weights, and thus the coreset posterior, change each iteration). The amount of tuning effort required makes the application constructed, all of the aforementioned methods require a secondary inference algorithm to take draws from Tw- Further, since (ww) is not known general, it is not tractable to use these methods to bound the marginal evidence Z.
this method too slow to be practical. Once the coreset is
2.2 Hamiltonian dynamics
In this section we provide very brief overview of some important aspects of a special case of Hamiltonian dynamics and its use statistics; see (35) for more comprehensive overview. The differential equation below in Eq. (T) describes how (deterministic) Hamiltonian system with position 0, E IRd, momentum Pe Rº. differentiable negative potential energy log π (0,), and kinetic energy 2PP P: evolves over time R:
dp: V log (0.) dt
d0c dt
Pt.
For t R. define the mappings H2 Rdd + 2d that take (02. Ps) + dynamics in Eq. (T). These mappings have two key properties that make Hamiltonian dynamics useful in statistics. First, they invertible, and preserve volume in the sense that (det. VH, In other words, they provide tractable density transformations: for any density q on R2d and pushforward on 2dd under the mapping H,, we have that (e(,)) :) = (H, (.,-)). Second, the augmented target density (0, p) on 2dd corresponding to independent draws from and (0,
under the
1 #0, p) x π (0) exp (-20² p)
is invariant under the mappings H₁ i.e., ((H+(-,)) = ((.,)) Given these properties, Hamiltonian Monte Carlo [35] (3) constructs Gibbs sampler for ग that interleaves Hamiltonian dynamics with periodic stochastic momentum refreshments ~ NN,0,), Upon completion, the component of the samples can be dropped obtain samples from the desired target æ.
In practice, one approximately simulates the dynamics in Eq. (I) using the leapfrog method, which involves interleaving three discrete transformations with step size O,
Pk+1 = Pks - V log (0a) 0k+1 02 + ePp+1
Pk+1 Pu+1 V log (0k+1).
(2)
Denote the map constructed by applying these three steps in sequence T. Rd2d transformations in Eq. 2 are all shear, T, is also volume-preserving, and for small enough step size nearly maintains the target invariance. Note also that evaluating single application of T, is of O(Nd) complexity, which is generally expensive in the large-data (large-N) regime.
R2d, As the
2.3 VI via Hamiltonian dynamics
Since the mapping T, is invertible and volume- preeee-reserving, it is possible to tractably compute the density of the pushforward of reference distribution g(:;-) under repeated applications of it. In addition, this repated application of T. resembles the steps of Hamiltonian Monte Carlo (HMC) 35). which we know converges in distribution to the target posterior distribution. [21] 22 use these facts to construct a normalizing flow (32) VI family. However, there are two issues with this methodology. First, the O(Nd) complexity of evaluating each step T. makes training and simulating from this flow computationally expensive. Second, Hamiltonian dynamics on its own creates a flow with insufficient flexibility to match a target of interest. In particular, given density g(.,)) and pushforward q: (-..) under He, we have
Vt R, DKL. (q.l(²) = DKT. (gll=).
(3)
3

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In other words. Hamiltonian dynamics itself cannot reduce the KL divergence to म; it simply interchanges potential and kinetic energy. [2] address this issue by instead deriving their flow from tempered Hamiltonian dynamics: for an integrable tempering function IR + R,
dp dt
d0, dt
V log (6.) pt)
P:.
The discretized version of the dynamics in Eq. corresponds to multiplying the momentum by tempering value ag > 0 after the hthh application of T.. By scaling the momentum, one provides the normalizing flow with the flexibility to change the kinetic energy at each step. However, show later in Proposition that just tempering the momentum does not provide the required flow flexibility, even for simple representative Gaussian target.
related line of work uses the mapping Т. for variational annealed importance sampling 23 25
The major difference between these methods and the normalizing flow-based methods is that the auxiliary variable is (partially) stochastically refreshed via P ~ NO /) after applications of T.- One is then forced to minimize the KL divergence between the joint distribution of 0 and all of the auxiliary momentum variables under the variational and augmented target distributions.
Sparse Hamiltonian flows
In this section we present sparse Hamiltonian flows, new method to construct and draw samples from Bayesian coreset posterior approximations. We first present method and supporting theory for selecting the data points to be included in the coreset, then discuss building sparse flow with these points, and finally introduce quasi-refreshment steps to give the flow family enough flexibility match the target distribution. Sparse Hamiltonian flows enables tractable i.i.d. sampling, provides a tractable density and normalization constant, and is constructed by minimizing the KI. divergence to the posterior with simple stochastic gradient estimates.
3.1 Selection via subsampling
The first step in our algorithm to choose a uniformly random subsample of M points from the full dataset; these will be the data points that comprise the coreset. Without loss of generality, assume these are the first M points. The key insight in this work is that while subsampling with importance weighting does not typically provide good coreset approximations (26). uniformly random subset of the N log- likeelooddood potential functions .f fas still provides a good basis for approximation with high probability. Proposition 3.1 provides the precise statement of this result for representative example model Eq. ( In particular, Proposition 3.1 asserts that as long as we set our coreset size M to be proportional to d logz N, the optimal coreset posterior approximation will be exact, i.e., have 0 KL divergence to the true posterior, with probability at least N- & (logz N) Thus we achieve an exponential compression of the dataset, N + log, N, without losing any fidelity. Note that we will still need method to choose the weights Wi...... WM for the M points, but the use of uniform selection rather than one-at-a-time approach 227 substantially simplifies the construction. In Proposition 3.1 is the universal constant from [37 Corollary 1.2), which provides an upper bound on the number of spherical balls of some fixed radius needed to cover d-dimensional unit sphere.
Proposition 3.1. Consider a Bayesian Gaussian location model:
0 0~ I) and Vn E [N],
X,
LA N),,
(5)
where 0, Xn E Rd for d N. Suppose the true data generating parameter 0 0, and set M log2(A, (Aog vVdd/2)- + C where A d â log(1+ + d). Then the optimal coreset nw- for the model Eq. (5) built using uniform subsample of data of size M satisfies
P P (Tw. 00
lim sup N
S
N (log N)

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3.2 Sparse flows
Upon taking a uniform subsample of M data points from the full dataset, we consider the sparsified Hamiltonian dynamics initialized 0b. Pu g(:, for reference density Censity"gge ((()))
dpt V log Twe (Ox)
d0c dt
dt
Much like original Hamiltonian dynamics the full target density, the sparsified Hamiltonian dynamics Eq. (6) targets the augmented coreset posterior with density (0, p) on IR2d
Tw (0, x Ter (0) exp (-
Discretizing these dynamics yields leapfrog method similar to Eq. (2) with three interleaved steps.
Pk+1 Pb + E log Tw (0x) Ok+1 02 + EPH+1 Pk+1
Pa+1 + V log n. (0x+1).
Denote the map constructed by applying these three steps sequence Tw.e Rd - Rd Like the original leapfrog method, these transformations are both invertible and shear, and thus preserve volume; and for small enough step size e, they approximately maintain the invariance of म.. (6, p). However, since w only has the first M entries nonzero,
V log (0x)
Σ Wm v Vog (0k,,
and thus coreset leapfrog step can be taken in O(Md) time, as opposed to O(Nd) time in the original approach. Given that Proposition 3.1 recommends setting M dllog, we have achieved an exponential reduction in computational cost of rumning the flow.
However, before, the weighted sparse leapfrog flow is not sufficient on its provide flexible variational family. In particular, we know that Twre nearly maintains the distribution 7w as invariant. We therefore need way to modify the distribution of the momentum variable p. One option is to include a tempering of the form Eq. (4) into the sparse flow. However, Proposition shows that optimal tempering does provide the flexibility match simple Gaussian target
Proposition 3.2. Let Oz, Pt E R follow the tempered Hamiltonian dynamics Eq. 4 targeting = =N o²), > O, with initial distribution 8n ~ N(,, 1). Po. ~ NO., for initial center μ E R and momentum scale O. Let qt be the distribution of (0t, Pt). Then
1 n
inf
DKT. (gel [) 2 log
+3
40
Note that if (() identically then VI. > DRL. (grl [) D kL (g0l (T)
The intuition behind Proposition 3.2 that while adding a tempering 7()) enables one change the total energy by scaling the momentum, it does not allow one fine enough control on the distribution under the current flow approximation, cannot scale
of the momentum. For example, if [F(p] the momentum to force Esp] = as it should be under the augmented target; intuitively, we also need the ability shift or recenter the momentum as well.
'The reference g can also have its variational parameters optimize, but in this paper leave fixed.

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Algorithm 1 SparseHamFloo
Algorithm 2 Estimaate_Ela
Require: 80, Pu. w, A. L, R J < 0, and (8. p) + (fo: Po)
Require: 9, no. W. A, L, R. S
(0o, Po) ~ g(:.-)
for r = 1..... R do
Forward pass:
for l. .,...,,. L do
p. J < Sparsellamloow Po: w, C, X, I, R)
Sparse flow leapfrog:
Obtain unbiased ELBO estimate:
0.p - (w. p)
(n1, (m......n.
Unif (f1,2, ,,))
end for
log + log n (0) +
N
E s=1 Jn, (0)+
Accumulate log Jacobian determinant:
log. (p
0,
5
+ JJ + log |det @Ra. Quasi-refreshment:
dp (p. 0)l
log + g(0,, Po) -J
return log log
p pt RA, (p.0) end for return p,
3.3 Queii-eefre
Rather than resampling the momentum variable from its target marginal-whise which removes the ability evaluate the density of 0t, Pr-in this work we introduce deterministic quasi-refreshment moves that enable the flow to strategically update the momentum without losing the ability to compute the density and normalization constant of ot, Pt (i.e., we construct a normalizing flow (32]) Here we introduce the notion of marginal muasi-refreshment, which tries to make the marginal distribution of P: match the corresponding marginal distribution of the augmented target a... Proposition 3.3 shows that marginal quasi-refreshment is guaranteed to reduce the KI. divergence.
Proposition 3.3. Consider the state ot, Pt E Rd of the flow at step t, and the augmented target distribution 0, p ~ Suppose that we have bijection R Rdd IRd such that R(pe)
d p. Then
DEL (0c, R(px))(0, p) DKL (0t, DKL (pelle).
See Appendix A for the proof of Proposition 3.3 and a general treatment of quasi-refreshment; for simplicity, we focus on the type of quasi- refreshment that we use in the experiments. In particular, if we are willing to make an assump- tion about the marginal distribution of P: at step
-380000
-390000
ELBO
of the flow, we can introduce a tunable family of functions RΛ with parameters À that is flex-
-400000
Hamiltonian dynamics
410000
ible enough to set Rx(pt) L p for some X, and include optimization of À along with the coreset weights. It is important to note that this assump- tion on the distribution of Pe is not related the posterior т. As an example, in this work we as- A-1) for some unknown mean i and diagonal precision A, which enables us simply set
quasi-refreshmen
420000
430000
0
10
20
30
40
50
leapfrog number
sume that Pr ~ NM,, A 1)
Figure 1: ELBO across leapfrog steps.
Rx(x) (x (x- p p).
(8)
We then include À = (n, A) as parameters to be optimized along with the coreset weights w (each quasi-refreshment step will have its own set of parameters μ, A). Even when this assumption does not hold exactly, the resulting form of Eq. (8) enables the refreshment step to both shift and scale (i.e., standardize) the momentum desired, and is natural implement part of single optimization routine.
Fig. [ provides an example of the effect of qussi-refreshment in a synthetic Gaussian location model (see Section
for details). In particular, it shows the evidence lower bound (EIBO) as a function of leapfrog step number in trained sparse Hamiltonian flow with the quasi-refreshment scheme in

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Eq. (8). While the estimated ELBO values stay relatively stable across leapfrog steps in between quasi-refreshments (as expected by Eq. ). the quasi-refreshment steps (colored red) cause the ELBO to increase drastically. We note that the BL does not stay exactly constant because the Hamiltonian dynamies targets the coreset posterior instead of the true posterior, and simulated approximately using leapfrog steps. As the series of transformations brings the approximated density closer to the target, the quasi-refreshment steps no longer change the ELBO much, signalling the convergence of the flow's ' approximation of the target. It is thus clear that the marginal quasi-refrrehments indeed decrease the KL, as shown in Proposition 3.3
3.4 Algorithm
In this section, we describe the procedure for training and generating samples from a sparse Hamil- tonian flow. As a normalizing flow, a sparse Hamiltonian flow can be trained by maximizing the augmented ELBO using usual stochastic gradient methods (e.g. as in B2),, where the transformations follow Eq. 7 with a periodic mutas-reffresheent Here and the experiments we focus on the shift-and-scale quasi-refreshment in Eq. (8)
We begin by selecting a subset of M data points chosen uniformly randomly from the full data. Next we select a total number R of quasi-refreshment steps, and a number I of leapfrog steps between each quasi-refreshment The flow parameters be optimized consist of the quasi-refreshment parameters À = (/1) the M coreset weights w = (wm)==) and the leapfrog step sizes = (e.) note that we use separate step size e, per latent variable dimension i in Eq. 35 modification enables the flow to fit nonisotropic target distributions.
Sec. 4.2]. This
We initialize the weights to N/M (i.e., uniform coreset), and select initial step size for all dimensions. We use a warm start to initialize the parameters X, = (J),,A)) of the quasi-refreshments. Specifically, using the initial leapfrog step sizes and coreset weights, we pass batch of samples from the reference density ((: through the flow up to the first quasi-refreshment step. We initialize M1. A1 to the empirical mean and diagonal precision of the samples at that point. We then apply the initialized first refssrrefreshment to the momentum, proceed with the second sequence of leapfrog steps, and repeat until we have initialized all quasi-refreshments
I....,R.
Once the parameters are initialized, we log-transform the step sizes, weights, and quass-reeressmmnt diagonal scaling matrices make them unconstrained during optimization. We obtain unbiased estimate of the augmented ELBO gradient by applying automatic differentiation [8 39] to the ELBO estimation function Algorithm[]] and optimize all parameters jointly using gradient- stochastic optimization technique such as SGD [40] [41] and ADAM [42. Once trained, we can obtain samples from the flow via Algorithm
4 Experiments
In this section, we compare our method against other Hamiltonian based VI methods and Bayesian coreset construction methods. Specifically, we compare the quality of posterior approximation, as well as the training and sampling times of sparse Hamiltonian flows (SHF). Hamiltonian importance sampling (HIS) 21), and unadjusted Hamiltonian annealing (UHA) (3 using real and synthetic datasets. We compare with two variants of HIS and UHA: "-Full," in which we train using in- fnowlow 23), but compute evaluation metrics using the full-data flow; and "-Coresee," in which we base the flow on uniformly subsampled coreset. We also include sampling times of adaptive HMC and NUTS 443 Alg. 5 and 6] using the full dataset. Finally, we compare the quality of coresets constructed by SHF to those obtained using uniform subsampling (UNI) and Hilbert coresets with orthogonal matching pursuit (Hi lbert-OMP) [28 44) All experiments are performed on machine with an Intel Core 1270H processor and 32GB memory. Code is available at https: thub www.eeaitongCheeeeeeeec Sparse- Hamiltoni an Flows Details of the experiments in Appendix B
minibatching as suggested by (2122,
4.1 Synthetic Gaussian
We first demonstrate the performance of SHF on synthetic coassian-locationn model,
0 ~ I(0 and
Vn E [N], X, us NN(0,c)) cI),
7

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Figure 2: ELBO (Fig. 2a). KL divergence (Fig. 2b). relative 2-norm mean error (Fig. 2c relative
Frobenius norm covariance enror (Fig. 2d), energy distance (Fig. [eh, and IMQ KSD 45
(Fig
g 21 for
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(c)
Figure 3: Density evaluation (Fig. 3ap and sample generation time (Fig. 3b (100 samples), and ELBO
versus time during training (Fig. Bcb for synthetic Gaussian. The lines indicate the median, and error regions indicate 25th to 75th percentile from 5 runs.
where 0, X, Rdd We set 100, d 10, N 10, 000. This model has closed from posterior distribution N
Ees Xa cf More details may be found in Appendix B.
Fig. 2a compares the ELBO values of SHF, HIS, and UHA across all optimization iterations. We can see that SHF and UHA-Full result in the highest ELBO, and hence tightest bound on the log normalization constant of the target. In this problem, since we have access to the exact posterior distribution in closed form, we can also estimate the 0-marginal KL divergence directly, as shown in Fig. 26 Here we see the posterior approximation produced by SHF provides significantly lower KI. than the other competing methods. Figs. 2c to 2 show, through number of other metrics, that SHF provides a higher quality posterior approximation than others. is worth noting that while the relative covariance сиror for SHF takes long converge, both the relative mean eтror and KL divergence of SHF. This means that for this particular problem, method finds the centre of the target before fine tuning the covariance. We also note that number of metrics go up for UHA-full because it operates on the augmented space based on sequence of distributions that bridge some simple initial distribution and the target distribution. Therefore, it is not guaranteed that all steps of optimization improve the quality of approximation on the marginal space of the latent variables of interest, which is what the plots in Fig. 2 show.
observe monotonic downward trend in
Figs. (3a and 3h show the time required for each method to evaluate the density of the joint distribution of 0, p and to generate samples. It is clear that the use of coreset improves the density evaluation
8

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Figure 4: Estimated KL divergence (Fig. 4ap. relative 2-norm mean error (Fig. 4b relative Frobenius
norm covariance error (Fig. (4c), energy distance (Fig, 4d), and IMQ KSD (Fig. 44) versus coreset
size. The lines indicate the median, and error regions indicate 25th to 75th percentile from 5 runs.
and sample generation time by more than an order magnitude. Fig. 3e compares the training times and UHA-Coreset (recall that due to the use of subsampled minibatch flow
of SHF, dynamics, HIS-Full and UHA-Ful1 share the same training time as their Coresse versions). The relative training speeds generally match those of sample generation from the target posterior.
Finally, Fig. compares the quality coresets constructed via SHF, uniform subsampling (UNI), and Hilbert coresets with orthogonal matching pursuit (Hilbert-OMP Note that in this problem, the Laplace approximation is exact (the true posterior is Gaussian), and hence Hilbert-OMP constructs coreset using samples from the true posterior. Despite this, SHF provides coresets of comparable quality, in addition to enabling tractable i.i.d. sampling, density evaluation, normalization constant bounds, and straightforward construction via stochastic optimization.
4.2 Bayesian linear regression
In the setting of Bayesian linear regression, consisting of features In E IR and response yn E I. and a model of the form
given set of data points (æn, Un Wn) each
[B log log ~ I),, Vn E [N], Yn In, In,3, indep N ([1 3,0))
where 3 E Re+1 is a vector of regression coefficients and E IR is the noise variance. The datase that we use consists N 100, 000 flights, each containing p = features (e.g., distance of the flight, weather conditions, departure time, etc), and the response variable is the difference, in minutes, between the scheduled and actual departure times. More details can be found in Appendix B.2
Since we no longer have the posterior distribution in closed form, we estimate the mean and covariance using 5000 samples from Stan [46] and treat them as the true posterior mean and covariance. Figs.pa to show the marginal KL, relative mean error, and relative covariance error of SHF, HIS, and UHA, where the marginal KL estimated using the Gaussian approximation of the posterior with the estimated mean and covariance. Here we also include the posterior approximation obtained using the Laplace approximation as baseline. We see that SHF provides the highest quality posterior approximation. Furthermore, Fig. [5d] shows that SHF provides significant improvement in the marginal KL compared with competing coreset constructions UNI and Hi 1beettOMP, This due to the true posterior no longer being Gaussian; the Laplace approximation required by Hiertr--ton PMP fails to capture the shape of the posterior. Additional plots comparing the quality of posterior approximations using various other metrics can be found in Appendix B.2
SelectPields "This dataset consists of airport data foomas/s////ew..tasstttt transtats bts St.e asp? gnoyr_ VQ FGJ and weather data from

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Ittrttin
Itcration
Conesot size
(e)
(f)
(h)
Figure 5: Linear (top) and logistic (bottom) regression results: Gaussian approximated KL divergence versus training time (Figs. [5aand and (e),, relative 2-norm mean error (Figs, Frigss/5Jaandd and 55), relative Frobenius norm covariance error (Figs, and 5g). and Gaussian approximated KL divergence versus coreset size (Figs. 5d and 5h). The lines indicate the median, and error regions indicate 25th to 75th percentile from 5 runs.
4.3 Bayesian logistic regression
In the setting of Bayesian logistic regression, we are given set of data points (Ins Bn) each consisting of features E IR' and label yn E {0. 1}, and model of the form
Vi E i 1), 3g iid Cauchy(O, 1), Vn E [N], Yn
indep Bern ((1 exp (- 2x] =;]))) +))
where 3 E IR++1 The same airline dataset is used with the labels indicating whether flight is can- celled. Of the flights included, 1.384% were cancelled. More details can be found in Appendix B.3
The same procedures as in the Bayesian linear regression example are followed to generate the results in Figs. Fo 5h To account for the class imbalance problem present in the dataset, we construct all subsampled coresets with half the data having label 1 and the rest with label O. The results in Figs, 5c to 5h are similar to those from the Bayesian linear regression example; SHF provides high quality variational approximations to the posterior. Additional plots comparing the quality of posterior approximations using various other metrics can be found Appendix B.3
Conclusion
This paper introduced sparse Hamiltonian flows, a novel coreset-based variational family that enables tractable i.i.d. sampling, and evalution of density and normalization constant. The method randomly subsamples small set of data, and uses the subsample to construct flow from sparse Hamiltonian dynamics. Novel quasi-refreshment steps provide the flow with the flexibility to match target posteriors without introducing additional auxiliary variables. Theoretical results show that, representative model, the method can recover the exact posterior using subsampled dataset of the size that logarithm of its original size, and that quasi-refreshments are guaranteed reduce the KL divergence to the target. Experiments demonstrate that the method provides high quality coreset posterior approximations. One main limitation of our methodology that the data must be "compressible" in the sense that log- likelihoohoo functions of a subset can be used to well represent the full likelloooo. If the data do not live on some underlying low-dimensional manifold, this may not be the case. Additionally, while our quasi-refreshment is simple and works well in practice, more work is required to develop wider variety of eeneral-purpose reass-nte moves. We leave this for future work.
10

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Acknowledgments and Disclosure of Funding
All authors were supported by a National Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant, NSERC Discovery Launch Supplement, and gift from Google LLC.
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contribu- tions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] In Section 3.3 we assume that the momentum distribution is Gaussian. In Section 5 mention that direction of future work is develop
general quasi-refreshment moves that do not require such assumptions.
(c) Did you discuss any potential negative societal impacts of your work? [No] This paper presents an algorithm for sampling from Bayesian posterior distribution the large-scale data regime. There are possible negative societal impacts of downstream applications of this method (e.g. in- ference for particular model and dataset). but we prefer not to speculate about these here. (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes] All proof's provided in the appendix
3. If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We have included anonymized
code in the supplement.
(b) Did you specify the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] All details are provided in the main paper and in Appendix [B]
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] All results come with error bars.
(d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Hardware is listed at the beginning of the experiments.
4. If you using existing assets (e.g.. code, data, models) curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] Footnotes to the source URL and bibliography citations are provided.
(b) Did you mention the license of the assets? [No] None of the real datasets we use specifly a particular license that the data were released under.
(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] We do not produce any new assets.
(d) Did you discuss whether and how consent was obtained from people whose data you're 're us- ing/curating? [No] Our data does not pertain to individuals.
(e) Did you discuss whether the data you are using/curating contains personally identifiable informa- tion or offensive content? [No] Our data does not pertain to individuals.
1f you used crowdsoureing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable?
[N/A]
(b) Did you descrihe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on
participant compensation? [N/A]
13


Reviews:
[{"review_id": "_ZYCQCa8iYX", "reviewer": "Reviewer_pCaB", "summary": "The paper introduces a new method for constructing Bayesian coresets. The authors demonstrate that a single uniform subsampling of data points is in principle sufficient to obtain an exact coreset, and introduce the sparse Hamiltonian flow to efficiently construct and sample from the corresponding coreset posterior approximation. Notable improvements over other coreset methods are reported in several experiments.\n", "questions": "### Detailed comments and questions:\n\nLine 110. After having introduced Hamiltonian Dynamics, the author state that here that it is possible to use it as the basis of a normalizing flow. It would be informative if they could also explain what the advantages of using a Hamiltonian flow are compared to other choices of normalizing flows. I assume this might be explained in [21,22], but it would be helpful if it was reiterated here.\n\nLine 142. The model covered in Proposition 3.1 is referred to as a \"representative example model\". The authors should elaborate on why this is a representative (or even relevant) example model. Can anything be said about the compression for the posterior of other models? Can anything be said for finite N? In particular, it would be interesting to know whether anything could be said about the choice of M in the general setting.\n\nLine 197. \"In this work we assume that p_t ~ N(mu, Lambda^-1)\"\nDoes this mean that we are ultimately assuming that the posterior can be approximated by a diagonal Gaussian? In that case, how is this different from the assumption made in a simple parametric VI setting? In the introduction, the authors explicitly mention the “simple parametric families” as a contrast to the current work. It would be helpful if the contrast between the two was explicitly explained for the case when this parametric assumption of p_t is made.\n\nLine 266. \"Figs. 2c and 2d demonstrate that this reduction in KL divergence is primarily due to a lower relative error in the approximate posterior mean provided by SHF.\"\nTo which extent is this explained by the fact that the posterior is Gaussian in this case? It seems like an idealized setting for the proposed method, given the assumption of Gaussianity of p_t.\n\nIn the Experiments section, the choice of baselines used in the different figures was not clear to me. Why there was a difference in the baseline methods included in Fig 2/3 and Fig 4?\n\nI would have liked to see simpler Bayesian Inference baselines such as Laplace and simple mean-field VI included as baselines as well. This would make it easier for someone not intimately familiar with coreset methods to judge how big an impact these methods have over conventional approaches.\n\nIn the Conclusion section, it would have been helpful with a discussion of the limitations of the proposed approach.\n\n\n### Minor details\n\nLine 81. \"While theoretically not expensive, interleaving MCMC and gradient descent steps is hard to implement and tune, and is too slow to be practical. \"\nThis sentence was not entirely clear to me. First, you state that it is theoretically not expensive, and that it is hard to implement and tune, but then you conclude that it is too slow to be practical, which seems to contradict the first part of the sentence. Please clarify whether the limitation here is fundamental (e.g. efficiency wise), or whether it is practical (difficult to implement and tune).\n\nLine 149. \"Let C be the universal constant from [37], Corollary 1.2.\nThis constant appears a bit unmotivated here right before Propoposition 3.1. It might increase readibility if it was either explained in greater detail, or otherwise moved to the end of the sentence in 152.\n\nLine 163. \"since w only has the first M entries nonzero\"\nWas the convention of placing the nonzero elements first mentioned anywhere? Perhaps I missed it.\n", "limitations": "The authors do not discuss the limitations of their method.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Originality:\nTo my knowledge, the presented method is original. The authors provide a footnote citing concurrent work based on similar ideas, but the combination of Hamiltonian flow approximations to the coreset posterior seems unique to his work. One could argue that the authors should provide a dedicated Related Work section to elaborate on the connections to earlier work.\n\n### Quality:\nThe paper is technically sound, and the claims are carefully developed and well supported. The paper could be further improved with some reflection on the limitations of the approach.\n\n### Clarity:\nThe manuscript is well structured and very clearly written, with helpful introductions to the methodological ingredients that it builds upon.\n\n### Significance:\nThe paper constitutes a significant contribution within research on Bayesian coresets, both in terms of methodology and measured in terms of the performance improvements over other methods. I am not certain how large a contribution it will have to the field of Bayesian inference in general. This would have been easier to assess if the authors had broadened the scope of their baselines to other Bayesian inference procedures.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Kj7xJtGm1s", "reviewer": "Reviewer_hoPP", "summary": "“Bayesian Inference via Sparse Hamiltonian Flows” combines three techniques to make Bayesian Inference faster and more accurate. It combines a) subsampling of the data (core sets), b) sparse flows and c) quasi-refreshments.\n\nThe paper provides theoretical evidence for why these subcomponents reduce the runtime or increase performance (see section 3) and empirical evidence in three different settings. The Sparse Hamiltonian Flows (SHF) clearly and strongly outperform the alternatives in most experiments. \n\n*UPDATE*: thanks for addressing all of my concerns. I update my score from 7 to 8. Great work. Keep it up!\n", "questions": "- Limitations: To my understanding, SHF makes some assumptions that imply limitations. For example, SHF chooses a random subset of data points. In cases where a random subset of data points is not representative of the entire dataset, SHF might be fast but not useful. Is this correct? Are there other limitations of SHF that are not explicitly mentioned?\n- Scale (related to limitations): I expect that more complicated data types will run into problems with random subsets more easily. Thus SHF might not be an appropriate solution for more complex models and data. I’d like to see a more explicit interaction with these points, either by actually running an additional experiment or by stating the implicit assumptions and following consequences in more detail. \n- Subsets: I’m not sure I fully understand the way in which the random subset is chosen. Is the subset of M values chosen uniformly at random once at the beginning of the process or does the subset change over time? \n- Figure 1: I see that during the leapfrog steps (e.g. every 10th iteration), the ELBO jumps up towards a better state. This is in line with the theory. However, from steps 0-9 the ELBO decreases rather than increases. I don’t understand why that is the case. My naive assumption would be that the blue lines should also go up just not as drastically as the red lines. \n", "limitations": "I think, the limitations of the method were discussed insufficiently and should be addressed as described in questions 1 and 2. \n\nI’ll use the rest of the section for high-level comments.\n- In its current form, the paper convinces me that SHF decreases runtime and increases performance for datasets with low complexity. The authors show this with their theoretical analysis and empirical experiments. Furthermore, the paper is well-written and the presentation is good. All of this combined already warrants publication in my opinion. \n- The assumptions that SHF makes and the implied limitations are underexplored. I expect that SHF will have a hard time with more complex models and data because it assumes that a random selection of data points is representative of the entire dataset. I think a good response or an additional experiment in this direction would convince me to raise my score further. Note, that I think the paper would be improved, even if the method is more limited than expected. Stating limitations helps readers and practitioners because it defines the scope of possible use cases more clearly. \n- I want to help where I can. In case something is unclear, feel free to ask follow-up questions. \n", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "In short, I think the paper is good and should be published with minor revisions. \n\n*Strengths:*\n- The paper is very well-written and clear\n- The suggested combination of methods clearly and strongly improves performance compared to the alternatives\n- The paper provides a theoretical analysis of why and in which manner the performance improves due to SHF. \n\n*Weaknesses:*\n- The experiments are all in fairly simple settings. The results already convince me that the method is very strong and warrants publication but a more complex experiment would increase this conviction (see questions). \n- The paper says little about its limitations (see questions).\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Ev9ZYoRkA3m", "reviewer": "Reviewer_ofer", "summary": "This paper proposes a Bayesian inference methodology incorporating coresets with Hamiltonian flows. The paper demonstrates theoretically the challenges that both coresets and variational inference via Hamiltonian dynamics face, and proposes a fix for both in their algorithm \"Sparse Hamiltonian Flows\". Their method first selects a coreset and then follows a sparsified Hamiltonian flow with quasi-refreshments, which allows the flow to update the momentum. Important or argumentative claims are backed up with theoretical proofs. Experiments on a variety of regression problems demonstrate the superiority of their algorithm over current state-of-the-art coreset compression and variational-flow-based methods.", "questions": "- I am under the impression that the quasi-refreshment step could also be incorporated into some of the other baselines considered in the experiments, and not just for SHF. If this is the case, how would the baselines then perform?\n- In Figure 2, relative cov error seems to take a long time to converge, compared to UHA-Full. Why did the purple UHA-Full suddenly go up after ~2000 iterations? It seems like at that point, the approximation is just as good as SHF.", "limitations": "Yes", "rating": 9, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- Important claims and new insights on coresets and Hamiltonian flows are backed up with theoretical proofs. Where may be difficult to prove in the general case, such as Proposition 3.1, a representative example is given and the claim is proven on it. \n- The proposed idea is very novel and addresses important drawbacks that current coreset methods suffer from.\n- A thorough and clear review of related and past methods is provided\n- Experiments are well-conducted and a variety of datasets, both synthetic and real, are explored\n\nWeaknesses:\n- Perhaps some other choices of $\\rho_t$ and $R_\\lambda(x)$ could be explored so that a potential user could understand the sensitivity of the algorithm to these choices", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept