Streaming Bayes GFlowNets
Authors: Tiago Silva
Daniel Augusto de Souza
Diego Mesquita
Venue: NeurIPS 2024 poster (2024)
Keywords: GFlowNets
Streaming VI
Primary area: probabilistic_methods

Abstract: Bayes' rule naturally allows for inference refinement in a streaming fashion, without the need to recompute posteriors from scratch whenever new data arrives. In principle, Bayesian streaming is straightforward: we update our prior with the available data and use the resulting posterior as a prior when processing the next data chunk. In practice, however, this recipe entails i) approximating an intractable posterior at each time step; and ii) encapsulating results appropriately to allow for posterior propagation. For continuous state spaces, variational inference (VI) is particularly convenient due to its scalability and the tractability of variational posteriors, For discrete state spaces, however, state-of-the-art VI results in analytically intractable approximations that are ill-suited for streaming settings. To enable streaming Bayesian inference over discrete parameter spaces, we propose streaming Bayes GFlowNets (abbreviated as SB-GFlowNets) by leveraging the recently proposed GFlowNets --- a powerful class of amortized samplers for discrete compositional objects. Notably, SB-GFlowNet approximates the initial posterior using a standard GFlowNet and subsequently updates it using a tailored procedure that requires only the newly observed data. Our case studies in linear preference learning and phylogenetic inference showcase the effectiveness of SB-GFlowNets in sampling from an unnormalized posterior in a streaming setting. As expected, we also observe that SB-GFlowNets is significantly faster than repeatedly training a GFlowNet from scratch to sample from the full posterior.

Streaming Bayes GFlowNets
Tiago da Silva Getulio Vargas Foundation tiago. tiago.henrien wwwriqueereeeee br
Daniel Augusto de Souza University College London daniel. souza 21@ucl..m ac. uk
Diego Mesquita Getulio Vargas Foundation diego meesuutttfgy br
Abstract
Bayes' rule naturally allows for inference refinement in streaming fashion, with- out the need to recompute posteriors from scratch whenever new data arrives. In principle, Bayesian streaming is straightforward: we update our prior with the available data and use the resulting posterior as a prior when processing the next data chunk. In practice, however, this recipe entails i) approximating an intractable posterior at each time step; and ii) encapsulating results appropriately allow for posterior propagation. For continuous state spaces, variational inference (VI) is particularly convenient due to its scalability and the tractability of variational posteriors. For discrete state spaces, however, state-of-the-ard VI results in ana- lytically intractable approximations that are ill-suited for streaming settings. To enable streaming Bayesian inference streaming Bayes GFlowNets (abbreviated as SB-GFlowNets) by leveraging the re- cently proposed GelowNets- - a powerful class of amortized samplers for discrete compositional objects. Notably, SB-GFlooNNet approximates the initial posterior using standard GFlowNet and subsequently updates it using tailored procedure that requires only the newly observed data. Our case studies linear preference learning and phylogenetic inference showcase the effectiveness of IB-GFlowNett: in sampling from an unnormalized posterior in a streaming setting. As expected, also observe that SB-GFlow Nets is significantly faster than repeatedly training a GFlowNet from scratch to sample from the full posterior.
discrete parameter spaces, we propose
Introduction
One of the foundations of the Big Data revolution in the sciences and engineering is the use of streaming data that is continuously collected and meant be processed as it arrives. Many of the large statistical models in use were first formulated for the batched i.i.d. data setting and updating such models in this setting data at each update to avoid forgetting. [19, 39] In this context, Bayesian inference appears as a natural starting point to learning in the streaming data setting due to its innate property of coherence [5]. which allows learning can happen continuously, independently of how the data is chunked into packages or ordered. More specifically, given prior distribution p(0) and data generating likelihood p(u: 0), the posterior distribution over set of data p(0 y1, M2) x P(y2 0)p(11 O))(O), can be written as p(y2 ()) y1), where first we compute the posterior over y/1 and then used prior to compute the posterior over y2. As conveniently summarized by Lindley [33]: "Today's 's posterior is tomorrow's 's prior".
challenge of its own, it would naively require
revisit all past
Nonetheless, Bayesian inference is notoriously known for being hard to obtain closed-form solutions. The gold standard of approximate methods, Markov chain Monte Carlo (MCMC), requires that the prior and likelihood distributions probability density/mass functions can be evaluated, however,
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

---

as it can only obtain samples from posterior, using the previous posterior as the prior of the next time-step requires additional work [6]. Thus, variational inference (VI), another popular approximate Bayes method, appears as the best fit as its foundation lies in directly parameterizing the approximate posterior distribution instead of relying on empirical distributions of samples.
Importantly, many problems of interest are built upon a discrete set of parameters. In Bayesian phylogenetic inference (BPI), for example, we are concerned with topologies of phylogenetic trees describing the evolutionary genetic history of population. This is a especially compelling application of streaming Bayes, allowing researchers to update posteriors over the phylogenetic trees as they de- code new nucleobases in genetic sequences without reprocessing previously decoded nucleobases. In practice, phylogenetic analyses might involve hundreds of thousands of nucleobases.
Nonetheless, popular methods for VI over discrete posteriors typically rely on gradient- opti- mization (e.g.. maximize evidence lower bounds), requiring the modeling of discrete parameters as transformed versions of latent continuous ones [20, 25, 36]. For methods using VI, a discrete posterior distribution with support over set of size N would require variational posteriors of dimension N, implying either the storage of large covariance matrices of order N in the case of continuous reparameterizations, or the use of massively simplifying assumptions such as mean-field reducing the expressivity of the variational posterior. As an example of support size for discrete distributions, the number of graphs with N nodes grows asymptotically as v2N(8 3//N [16].
In this work, we leverage Generative Flow Networks (GFlowNets) [2, 3, 30] efficiently address the implementation of approximate Bayesian models defined on a discrete set of parameters within streaming data setting. In summary, GelowNets are family of amortized VI methods for high- dimensional discrete parameter spaces, that learn the policies of a finite-horizon Markov decision process (MDP) by minimizing a loss function enforeing a balance condition. The condition, in turn, provably ensures that the samples generated from the MDP are correctly distributed. Notably, despite their successful deployment in solving a wide range problems ([11, 12, 22-24, 32, 34, 64]), previous approaches assumed that the target (posterior) distribution did not change in time. Hence, this is the first work handling the training of GFlowNets in dynamic environments.
More specifically, we propose two novel training schemes to enable GFlowNets on streaming settings. The first consists of directly enforcing a streaming balance condition, which induces least-squares loss that can be optimized in an off-policy fashion, potentially avoiding issues such as mode collapse. Alternatively, we leverage the well-known relationship between GFlowNets and VI to develop an algorithm that often exhibits faster training convergence when the target distribution is not very sparse. We also analyze theoretically how local errors accumulate through posterior propagation, providing upper bounds on the approximation quality of models learned using both of our update schemes.
on-policy Divergence-minimin
In summary, our main contributions are:
1. We propose streaming balance condition and corresponding provably correct algorithm enabling the training of GFlowNets in a streaming data setting (Algorithm 1), without the need to revisit past data;
2. We devise an alternative VI algorithm employing fowvvariance gradient estimators to train to update GFlowNets in streaming fashion (Section 3).
3. We theoretically analyze how inaccuracies of individual updates propagate to subsequent posterior approximations, bounding the approximation errors accumulated through repeated streaming updates of the model (Section 4);
4. We demonstrate the correctness and effectiveness of our method in a series of streaming tasks, such as Bayesian linear preference learning with integer-valued features (Section 5.2) and online Bayesian phylogenetic inference (Section 5.3).
2 Preliminaries
Notation and definitions. Let X' be a set of compositional objects (e.g., trees with nine nodes) and S 2 X' be an extension of X (e.g., forests with nine nodes). Define g = (S, A) as a directed acyclic graph (DAG) with nodes in S, adjacency matrix A, and the following properties: (i) G is weakly connected; (ii) there an unique state 8, S \ X' without any incoming edges; and (iii) there are edges leaving x for every X. We call S the set of states, G the state graph, and
2

---

X' the set of terminal states; is called the initial state. A forward policy over G is function Pr(u, w): 8 x S S-R, such that defines a probability distribution over S with support on v's children in G. A backward policy is a forward policy over GT = (S, AT). To alleviate notation, we denote the set of trajectories that ends in æ as {r T æ), and define a probability distribution in this set PF (F(T) = [1, PF p (8,, 81+1) and the probability of the backwards trajectory condition on the end point x as PB(T x) = [ PB
GelowNets. We define a GFlowNet as a tuple G - (G,PF, ,D,, 2) specifying a state graph G. forward pp and backward pg policies, and an estimate Z of normalizing constant. A GelowNet induces probability distribution over X' through its forward policy, pT (x) == Er (T( When there is no risk of ambiguity, we will generally omit & from this notation. We also drop Z when training the GFlowNet with a criterion that does not require it. Following the common convention, we parameterize the forward network by neural network, and the backward policy at each state left fixed as a uniform distribution.
The task GelowNet is trained for is to match its induced distribution PT with target distribution T (z) o F(r), where ((x) is an unnormalized distribution. In practice, the agreement between pT and can be enforced by balance condition over trajectories, avoiding references pT. For instance, we may enforce trajectory balance (TB) condition, ZPF(T) PB(T ())(() for all ~e x, to estimate the parameters of the GFlowNet by minimizing
log ZPF(T) x) x)
2
CTB(G)= E T
(T
where & is base policy with full support the space of complete trajectories, i.e., trajectories starting at 8, and terminating at sy. Then, we can use the following unbiased estimate to compute the GFlowNet's 's marginal distribution over terminal states PT:
(F(T)
PF(T.)
PT (I) E, (t)pa((a))
Σ ISPS (B(T
(2)
(B((Tx)
K
For a thorough review of GFlowNets, please refer to [3].
GFlowNets and VI. Malkin et al. [38] showed that the training of GFlowNets may be framed as a variational inference on the target Ps (T) ((T)))(T(z) with pF as the proposal distribution. Through the data processing inequality, they showed that the minimization of the Kullback-Leiin (KI.) divergence between trajectory-
distributions pv and ps incurred in a marginal pt over X' matching the target, namely, Dx [Pr [PB] 2 Dx [pT] Then, they demonstrated the equivalence between the gradients of the on-policy TB loss and the KI. divergence, and that the latter a sound learning objective for GFlowNets. Here, we build upon this construction to design KL divergence-basee algorithm for updating GelowNets in a streaming context.
Problem description. We assume that the data is drawn from a distribution f(-lx) indexed by parameter z E X and we define π (x) as a prior distribution over X. Let (D,) be sequence of continually collected independent data sets and Ti (z( D₂) x f(D1..... De (z) = म. (œ) be the posterior conditioned on the first data sets. Note that, due coherence, we can also write te (2:) = (D, -
3 Streaming Bayes GFlowNets
In this section, we propose streaming Bayes GFlowNets (SB-GFlowNets)) as an extension of GFlowNets to handle streaming data. In essence, an SB-GFlowNet G avoids evaluating क.+ (2r).
which would require revisiting all the previous data, by maintaining an SB-GFlowNet G. which
targets previous posterior #t (z). allowing the target distribution of G.41 be f(D |=)pr" (f) (z). (x).
where PT (x) is the induced distribution of Gr, thus avoiding the storage of previous data sets.
We propose two strategies for matching the target distribution, enforcing the streaming balance (SB) condition (Section 3.1) and applying KL streaming updates (Section 3.2). Importantly, both methods require only newly observed data Di+1 and the previously trained model, G, to update the posterior approximation, without revisiting past data batches Di... These conditions are extensions of TB condition and KI. criterion proposed for the batched data case, each with its own strengths and weaknesses. Subsequently, we provide both a theoretical analysis (Section 4) and experimental validation (Section 5) for DB-GFLowNets.
3

---

3.1 Streaming balance condition.
In streaming eettings,trainig GFlowNet to sample from T441 x f(D,.... (1(=))(()) by naively enforcing the TB condition entails numerous evaluations of the complete likelihood. However, as- suming we have previously trained a GFlowNet to sample from क (z)) we propose a more convenient balance condition that does not refer explicitly to previously-seen data.
Definition 1 (Streaming balance condition). Let G, (G Pr PB Ze) be a GelowNet trained to sample proportionally to the posterior 7t (z). The streaming balance (SB) condition for the GFlowNet
(t) (t)
G(+1 PF (G, PB Ze+1), conditioned on G2, is defined
(++1) (++1)
Z+IPP ((+1) (T) [(P(+1l (6)
(T)
PB (T(z),
(6) (r|x)
PB
in which (D+++ 1 (xr) is the model's likelihood. When the backward policy does not depend on i, e.g.,
is fixed as an uniform policy, Equation 3 reduces to Zi+1P$ (F) Zipp Z (D(1))
Intuitively, if we consider the TB conditions for the GetowNets G, and G.+1:
2ppp (T) #. ( (8) (r|æ) and
Z++1PF (1+1) (7)
(t- (T|2),
we can re-arrange the identity for G2 to obtain the unnormalized posterior time I, i.e. me = 7 (1 and, then, apply this the TB condition of G1+1 yielding Equation 3.
Naturally, the SB condition gives rise to loss function by considering the rog-squured ratio between the left- and right-hand sides of Equation 3, namely:
Z.+1PF (1+1
(T)
PB (t)
(T|x)
CsB(Gr+1iGr = E T
log
(4)
1) (T|z)
Zipp
[(D++ (x))
PB
for a distribution & of full-support over trajectories. Importantly, Proposition 1 ensures that this approach results in a model sampling proportionally to (x)) In concurrent work, Venkatraman et al. [54, Eq. (8)) proposed a similar learning objective, termed relative trajectory balance, to update diffusion models according novel and arbitrary likelihood function.
Proposition (Soundness of LSB). Assume p (x)) o (z(z). Then, if LSB(Gr44i Gt) then PT (++1) (x) samples objects from X proportionally to (x+1(x).
Algorithm describes the training of SB- GFlowNet by minimizing CsB.
3.2 Divergence-based updates of SB-GFlowNets.
In practice, estimating log Z, by learning is not straight-forward and, if not done properly, may severely damage the quality of the approximation. Divergence-based objectives provide ways train GFlowNets without relying estimates of log Z,. Indeed, as show in our experimental results (Section 5), minimizing a divergence- based objective often leads to a better approximation than conventional approaches. However, unlike TB which allows the use of arbitrary base policies, these approaches normally require the use of Py (x) as the base policy which may lead to mode collapse depending target distributions [6, 38].
So, as discussed in Section 2, recall that a GFlowNet may equivalently be interpreted as a hierarchical variational model using pr(r) as an approximation to (T(T) x ( (r (z) allowing GFlow to be trained by minimizing any divergence between pr and ps [38]. Based on this insight, we propose the following divergence-basee based training criterion for streaming GFlowNets:
Definition 2 (Divergence-basee) based streaming update). Let G₁ (G.P)) (t) p(0) be GFlowNet balanced
(t+1) (t+1)) and target distribution 1 define the sampling proportional to t.. For G = (G,PF (t+1) :Pn
unnormalized distribution p(T) x PF (T) (z) over trajectories. Then, if E denotes equality up an additive constant,
(T) (4)
LKI G.) =E
T-p (+1 log
Pp (T)
S DEL PF (c+1) (Ct (T)l((r)
(5)
PF
is called the KL's streaming criterion.

---

Note that when LK vanishes, PF (T) Then,
PF 10) (D) for all m and x E X.
(1+1) PT (x)
Σ ( T-2
PF (T) p (x)s(D.))).
Consequently, minimizing LK a sound objective for learning to sample from Te+1 when PT (z) (z)
(t++1) with that of PT when using either (a)).. In Section 4, we quantitatively relate the accuracy of pT
Equation (4) or Equation (5) as learning objectives for the SB-GFlowNet.
Low-variance gradient estimators for DKL. Score matching estimates of the divergence's s gradients, VoDKL, are of high variance, negatively affecting the convergence speed of the trained model [42, 46, 58]. To circumvent this issue, we rely upon the REINFORCE leave-one-out (RLOO) gradient estimator [41], which employs sample-dependent control variate to significantly reduce the noiseness of the estimated gradients. More specifically, define ((T = " (P)((D
and let ..,.... Tk be independently sampled trajectories from PF (1+1) Also, denote by 0 the parameters of PF
(++1), In this context, the RLOO estimator for the gradient of KL's streaming criterion is
1 E Σ Voy(T)) ISiSK
((T))
(T;))
V Vogopp
(++1) (T),
(T),
(6)
k
ISiSK
Isiskijii
which is an unbiased estimate for Voll x sented as vector-Jacobian product. Thus, it can be swiftly computed in standard reverse- mode au- todifferentiation packages [47], adding negligible computational overhead the algorithm.
[T(T]].
Importantly, RLOO is straightforwardly repre-
Theoretical analysis
While posterior propagation computationally convenient, preventing training from scratch repeat- edly. we should also expect errors to propagate through updates. To better understand the behavior of SB-GFlowNets,, we analyze how choosing sub-optimal SB-GFlowNet at time influences our approximation's 's quality at time t + 1. We quantify goodness-of fit of SB-GFlowNets' sampling distribution wrt target both in terms of TV and of their expected distance in log space. Since the SB loss and KL updates incur different parameterizations, we analyze separately the cases in which SB SB-GFlowNets are trained using each loss.
Overall, we establish that the accuracy of a SB-GFlowNet's sampling distribution pf at time t + 1 depends how close the forward policy Pr (t-1) is from being optimal, the size of the new data chunk D++1, and on the goodness-of-fit of the previous estimate PT As expected, our analysis suggests that the negative effects of poorly learned p( are negligible when the size of a data chunk is relatively large. Also, when the previous SB-GFlowNet Ge is poor approximation to the true posterior ne we discuss the benefits using an earlier and potentially more accurate checkpoint G, with < as reference for training Gr+1.
4.1 Analysis for SB loss-based training
We first analyze how errors propagate when training SB-GFlowNets by directly enforcing the SB condition, i.c., minimizing Equation 4. To this effect, recall using the SB loss implies learning an estimate the partition function Z, as well as the forward policy PF at each time step t. In partic- ular, Proposition measures the consequences of an inaccurately learned p and of insufficiently minimizing the SB loss on the (1 + 1)th approximation.
.P Z.+1) be pair of GFlowNets. Let also TT+1(x) x (D and recall that n: + R4 our prior dis- tribution. Defining Zī := = x(x) I. f(D.(æ) (o)))fffol for any ks O, and letting (tt+)) ((++)) 2e41) be the optimal solution to Equation satisfying the SB condition. Then,
Proposition 2.
Let G, (G.P)) (t) (t)
PB Ze) and G (G.Ptt)
(t+1)
(t+1).
F ( (t+1) L3 (T Tet1 s 5811
(p
PT
(++1) s141 +
log
Zet1
Z
F +
LS
PT
Te
+
Ze log
Z
Estimation error
Accuracy of
5

---

where of (p. 9) = (Ez~ [loge<] log (())((()))))))))))) (z))
As expected, Proposition 2 underlines the importance of starting with a good approximation from the ith stage and of properly solving the learning problem the (t + 1)th stage in order to obtain an accurate approximation to Tt-1- In particular, this result reveals the detrimental effect of an inadequately estimated partition function on the quality of GFlowNet's streaming updates.
Complementary, Proposition 3 provides an upper bound the TV distance between pF p(++1) and
Pg (T) = (t++2) (r)) ( Besides corroborating the conclusions of Proposition 2, the ex- plicit dependence of the TV upper bound on of Dz+1 allows for analyzing it as a function of the newly observed data set's likelihood f(D++1 (ær). In particular, consider the case in which [D++1] + oo. Then, since f(Dc+1læ) + uniformly on x and, for many models, f(D++1 (()++1/)///) f(D++ll))) - oo for y & arg max, (D(xx) Then, the accuracy of pt? becomes negligible to the overall approximation of the full posterior when the newly observed data set D4+1 is relatively large and the second term in Proposition 3 approximately vanishes.
Proposition 3. Let (p g) == 1 ExEx p(()) q(x)) be the TV distance between probability distributions p and q. Then, under the same setting of Proposition 2,
2
TV (pt (++1) met1) S TV (PT (e+1)
f(Di+al))
Σ
Z:
pp(())
PT
(x)
Zi
nt(2)
Z+41
Zi+1
&EX
in which & E arg max J (Dee1 (2)) is the maximum likelihood instance in X for D++1-
4.2 Analysis for KL streaming criterion-based training
We now turn to the case in which SB-GFlowNets are trained with KL streaming updates, sequentially minimizing Equation 5 - which case we only learn the forward a backward policy networks. More specifically, Proposition 4 provides an upper bound on the TV distance between pT (1+11 and T as a function of f(D pT (''s closeness to ne, and the learning objective.
Proposition 4. Recall that p(T) oc PF (T))(DL+1lz). Thus, under the notations of Proposition 2,
(4)
PT T () E yr [S(D:l)) Bynr, [D(r)1))]
m(()((D(21l))
(+1) TV (T
TT+1) S 2
DKL
(1+1) PF
lp
+TV
(7)
Estimation error
Accuracy of D
In spite of considering a different learning objective compared to Proposition 3, Proposition 4 rein- forces the importance of properly optimizing the loss in each time step - due its compounding impact on SB D--GFloWNet's accuracy. Nonetheless, the quality of ग. is roughly negligible when the ob- served data set f(D++1) dominates the shape of pT ())((141)))) - and of (De.))(
(1)
Experiments
We show that Be-GFlowNets can accurately learn the posterior distribution conditioned on the stream- ing data for one toy and two practically relevant applications. To start with, Section 5.1 illustrates the applicability of SB-GFlowNets in the context of set generation. Nextly, Section 5.2 showcases the correctness of BB-GFlowNNess in the problem of Bayesian linear preference learning with integer- valued features [8]. Then, Section 5.3 indicates the potential of SB-GFlowNets for carrying out online Bayesian inference over phylogenetic trees [13, 15, 59). Finally, Section 5.4 touches on the problem of Bayesian structure learning with streaming data. To the best of our knowledge, IB-GFlowNets the first method enabling variational inference over discrete parameter spaces within streaming setting, as previous approaches either relied on intractable continuous relaxations [20, 25, 36] or scaled poorly in the size of the target's support [56]. We provide further implementation details regarding model configurations and experimental settings in Appendix C.
5.1 Set generation
Problem description. We first remark that, if Ri,..., Ra+1 are positive functions on X, the problem of streaming Bayesian inference may be generalized to the problem of sampling from

---

231
XBr
X33
25
05
13
Figure 1: SB-GFlowNeets learn an accurate approximation to a changing target distribution in streaming setting for the task of set generation. Each plot depicts the target and learned distributions from the first (left-most) to the last (right-most) streaming update.
X' proportionally to the product II I1:-1 R; by having only access to R&+1 and GFlowNet approximating I Rg Indeed, in the context Bayesian inference, this correspondence is achieved z(D,lz) for i
and (x)x) f(D, (x)) Thus, to demonstrate the effectiveness of SB-GFlowNets and highlight the benefits and pitfalls of each proposed training scheme, we consider the set generation task a popular toy experiment in the GFlowNet literature [3, 44, 53]. In this task, X' is the space of sets with S elements extracted from fixed deposit I= {l....: d). Then, for i E (1,...,K}, we define f:: I + IR and let log R2(x) EeEr fi(e) be the ith log-reward of set 2.
by defining (x
Experimental setup. We fix 24 and 5 = 18. To define R4, independently sample fi(d) from an uniform distribution on [-5,5] 5]. Correspondingly, we define R(a) R/y" as the tempered reward, noting that R2(a) becomes increasingly sparse as O.
tween Table highlights the differences be- training schemes in terms of the target's 's temperature. For relatively sparse tar- gets (with a < 1), the benefits of off- off- off-policy sampling enacted by the minimization the TB loss lead better performance relatively to the KL-minimizing algorithm. Indeed, the on-policy exploration employed by the latter potentially hinders the model's capabilities of finding the sparsely distributed high-probability regions of the target, slowing down the training convergence and potentially leading to mode collapse [38]. Also, albeit one could implement an IS estimator for off-policy training under the KL criterion, our early experiments suggested that the increased gradient variance outweighed the gains from enhanced exploration. Ultimately, the choice of an appropriate learning objective for training GFlowNets will depend on the application. For training SB FB-GGFloNNets, Figure shows that both TB and KL minimization result accurate approximations the changing posterior.
Results.
Table I: TV between the GelowNet and the target. TB outperforms KL for sparse targets (small a).
a 1.00
0.75
0.50
TB 0.21
0.28 +0.10
0.36
KL. 0.13 L0.03
0.17 10.04
0.55
5.2 Linear preference learning with integer-valued features
Problem description. Consider a collection of instances }yr} endowed with a transitive and complete
preference relation 2; we assume that each y; E {1, 0} is binary feature vector. Naturally, the preference relation ž represented as a mapping u such that y > y/ if and only if u(y) u(y'), ) uncovering is the major goal of preference learning methods. Similarly to [8, 21], we here assume that is linear function, u(y) x y, for integer-valued vector and that have access to data set y = {( 3/2 p+)}; } denoting whether yi is preferred to yi2 (p. 1) or otherwise (p: = 0) for fixed individual. Subsequently, we define the probabilistic model
p(p: = 1|x, (Mi yia)) = o (u(ya) u(Ma)) = a (x" (ya - 112))
in which o is the sigmoid function, and prior distribution π (αx) over x [18]; the intuition is that a larger difference between Y/1 and y12' utilities makes the event in which ya is preferred over y12 more likely. Our goal is to infer the individual's preferences based on the posterior x (rlly) for some unseen pair (D1, új), i.e., estimate the predictive distribution p ( 82).y).
Experimental setup. We assume E [[0,4]] and d = 24. At each iteration of the streaming process, we sample a novel subset of the 2d-1, (2d - 1) pairs of d-dimensional binary feature vectors and use them to update the GelowNet. The prior on x is factorized truncated Poisson with À 3.
7

---

232 XIP
10
18
06
05
Figure SB-GFlowNet accurately learns the posterior over the utility's parameters in streaming setting. Each plot compares the marginal distribution learned by SB-GFlowNet (horizontal axis) and the targeted posterior distribution (vertical axis) at increasingly advanced stages of the streaming process, i.e., from n (-121) (left-most) to T (Pis) (right-most).
Results, Figure shows SB-GFlowNet correctly samples form the posterior in a streaming setting. Note, in particular, that the GFlowNet maintains a high distributional accuracy throughout the stream- ing iterations. Correspondingly, the predictive log-likelihon oogllikelhood of fixed held out data set mono- tonically increases as function of the amount of data consumed by the Se-,FlowNet, as shown Figure 3; this behavior, also exhibited by the true posterior [55], emphasizes the similarity between the learned and targeted distributions.
Figure 3: Predictive performance of -esGesowNets in terms of pred. NLL and avg. MSE. SB-GFlowNets behaves similarly to the ground-truth, wrt how the NLL evolves as function data chunks.
5.3 Online Bayesian phylogenetic inference
Problem description. Bayesian phylogenetic inference (BPI) [29, 60] aims to infer structural properties of evolutionary trees given molecular sequences such as DNA and RNA. Formally, let {S,...., Sx} } be collection of N DNA sequences of size M, S; E {A,T, (A,TCC,GT GY", one for each bio- logical species. We define phylogeny as a tuple = (L. b) comprising rooted tree topology and its non-negative branch lengths b. In this context, rooted tree topology is complete binary tree having N labeled leaves corresponding to the considered species and
unlabeled internal nodes corre- the space of phylogenies, adopt
sponding to their ancestors. To carry out Bayesian inference Jukes Cantor's 's nuclcotide substitution model (JC69; [26]) to define a observational model over the observed DNA sequences; calculate the likelihood, we use Felsenstein's algorithm [15].
Crucially, despite the popularity BPI methods, the development of new sequencing technologies has swiftly led to the enlargement of already sizeable sequence databases. In this scenario, maintaining an up-to-date estimate of the posterior became an increasingly difficult task due to the necessity of re-estimating the full posterior from scratch every time a new batch of data is collected. In the following, we show that GFlowNets, which have recently shown SOTA performance BPI [64], can also accurately update the posterior phylogenetic trees given additional sequences.
Experimental setup. We generate the data by simulating JC69's s model for a collection of N species and a substitution rate of 5-1001 (see [59], Chapter 1). At each iteration, we sample a new DNA subsequence of size 102 for each species and update SB-GFlowNet according to Algorithm For Table 2, [D1 = 10 and 'D2l = 102. The prior is an uniform distibution over phy phylogenies.
Figure 4 (left) shows that the learned posterior distribution gets increasingly concen- trated the true phylogenetic tree; this behav- ior, which is inherent to posteriors over phyloge- nies under uniform priors [50], emphasizes the similarity between the learned and targeted dis- tributions for DB-GGFowNets. Figure 4 (right), on the other hand, also suggets that the model's accuracy decreases as a function of the number of streaming updates. Nonetheless, we believe that this is predominantly caused by the poste- rior's increasing sparsity due to the expanding data sets [50], which makes it more difficult for the GFlowNet to learn good approximation [1 [1,,, instead of by an intrinsic limitation of SB-GFlowNE Nets. An investigation of this phenomenon and of the conditions upon which re-train the SB-GFlowNet
Results.
Figure 4: SB-GFlowNet's accurate to the true pos- terior in terms of the probability of the true phylogeny (left) and of the learned modcl's accuracy (right).

---

Table SB-GFlowNet significantly accelerates the training of GetowNets in a streaming setting. Indeed, BB-GFloWNets achieve an accuracy comparable to a GFlowNet trained from scratch to sample from T2( (D::2) in less than half the time (measured seconds per 20k epochs). Number of leaves
Model
7
9
11
GFlowNet
2846.88
3779.11
4821.74
SB-GFlowNet (ours)
1279.68
1714.49
2303.99
Relative accuracy gain (TV)
0.00=0.01
-0.02+0.01
0.00=0.01
based on an earlier checkpoint due to the accumulated unreliability of the streaming updates, as described by Proposition 4, is left application-dependent and is thereby left to future endeavors. Finally, Table 2 shows that GFtow which avoids evaluating the full posterior, more than halve the training time required by standard GFlowNet, while achieving similar performance in terms of the total variation (T'V) distance between the learned and target distributions.
5.4
Bayesian structure learning
Bast Pestperroo
1
1-1
1-4
xurt 1
1-6
50
28
X
x
Sewinng
Figure 5: SB-GFlowNets accurately learns a distribution over DAGs for causal discovery in each time step. each update, an additional dataset of 200 points was sampled from the true model. For this problem, we implemented a DAG-GFlowNNt on 5-variable data sets, similarly to [11, Figure 3].
Problem description. Let X E Raxd be a data set distributed according to a linear Gaussian structural equation model (SEM), i.e., X 3x e, in which 3 Rdxd (sparse) matrix and E Raxd are nd i.dd. samples from normal distribution. We assume that the adjacency matrix induced by 3. [3 # 0] (L. 1. characterizes directed acyclic graph on the dataset's variables. In this case, the linear Gaussian SEM represents Bayesian network, and it is the goal of Bayesian structure learning algorithms to find such network based on the observed X [11]. Although this problem has been studied beyond the constraints of linear models governed by Gaussian distributions [12), we focus on simplified setting in this section. In particular, given sequence (X,)121 of i..dd realizations of a linear Gaussian SEM, we define a belief distribution over Bayesian networks N = ((I,...,d) d), A) on variables (I....,d) d) and adjacency matrix as
log (e)N)=
8 (B)X,) and R1.(N) II R((N), SSiS
(9)
9: [
in which {(3)x,) is the log-likelihood of X, under the linear Gaussian SEM parameterized by B. SB-GFlowNets can naturally handle streaming inference for this model by casting each Rd subposterior distribution - in the same fashion of the set generation task in Section 5.1.
Experimental setup. Likewise Deleu et al. [11, Figure 5], we evaluate SB-GFlowNets on models with d variables. Also, we sample new 200-sized data set from a fixed linear Gaussian SEM for each streaming update of the GFlowNet. To ensure that the generated samples are valid DAGs, we adopt Delcu et al. [11]'s DAG-GFlowNet.
Iubibitty mase theme IMM
0.20
01.31
Results. Our results corroborate the findings of Section 5.3. On the one hand, Figure 5 highlights that our SB-GF 3B-GFlowNet can accurately learn distribution over DAGs across a range of streaming updates. On the other hand, Figure 5 shows that the probability mass assigned by SB-GFlowNer to the true DAG responsible for the data-generating process increases as more samples are incorporated into the belief distribution. Hence, SB-GFlowNets abide by the desiradata of streaming algorithm for Bayesian inference.
Figure 6: The probability mass on the true DAG in- creases more samples added to SB GFlowNet.

---

6 Related works
Applications of GFlowNets. GFlowNets [2, 3, 30] were originally proposed as reinforcement learning algorithms to train stochastic policy sample states proportionally a prescribed reward function, From this point on, GFlowNets were widely applied to problems as diverse as probabilistic modeling [23, 61], combinatorial optimization [62], Bayesian causal discovery [11, 12] and phylogenetic inference [64], symbolic regression [32], stochastic control [30], language modeling [22]. and Bayesian deep learning [34]. Recently, there is an increasing literature concerned with the possibility of composing multiple pre-trained GFlowNets to fit them to specific applications [17] and to accelerate training convergence [31]. In this context, we show how the composition of two GFlowNets through the SB condition leads to an efficient learning algorithm for Bayesian inference under a streaming setting.
Streaming Variational Bayes. Streaming, Distributed, Asynchronous (SDA) Bayes [6] was originally presented as a general framework for implementing variational inference algorithms in streaming setting. Similarly to work, it was based upon the principle that, if the variational distribution (xx) is good approximation to the unnormalized posterior (1)), then ((++1( may be optimized to approximate q( (x)
instead of f(Dx)=)&{(z). This framework was instantiated to accommodate, for example, the learning of Gaussian processes [7], of tensor factorization [14], of feature models [51] and, jointly with SMC, nonlinear state-space models [63]. Nonetheless, our work is the first one to enable the training of GFlowNets in a streaming setting and, due to the relationship between GFlowNets and variational inference algorithms presented by [38, 65], may be viewed as an instance of -Dy-Bayes tailored to inferential problems on combinatorial support. In the realm of approximate Bayesian inference.
Aahound-Sadeed et al. [1], Berner et al. [4], Sendera et al. [52], Zhang and Matsen [60] propose advancements in diffusion-based sampling methods, which we believe could be extended to the context of approximate streaming Bayesian inference via the techniques presented in this work. On the other hand, Mittal approach for approximate Bayesian inference that amortizes over exchangeable data sets to handle posterior inference in novel data. Also, Richter and Berner [48], Richter et al. [49] develop low-variance gradient estimator for carrying out variational inference derived from the log-variance loss, which could be employed as an alternative to our KL-based objective for streaming updates of GFlowNets. On broader scale, Cranmer et al. [9] reviews approximate Bayesian methodology under the lens of simulation-bas based inference.
et al. [40] introduces neural network-basse
7 Conclusions, limitations, and outlook
Conclusions. We proposed the first method for carrying out approximate Bayesian inference over discrete distributions within a streaming setting, called SB-GFlowNet. We proposed two training/update schemes for SB-GF SB-GFlowNet, as well as a theoretical analysis accounting for the compounding effect of errors due to posterior propagation. Our experimental evaluation showcases SB-GFlowNet's 's effectiveness in accurately sampling from the target posterior - while still achieving a significant reduction in training time relative to standard GFlowNets.
Limitations. Proposition 2 and Proposition suggest that an inappropriate approximation to ne may be propagated through time and lead to increasingly inaccurate models. This phenomenon, known as catasthropic forgetting in the online literature [19, 39], may eventually demand retraining of the current SB- GFlowNet based traditional variational methods, where the expressiveness of the approximation family is explicitly chosen, the expressiveness implied by different parametrizations of GFlowNets is not clear. We believe this is a fruitful avenue for future investigation.
earlier checkpoint on the full posterior. We also note that unlike
Outlook and future works As our proposal provides generic and efficient streaming variational inference solution for discrete parameters, we believe it will allow the use of streaming discrete Bayesian inference to more datasets and to less explored research domains. In this work, we proposed two training criteria based on the trajectory balance condition and the KL criterion, nonetheless, these are not the only training schemes for GFlowNets, we believe other proposed criteria such as detailed balance [3] or the sub-trajectory balance [35] could be adapted for the streaming setting as well. Additionally, due to the flexibility of GFlowNets in sampling unnormalized distributions, our proposal can be extended to different divergences and generalized likelihoods to improve the predictive quality of the learned posteriors [28]. Finally, the problem of updating GelowNet when the size of the generated object changes, c.g., when new species is observed during BPI, remains open.
10

---

Acknowledgements
This work was supported by the Fundação Carlos Chagas Filho de Amparo Pesquisa do Es- tado do Rio de Janeiro FAPERJ (SEI-26000300070//0033 the São Paulo Research Foundation FAPESP (2023/00815-6) the Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq (404336/2023-0) and the Silicon Valley Community Foundation through the University Blockchain Research Initiative (Grant #2022-199610).
References
[1] T. 1.Akhound-SS Akhound-Sadegh, Rector-Brooks, J. Bose, S. Mittal, P. Lemos, C.-H. Liu, M. Sendera, S. Ravanbakhsh, G. Gidel, Y. Bengio, N. Malkin, and A. Tong. Iterated denoising energy matching for sampling from Boltzmann densities. arXiv preprint arxiv:2402.06121, 2024.
[2] E. Bengio, M. Jain, M. Korablyov, D. Precup. and Bengio. Flow network based generative models for non-iterative diverse candidate generation. In Advances in Neural Information Processing Systems (NeurIPS), 2021.
[3] Y. Bengio, Lahlou, T. Deleu, E. Hu, M. Tiwari, and E. Bengio. GFlowNet foundations. Journal of Machine Learning Research (JMLR), 24(210), 2023.
[4] J. Berner, L. Richter, and K. Ullrich. An optimal control perspective on diffusion based generative modeling. arXiv preprint arxiv:2211.01366 2024.
[5] G. Bissiri, C. Holmes, and Walker. general framework for updating belief distributions. Journal of the Royal Statistical Society: Series B (Methodological), 78(5), 2016.
[6] T. Broderick, N. Boyd, A. Wibisono, A. C. Wilson, and M. I. Jordan. Streaming variational Bayes. In Advances in Neural Information Processing Systems (NeurIPS), 2013.
[7] T. D. Bui, C. Nguyen, and R. E. Turner. Streaming sparse Gaussian process approximations. Advances in Neural Information Processing Systems (NeurIPS), 2017.
[8] T. Cole. Algorithm AS 281: Scaling and rounding regression coefficients to integers. Applied Statistics, 42(1), 1993.
[9] K. Crammer, J. Brehmer, and G. Louppe. The frontier iimulation-bssed inference. Proceedings of the National Academy of Sciences, 1 117(48), 2020.
[10] I. Csiszár and J. Kömnr. Information theory: coding theorems for discrete memoryless systems. Cambridge University Press, 2011.
[11] T. Deleu, A. Góis, C. C. Emezue, M. Rankawat, S. Lacoste-Julien, S. Bauer, and Y. Bengio. Bayesian structure learning with generative flow networks. In Conference on Uncertainty in Artificial Intelligence (UAI), 2022.
[12] T. Deleu, M. Nishikawa-Toomeee J. Subramanian, N. Malkin, L. Charlin, and Y. Bengio. Joint Bayesian inference of graphical structure and parameters with single generative flow network. In Advances in Neural Information Processing Systems (NeurIPS), 2023.
[13] V. Dinh, A. E. Darling, and F. A. Matsen, IV. Online Bayesian phylogenetic inference: Theoret- ical foundations via sequential Monte Carlo. Systematic Biology, 67(3), 2017.
[14] Fang, Z. Wang, Pan, J. Liu, and S. Zhe. Streaming Bayesian deep tensor factorization. In International Conference on Machine Learning (ICML), 2021.
[15] J. Felsenstein. Evolutionary trees from DNA sequences: A maximum likelihood approach. Journal of Molecular Evolution, 17, 1981.
[16] Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, edition, 2009.
[17] T. Garipov, S. D. Peuter, G. Yang, V. Garg, S. Kaski, and T. S. Jaakkola. Compositional sculpting of iterative generative processes. Advances in Neural Information Processing Systems (NeurIPS), 2023.

---

[18] J. González, Z. Dai, A. Damianou, and N. D. Lawrence. Preferential Bayesian optimization. In International Conference on Machine Learning (ICML), 2017.
[19] I. J. Goodfellow, M. Mirza, X. Da, A. C. Courville, and Y. Bengio. An empirical investigation of catastrophic forgeting in gradient-based neural networks. In International Conference Learning Representations (ICLR), 2014.
[20] J. Han, F. Ding, X. Liu, L. Torresani, J. Peng, and Q. Liu. Stein variational inference for discrete distributions. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2020.
[21] J. C. Hornberger, H. Habraken, and D. A. Bloch. Minimum data needed on patient preferences for accurate, efficient medical decision making. Medical Care, 33(3), 1995.
[22] E. J. Hu, M. Jain, E. Elmoznino, Y. Kaddar, G. Lajoie, Y. Bengio, and N. Malkin. Amortizing intractable inference in large language models. arXiv preprint arxiv:23 2023.
[23] E. Hu, N. Malkin, M. Jain, K. E. Everett, A. Graikos, and Y. Bengio. GFlowNet-EM for learning compositional latent variable models. In International Conference on Machine Learning (ICML), 2023.
[24] M. Jain, S. C. Raparthy, A. Hemandez-GGarchn J. Sector-Brooks, Y. Bengio, S. Miret, and E. Bengio. Multi-objective GFlowNets. In International Conference Machine Learning (ICML), 2023. 2023.
[25] E. Jang, S. Gu, and B. Poole. Categorical reparameterization with Gumbel-softmax. In International Conference on Learning Representations (ICLR), 2017.
[26] H. Jukes and R. Cantor. Evolution protein molecules. Mammalian Protein Metabolism. Academic Press, 1969.
[27] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arxiv:14/2.6980, 2014.
[28] J. Knoblauch, J. Jewson, and T. Damoulas. An optimization-centtr view on Bayes' rule: Reviewing and generalizing variational inference. Journal of Machine Learning Research (JMLR), 23(132), 2022.
[29] O. Kviman, R. Molén, and J. Lagergren. Improved variational Bayesian phylogenetic inference using mixtures. arXiv preprint arxiv:2310.009: 2023.
[30] S. Lahlou, T. Deleu, P. Lemos, D. Zhang, A. Volokhova, A. Hernández-Gareía, L. N. Ezzine, Bengio, and N. Malkin. A theory of continuous generative flow networks. In International Conference on Machine Learning (ICML), 2023.
[31] E. Lau, N. M. Vemgal, D. Precup. and E. Bengio. DGFN: Double generative flow networks. In NeurlPS 2023 Generative Al and Biology (GenBio) Workshop, 2023.
[32] S. Li, I. Marinescu, and S. Musslick. Gfn-sr: Symbolic regression with generative flow networks. arXiv preprini arxiv:2312.00330 2023.
[33] D. V. Lindley. Bayesian statistics: review. SIAM, 1972.
[34] D. Liu, M. Jain, B. F. P. Dossou, Q. Shen, S. Lahlou, A. Goyal, N. Malkin, C. C. Emezue, D. Zhang, N. Hassen, X. Ji, K. Kawaguchi, and Y. Bengio. GFlowOut: Dropout with generative flow networks. In International Conference on Machine Learning (ICML), 2023.
[35] K. Madan, J. Rector-Brooks, M. Korablyov, F. Bengio, M. Jain, A. C. Nica, T. Bosc, Y. Bengio, and N. Malkin. Learning GFlowNets from partial episodes for improved convergence and stability. In International Conference on Machine Learning (ICML), 2022.
[36] C. J. Maddison, A. Mnih, and Y. W. Teh. The concrete distribution: A continuous relaxation of discrete random variables. In International Conference on Learning Representations (ICLR), 2017.
12

---

[37] N. Malkin, M. Jain, E. Bengio. C. Sun, and Y. Bengio. Trajectory balance: Improved credit assignment in GFlowNets. In Advances in Neural Information Processing Systems (NeurIPS), 2022.
[38] N. Malkin, S. Lahlou, T. Deleu, X. Ji, E. Hu, K. Everett, D. Zhang, and Y. Bengio. GFlowNets and variational inference. International Conference on Learning Representations (ICLR), 2023.
[39] M. McCloskey and N. J. Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of Learning and Motivation, volume 24. Academic Press, 1989.
[40] S. Mittal, N. L. Bracher, G. Lajoie, P. Jaini, and M. A. Brubaker. Exploring exchangeable dataset amortization for Bayesian posterior inference. In ICML 2023 Workshop Structured Probubilistic Inference/ / Generative Modeling, 2023.
[41] A. Mnih and D. Rezende. Variational inference for Monte Carlo objectives. In International Conference on Machine Learning (ICML), 2016.
[42] S. Mohamed, M. Rosca, M. Figurnov, and A. Mnih. Monte Carlo gradient estimation in machine learning. Journal of Machine Learning Research (JMLR), 21(13 2020.
[43] M. Newman. Networks. Oxford University Press, 2018.
[44] L. Pan, N. Malkin, D. Zhang, and Y. Bengio. Better training of GFlowNets with local credit and incomplete trajectories. In International Conference on Machine Learning (ICML), 2023.
[45] I.. Pan, D. Zhang, M. Jain, L.. Huang, and Y. Bengio. Stochastic generative flow networks. In Conference Uncertainty in Artificial Intelligence (UAI), 2023.
[46] M. Papini, D. Binaghi, G. Canonaco, M. Pirotta, and M. Restelli. Stochastic variance-reduced policy gradient. In International Conference on Machine Learning (ICML), 2018.
[47] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang. 7. Vito,, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, I.. Fang, J. Bai, and S. Chintala. PyTorrch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems (NeurIPS), 2019.
[48] L. Richter and J. Berner. Improved sampling via learned diffusions. arXiv preprint w0x.0 2023.
[49] L. Richter, A. Boustati, N. Nüsken, F. Ruiz, and O. D. Akyildiz. Vargrad: a Low-veriance gradient estimator for variational inference. Advances in Neural Information Processing Systems (NeurIPS), 33, 2020.
[50] A. RoyChoudhury, A. Willis, and J. Bunge. Consistency of phylogenetic tree maximum likelihood estimator. Journal of Statistical Planning and Inference, 161, 2015.
[51] R. Schaeffer, Y. Du, G. K. Liu, and I. Fiete. Streaming inference for infinite feature models. In International Conference on Machine Learning (ICML), 2022.
[52] M. Sendera, M. Kim, S. Mittal, P. Lemos, L. Scimeca, J. Rector- Brooks, A. Adam, Y. Bengio, and N. Malkin. On diffusion models for amortized inference: Benchmarking and improving stochastic control and sampling. arXiv preprint arxiv:2307.01 198, 2024.
[53] M. W. Shen, E. Bengio, E. Hajiramezanali, A. Loukas. Cho, and T. Biancalani. Towards understanding and improving GFlowNet training. In International Conference on Machine Learning (ICML), 2023.
[54] S. Venkatraman, M. Jain, L. Scimeca, M. Kim, M. Sendera, M. Hasan, Rowe, Mittal, P. Lemos, E. Bengio, A. Adam, J. Rector-Brooks, Y. Bengio, G. Berseth, and N. Malkin. Amortizing intractable inference in diffusion models for vision, language, and control, 2024. URL https:/ //tpr//aavvv////aa//aa/eeaaaaiaa org/ 2405 20971.
13

---

[55] A. M. Walker. On the asymptotic behaviour of posterior distributions. Journal of the Royal Statistical Society: Series B (Methodological), 31(1), 1969.
[56] M. Wojnowicz, S. Aeron, E. L. Miller, and M. Hughes. Easy variational inference for categorical models via an independent binary approximation. In International Conference Machine Learning (ICML). 2022.
[57] K. Xu, W. Hu, Leskovec, and S. Jegelka. How powerful are graph neural networks? Interna- tional Conference Learning Representations (ICLR), 2019.
[58] Xu, F. Gao, and Q. Gu. An improved convergence analysis of stochastic variance-reduced policy gradient. In Conference on Uncertainty in Artificial Intelligence (UAI), 2020.
[59] Z. Yang. Molecular Evolution: A Statistical Approach. Oxford University Press, 2014.
[60] C. Zhang and F. A. Matsen, IV. Variational Bayesian phylogenetic inference. In International Conference on Learning Representations (ICLR), 2019.
[61] D. Zhang, N. Malkin, Z. Liu, A. Volokhova, A. Courville, and Y. Bengio. Generative flow networks for discrete probabilistic modeling. In International Conference Machine Learning (ICML), 2022.
[62] D. Zhang, H. Dai, N. Malkin, A. Courville, Y. Bengio, and L. Pan. Let the flows tell: Solving graph combinatorial optimization problems with GFlowNets. In Advances in Neural Information Processing Systems (NeurIPS), 2023.
[63] Y. Zhao, J. Nassar, I. Jordan, M. Bugallo, and I. M. Park. Streaming variational Monte Carlo. IEFE Transactions on Pattern Analysis and Machine Intelligence, 45(1), 2023.
[64] M. Zhou, Yan, E. Layne, N. Malkin, D. Zhang, M. Jain, M. Blanchette, and Bengio. PhyloGFN: Phylogenetic inference with generative flow networks. In International Conference on Learning Representations (ICLR), 2024.
[65] H. Zimmermann, F. Lindsten, J.-W, van de Meent, and C. A. Naesseth. A variational perspective on generative flow networks, 2022. URL https://arxivv www.aiwww......e.....eoee wwrg/abs/2110... 07992.
14

---

Algorithm Training a SB-GFlowNet by minimizing Lsn
Require: (Di)r2121 streaming data sets, [(((ar) likelihood model parametrized by x (I)) prior distribution over X'
Ensure: GT samples proportionally to (II,, f(D()z)) ((x)
G1 + ( Z1) =- arg min E (TT pn, Z)]
Roughly minimize LT via SGD
PF-PB.2
for t in (2...., T) do
Roughly minimize LsB via SGD
Ge + (pF PB Z1) = arg min T-pp-1) [Csn(T:pr.PB: PB: Z: G:)
pF.pa.2
end for
A Training of SB-GFlowNets
Algorithm outlines the training of RB-GGFlowNet by minimizing the SB loss. As we described in the text, the first GFlowNet is trained conventionally by approximately minimizing either the TB loss or the KI. divergence between the forward and backward policies. Then, the subsequent models are trained by the approximate minimization of either the SB loss or streaming balance criterion. Notably, the problem of streaming update may framed as the learning of GFlowNets with stochastic rewards [45], which are not determistically associated to the terminal states.
B Proofs
B.1 Proof of Proposition
We are assuming that p x m, and that (x<g([sB(T)) = 0 for a distribution & of full support. Thus, (SB(T) = 0 for all As consequence,
a PF
(T)
PB
(++1) (r(z) (r(x)
Z: (
PF (T)f(D(+1lz))
(10)
PB (T|x)
(1)
Ze+1
(8) By assumption, (pF , PB Z,) satisfy the trajectory balance condition with respect therefore,
Z: PF (T) = Pp (r))) Thus,
PF (T)
(11)
(1+1) (f|z)
PB
Finally, by summing over 7 r:
Pr (T)
(t+ (x) = B T (11))))))))) PT (x)
(12)
[e+ 1 (T(z)
PB
(x))
(13)
B.2 Proof of Proposition
Firstly, we note that (p.(p.))) q) is metric. Thus, by the triangle inequality,
T++ LS
PT MT+1 S 655 (pf (++1) p (((+1))
(++1)
+ -T (pt+3) Te+1)
(14)
The first term of the higd--handsside of the preceding equation corresponds to the estimation er- ror associated to the GFlowNet's learning problem. For the second term, note that Ø
((++)(())
(x)
Ze 1t (D++al))))) by assumption (5 (z(++]) satisfies the SB condition). Thus,
Z0n1
Z,
+ logp (1) (x){(D(+1lz) log ZiTe(z) f(D(-1lz) Zi+1
log log ((++)) (z))))(()))))) (x) - log (z)1 = log
Z
Z
Zi
PT (x)
(1)
log
log
log
2,+1
Z,
15

---

The result follows by further application of the triangle inequality to o namely,
(PT
p m0t1)
27 1/2
((++)) PT (e411
Zi+1 Z
Zt
PT
10
(x)
LS
Exwtet
log
+ log
log
Z
TT
(15)
log
Ze+1 Z
+
log
Ze
LS
(PT m
(t)
Zi
Proposition 2 is obtained by plugging Equation 15 into Equation 14.
B.3 Proof of Proposition
This result, which follows from reasoning similar to the one in Proposition 2 above, aims to show the dependence of the model's performance on the newly observed dataset. In this sense, notice that
TV ( 11+t1) x++1) & TV
(((t1) 1 p(+tt) + TV (p4+1), mmt1))
(16)
Thus, since p
2 (z) = Zi4 1-p (z)
TV p((+1) Te
1 Σ
(x) (2)1
-
PT
Ze (4)
Zi 7i+1
2
Σ
[(D++1lxx)
2,41
PT (2r)
Te (x(
(17)
ZEN
1 (D(+1
Σ
Ze
PT (x)
Z
Tr (3r)
Zz41
7i44
TEX
The result follows by plugging Equation 17 into Equation 16.
B.4 Proof of Proposition
This result follows from the successive application of the triangle, Pinsker's [10] and Jensen's inequality applied to the KL divergence. More specifically, first note that the optimal distribution under the KL. streaming criterion satisfies pf+11 x pt s(De+1l2)) 1z)). Then, by the triangle inequality,
TV p (
me+1)
TV (p(+)) p
p(+11) (+(+))((+)))
TV (p(+)),T44))
(18)
For the first term, note that
TV (e+1) p pT
((+1)
I 2 DxL
1
pT 11
| (1+2)
((+1)
S
2 2/
1
DEL
(e+1) PF
11p
(19)
(((+)) x by definition; recall that p(r) = pr ( 1z). Here, the first inequality follows since pf from Pinsker's s inequality and the second one, from the pata-procsssing inequality. For the second
term, note that
(2) EBEX P (v)
PT (x)((D++1z))
(z)
(20)
E WE [[(D++1
with a corresponding representation π&+1 as function n, and (D++1lzz The result follows by pluggin Equation 20 and Equation 19 into Equation 18.
C Experimental details
We provide below details for reproducing our experiments for each considered generative task. To approximately solve the optimization problem outlined in Algorithm 1, we employed the Adam optimizer [27] with learning rate of 10-3 for the pr 's parameters and 10-1 for log Z., following rec- ommendations from [37]. Also, we linearly decreased the learning rate during training. Experiments were in cluster equipped with A100 and V100 GPUs, using single GPU per run.
16

---

3
(
LL2
!!
D.20
(15) E #119 (Ssss""
1a 19- - E
#P 18)
#Pt8 e al (8
Figure 7: Illustration of the task of generating sets of size |S] with elements in 2,32. On each streaming update. a novel reward function R.!") is observed; a small value of a entails a more sparse and harder-to-samin -aaeertt---amle-rroon distribution. Terminal states X' illustrated in green and non-terninal states are depicted in blue. At the /th iteration, we learn generative model p sampling S proportionally to Misie R (S).
C.1 Set generation
Experimental setup. We fixed d 24 and S 18. To parameterize the forward policy, imple- mented an two-layer neural network with a 128-dimensional latent embedding. For the streaming updates, we fixed trained the model by minimizing the KL streaming criterion.
and randomly sampled the objects' utilities each novel iteration. We
GFlowNet's design. To generate set X, we iteratively add elements randomly sampled from to an initially empty z until æ has size St see Figure 7. The forward policy is parameterized as a two-layer neural network and the backward policy is fixed an uniform distribution.
C.2 Linear preference learning with integer-valued features
Experimental setup. We assume that x [[0, 4]] and d 24 and that the data was simulated from the observational model. At each streaming round, a novel and independent data set was simulated and the model was trained by minimizing the SB loss. To parameterize the forward policy, implemented an MLP with 2 64-dimensional layers receiving the padded parameter a as an input and returning probability distribution over [[o, 4]].
GFlowNet's design. The generative process implemented by the GFlowNet consists of, starting an initially empty state iteratively sampling a value from [[0, 10]] and appending this value to the current state until its length reaches d. To parameterize the forward policy, we use an MLP with two 64-dimensional layers that receives the padded state fixed-size input.
C,3 Online Bayesian phylogenetic inference
Experimental setup. We assume the observational data follow the J&C69 mutation model with an instantaneous mutation rate of À corresponding observational model conditioned on randomly sampled tree with 7 leaves, and this process was repeated at each streaming round. To illustrate the computational gains enacted by the implementation of 3B-GFlowNets in Table 2, considered updating GFlowNet trained initially large data set according to the newly observed and relatively small biological sequences, which is common challenge faced by practitioners. Then, by avoiding the additional log- likeeiidd evaluations, we achieved significantly faster inference.
10-3 The data was synthetically generated from the
GFlowNet's design. A state consists of a forest of complete binary trees. Initially, all leaves are roots of their own singleton trees and, at each iteration of the generative process, we select two trees and join their roots to newly added unlabelled node. This procedure is finished when all leaves are connected; see [64, Figure 1] for an illustration of this generative process. To parameterize pp, we graph isomorphism network (GIN; [57]); the backward policy is fixed as uniform.
C.4 Bayesian structure learning
Experimental setup. We sample each component of the error terms from zero-centered Gaussian distribution with standard deviation a = 10-2, Also, we select a random Bayesian network from a directed configuration model [43] and draw the components 3 from corresponding standard Gaussian distribution to define the true data- sattaggnerating process. To parameterize the policy network, we use an MLP with 2 28-dimensional layers receiving the DAG's flattened adjacency matrix input.
17

---

GFlowNet's design. We adopt Deleu et al. [[]]'s DAG-GFlowNet. In a nutshell, the generative process starts an edgeless graph and each transition either adds an edge to the current state or triggers a stop. To ensure the acyclicity of the generated samples, we follow [11, Appendix CJ and iteratively update a binary vector
(1,0) axd indicating which edges in the adjacency matrix can be safely added to the current state without forming cycles.
D On the permutation invariance of SB-GFlowNets
(a) Phylogenetic inference.
(b) Set generation.
Figure 8: Permutation invariance of SB-GFlowNets for phylogenetics (a) and set generation (b). When the first GFlowNet not adequately trained, the learned distribution after two streaming updates depends the ordering of the observed datasets (left (a), left (b)). In contrast, when both the first and second GFlow Nets are accurate, the resulting distribution is approximately invariant the data permutation (right (a), right (b)).
Exchangeability is natural property of each case study we presented in the main text. We thus ask: SB- Be-GFlowNNets permutation invariant? Clearly, the distribution learned by a SB-GFlowNet does not depend on the order in which the data sets are observed when the SB condition is satisfied at each streaming update; this is a direct consequence of Proposition On other hand, permutation in- variance is not guaranteed when the learning objectives are only imperfectly minimized; see Figure 8 for an extreme example. To the best of our knowledge, however, this sensibility to data ordering is property of every approximate streaming Bayesian inference method, c.g., [6, 13].
18

---

NeurIPS Paper Checklist
Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: Our claims are supported by proofs in Appendix B and experiments in Sec- tion 5.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine include aspirational goals as motivation long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
The middle paragraph in Section 7 discusses limitations.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed not penalize honesty concerning limitations.
19

---

3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full of assumptions and a complete (and correct) proof?
Answer: [Yes]
Justification:
Assumptions are clearly stated in the statements, and Appendix B.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
Justification: Details are provided Appendix C.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility instructions for how to replicate the results, access to a hosted model (e.g., in the case of large language model), releasing of a model checkpoint, other means that appropriate to the research performed.
also be provided via detailed
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how reproduce that algorithm.
(b) the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.
(c) the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce
20

---

the model (e.g., with an spen-source dataset or instructions for how to construct the dataset).
(d)
We recognize that reproducibility may be tricky in some cases, in which authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should possible for other researchers have some path reproducing verifying the results.
5. Open access to data and code
Question: Does the paper provide open access the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification: Provided in a zip file.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.co/
for more details.
While we encourage the release of code and data, we understand that this might be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https: ctaips.c//ublll. www.ccccgueseeeeeeeeee Sowps//Sooeeuuiisiiiioooooooo
for more details.
The authors should provide instructions data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing much information possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental Setting/Details
Does the paper specify all the training and test details (c.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary understand the results?
Question:
Answer: [Yes]
Justification: Provided in Appendix C.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to a level of detail that is necessary appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment Statistical Significance
21

---

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification:
In general, plots count enror bars and tables count standard deviation.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results accompanied by error bars, confi- dence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma error bars, but should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification: First paragraph of Appendix C.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary failed experiments that didn't make into the paper).
9. Code Of Ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics https: : http//euripp..//
Answer: [Yes]
Justification:
Our submission follows the NeurIPS ethical guidelines.
Guidelines:
22

---

The answer NA means that the authors have not reviewed the NeurlPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there special consid- eration due to laws or regulations in their jurisdiction).
10. Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification: We provide perspective broader impacts in Section 7, but do not foresee any direct negative societal impact.
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (c.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g.. deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional unintentional) misuse of the technology.
there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility ML).
11. Safeguards
Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, scraped datasets)?
Answer: [NA]
Justification:
We do not foresee any direct risk stemming from work.
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
23

---

We recognize that providing effective safeguards is challenging, and many papers do require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Question: Are the creators or original owners of assets (e.g.. code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [NA]
Justification: All code was made by the authors
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi seporswithcode thcode. com datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
this information not available online, authors are encouraged to reach out to the asset's s creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification:
No new assets.
Guidelines:
The answer NA means that the paper does release new assets.
Researchers should communicate the details of the dataset/coddemoodll as part of their submissions structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include an anonymized zip file.
Crowdsourcing and Research with Human Subjects
For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Question:
Answer: [NA]
Justification: No experiments with human subjects.

---

Guidelines:
The answer NA means that the paper does involve crowdsourcing research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification:
No experiments with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing research with human subjects.
Depending the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such the institution conducting the review.
25


Reviews:
[{"review_id": "K3Sd4gXWhJ", "reviewer": "Reviewer_xCCn", "summary": "The paper introduces a method for Bayesian inference in streaming data scenarios, particularly for discrete parameter spaces, called Streaming Bayes GFlowNets (SB-GFlowNets). The main contributions are:\n\n* Bayesian Streaming Inference: SB-GFlowNets enable the continuous updating of posterior distributions as new data arrives, without needing to recompute from scratch.\n\n* Addressing Intractability: The method addresses the challenge of approximating intractable posteriors in discrete state spaces, which is a limitation of existing variational inference (VI) techniques.\n\n* GFlowNet Utilization: SB-GFlowNets leverage GFlowNets, a class of amortized samplers, to approximate the initial posterior and update it incrementally with new data.\n\n* Case Studies: The effectiveness of SB-GFlowNets is demonstrated in linear preference learning and phylogenetic inference, showing its ability to sample from unnormalized posteriors efficiently in a streaming context.\n\n* Performance: The method is significantly faster compared to repeatedly training a GFlowNet from scratch for the full posterior.", "questions": "Sentence in lines 58-59 trails off.", "limitations": "Yes, limitations have been addressed.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "Two technically novel training algorithms for streaming inference on GFlowNets based on modified balance condition and VI with a control variate. Presentation was reasonably clear and the experimental results were convincing. Removes limitation of GFLowNets, an increasingly important family of models, to only batched training.", "weaknesses": "A weakness of the paper is that it may not sufficiently highlight the significance of streaming inference compared to batch inference. This may be apparent to Bayesian practitioners but not so to a general ML audience. Are there more compelling applications of streaming inference?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your suggestions to improve our manuscript. We hope our answers elevate your appraisal of our work. Should you feel that any points require additional clarification, we are more than willing to engage further.\n\n> A weakness of the paper is that it may not sufficiently highlight the significance of streaming inference compared to batch inference. This may be apparent to Bayesian practitioners but not so to a general ML audience. Are there more compelling applications of streaming inference?\n\nThank you for the suggestion. \n\nOne especially compelling application is Bayesian phylogenetics, which we consider in our experiments. Streaming Bayes would allow researchers to update posteriors over the phylogeny tree as they decode new nucleobases in genetic sequences – without reprocessing previously decoded nucleobases. This is particularly convenient since, while there are GPU-accelerated algorithms to compute the likelihood [1-2], “making things fit within a GPU” to minimize communication overhead is a major concern in practical Bayesian phylogenetics. In practice, phylogenetic analyses might involve hundreds of thousands of nucleobases.\n\nMore generally, in streaming data settings (i.e., ever-increasing datasets), updating approximate Bayesian posteriors in batch mode entails repeatedly estimating the true posterior from scratch. The major raison d'être for streaming Bayes methods is alleviating this computational bottleneck by reusing the previous posterior estimate as the prior of the next posterior estimate (\"Today's posterior is tomorrow's prior'). \n\nWe will incorporate this discussion in the introduction to make it more suitable for a broader ML audience.\n\n\n> Sentence in lines 58-59 trails off. \n\nThank you for pointing it out, we accidentally commented out part of this sentence. The intended sentence is as follows: “Notably, despite their successful deployment in solving a wide range of problems ([8 , 9, 19 – 21, 29, 31, 55 ]) , previous approaches assumed that the target (posterior) distribution did not change in time. Hence, this is the first work handling the training of GFlowNets in dynamic environments.”\n\n\n\n[1] Ayres et al., BEAGLE 3: Improved Performance, Scaling, and Usability for a High-Performance Computing Library for Statistical Phylogenetics, Systematic Biology, 2019\n\n\n[2] Suchard, Rambaut. Many-core algorithms for statistical phylogenetics. Bioinformatics, 2009.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4S7KMxhhy0", "reviewer": "Reviewer_w9BG", "summary": "This paper introduces Streaming Bayes GFlowNets, a method for performing approximate Bayesian inference over discrete parameter spaces in streaming data settings.\n\nSB-GFlowNets allow efficient updating of posterior approximations as new data arrives. The method reduces training time compared to retraining GFlowNets from scratch and maintains comparable accuracy. \n\nIn conclusion, SB-GFlowNets enable streaming variational inference for discrete parameters, opening up new applications of Bayesian methods to large-scale streaming data problems.", "questions": "- Do you think the accumulative error issue can be solved by cache previous gradient (borrowing tricks from continual learning)?\n- It would be great if you can just compare with some continual learning method as baselines.\n- In sparse target distributions, if not using KL-based training scheme, what are the possible schemes?", "limitations": "The authors have adequately addressed the limitations", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "- SB-GFlowNets significantly reduce training time compared to retraining GFlowNets from scratch for each new data batch. This is particularly valuable for large-scale streaming data problems.\n- The paper provides a theoretical analysis of how errors propagate through posterior updates\n- The method is demonstrated to work well on various tasks (set generation, Bayesian linear preference learning, and online phylogenetic inference)", "weaknesses": "- As mentioned in the paper's limitations section, inappropriate approximations to earlier posteriors may propagate through time, potentially leading to increasingly inaccurate models. This is a form of catastrophic forgetting\n- Due to the error accumulation issue, there may sometimes need to retrain the model from an earlier checkpoint or using the full posterior\n- As the paper mentioned, for very sparse target distributions, the performance can degrade, particularly when using the KL-based training scheme", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for thoughtful feedback. We hope our clarifications solve your concerns and elevate your appraisal of our work. Otherwise, we will gladly engage further during the discussion period.\n\n> Do you think the accumulative error issue can be solved by cache previous gradient (borrowing tricks from continual learning)?\n\nThanks for the question. The accumulative error issue as described in the paper relates to the issue of the past GFlowNets ($t^\\prime < t+1$) not being trained until they reach global optima, affecting the quality of the current GFN at time ($t+1$). As we briefly discussed in lines 196-198, our suggestion is to reuse a previous checkpoint at time ($S$)  which has better training loss and use all of the data after that point ($t +1 > t^\\prime > S$) to train a GFN for time $t^\\prime$ until $t+1$.\n\nThe closest parallel to this strategy in continual learning is _replay_. However, replay typically entails using data from past tasks/batches to avoid catastrophic forgetting. We would like to emphasize that there are fundamental differences between catastrophic forgetting and accumulative errors. The most straightforward is that forgetting may happen in continual learning even if we reach local optima consecutively for all data batches. In our case,  we show in the paper when the models converge in our proposed losses, there is no issue with accumulative error. \n\nThat being said, including arbitrary terms in our SB-GFlowNets generally entails losing our correctness guarantees. This is the case for replay terms, but also for regularization terms like the elastic weight consolidation. Nonetheless, it is interesting to note both our divergence-based loss in Eq. 5 can be written as \n\n$$\n\\mathbb{E}\\_{p\\_F^{(t+1)}}\\left[ - \\log \\pi\\_{t+1}(x) \\right] + \\mathcal{D}\\_{\\text{KL}}\\left[p\\_{F}^{(t+1)} \\| p\\_{F}^{(t)}\\right],\n$$\n\nwhere the KL term also penalizes forward policies that deviate from that of the previous time step.\n\nWe hope this clarification addresses your question. Otherwise, we would be really happy if you could elaborate further on possible connections with continual learning that we could explore during the discussion period.\n\n\n> It would be great if you can just compare with some continual learning method as baselines.\n\nTo the best of our knowledge, there is no continual learning method capable of learning discrete posteriors over discrete random variables. If you believe we are missing any relevant piece of work, we would gladly test against them and report our results during the discussion period and add it to the final manuscript.\n\n\n> In sparse target distributions, if not using KL-based training scheme, what are the possible schemes?\n\nAs discussed in the paper, the issue with sparse target distributions is that GFlowNets trained with KL-divergence based criteria may fail to represent all modes of the final posterior distribution; we refer to this phenomenon as mode collapse in the paper. An alternative scheme that circumvents this issue (and is another contribution of our work) is to minimize the streaming balance loss (Eq. 4) in an off-policy fashion. In summary, if the target distribution is not sparse, we expect the KL-based training to converge faster, but, on the other hand, streaming balance condition based training can be applied to sparse target distributions.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RZpKeAifsY", "reviewer": "Reviewer_ZbQa", "summary": "This paper introduces Streaming Bayes GFlowNets (SB-GFlowNets), which enables streaming Bayesian inference over discrete parameter spaces, relying on the expressive power of GFlowNets as amortized samplers over discrete compositional objects. The process is akin to Bayesian streaming, where the posterior updates with each new data stream and the posterior from the batch before acts as a prior for the current batch. The authors use a GFlowNet to fit the initial posterior and update it training only on new data stream batches. To do so, the authors propose two different solutions: enforcing a streaming balance condition at each streaming step, or relying on direct divergence-based updates (letting go of the estimation of the normalizing constant). For the latter, the authors show how to obtain low-variance estimators for the gradient of the proposed KL divergence loss. Finally, the authors provide a theoretical performance analysis of their two proposed variants along with a diverse set of experiments to showcase the performance in each case, and show the gain in training time in the streaming setting compared to retraining for all the data at once at each step.", "questions": "See weaknesses.", "limitations": "- The authors adequately addressed limitations throughout the paper (for instance posterior error propagation etc) and through a separate section in the conclusion.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The authors provide a complete study of all the important components within their work (experiments, runtime comparison to full retraining and practical performance enhancement etc). \n- The idea is interesting across many applications where data streams are processed continuously. \n- The paper is overall well written and easy to follow.", "weaknesses": "- I'm not sure where equation (9) comes from. Equating \\$\\mathcal{L}\\_{SB}(\\tau)\\$ to 0 would result in\n\\\\[\np\\_F^{(t+1)}(\\tau) = \\frac{Z\\_t}{Z\\_{t+1}} \\cdot \\frac{p\\_B^{(t+1)}(\\tau \\vert x)}{p\\_B^{(t)}(\\tau \\vert x)} \\cdot p\\_F^{(t)}(\\tau) \\cdot f(\\mathcal{D}\\_{t+1} \\vert x)\n\\\\]\nEven by assuming \\$(p\\_F^t, p\\_B^t, Z\\_t)\\$ satisfy TB \\$(\\mathcal{L}\\_{TB} \\rightarrow 0)\\$, I'm not sure how that would directly yield equation (10). The proofs for proposition 2 also seem to start from (9). Can you elaborate on that? \n\n- To what extent is the assumption that new data is independent of past data under the posterior predictive? This question is especially important in cases where you try to sample from the marginal likelihood over some object (for instance for structure learning), where you lose that independence under the posterior given only the structure (and not parameters for example). I suspect this might be the reason for performance degradation across streaming steps in the Bayesian phylogenetic inference case, given that Felsenstein's algorithm assigns a marginal likelihood. \n\n**Minor Comments** \n- Line 30, $\\{y_i\\}_{i=1}^2$ should be defined/introduced better.\n- Line 134, space after comma. \n- In line 142, in the second TB condition, it should be $p_F^{(t+1)}(\\tau)$ instead of $p_F^{(t)}(\\tau)$.\n- $p_F^{(t+1)}$ should be evaluated at $\\tau$ in equation (10).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for carefully reviewing our work. We have corrected all the typos you pointed out. \nWe hope our clarifications address your concerns and enhance your view of our work.\n\n\n> I'm not sure where equation (9) comes from ...\n\nThank you for catching this typo, some of the indices were written incorrectly but the result shown in the paper is correct. We have fixed it and added additional clarifications to the paper that we will now explain here.\n\nIndeed, as you’ve written (9) should have $\\frac{Z_t}{Z_{t+1}}$ instead of the written $\\frac{Z_{t+1}}{Z_{t}}$ and on (10) the left hand side should be $\\frac{p_{F}^{t + 1}(\\tau)}{p_{B}^{t + 1}(\\tau | x)}$.\n\nThen, by assuming that the balance condition is satisfied, we deduce that $Z_t\\cdot p_F^t(\\tau) = p_B^t(\\tau|x)\\pi_t(x)$, in other words, $\\frac{p_F^t(\\tau)}{p_B^t(\\tau|x)} = \\frac{\\pi_t(x)}{Z_t}$.\n\nBy doing this substitution on the corrected eq (9), we obtain the corrected eq (10): $$\\frac{p_{F}^{t + 1}(\\tau)}{p_{B}^{t + 1}(\\tau | x)} = Z_{t + 1} \\pi_{t}(x) f(\\mathcal{D}_{t + 1} | x).$$\n\nNow, as defined in the paper:\n\n$$p\\_{\\intercal}^{t + 1}(x)= \\mathbb{E}\\_{\\tau \\sim p\\_{B}^{t + 1}(\\cdot | x)}\\left[\\frac{p\\_{F}^{t + 1}(\\tau)}{p\\_{B}^{t + 1}(\\tau | x)}\\right],$$ which allows us to conclude, as written in the submitted version, $ p_{\\intercal}^{t + 1}(x) \\propto \\pi_{t + 1}(x)$.\n\nAs for the proof of proposition 2, we fix typos and added extra comments to improve the clarity of presentation. Once again we have written $\\frac{Z_{t+1}}{Z_{t}}$ instead of the correct $\\frac{Z_t}{Z_{t+1}}$. Additionally, this proposition does not use either eq (9) or (10) in the proof. Due to a typo, the equation in lines 532-533 looks similar to eq (9), however, the corrected version, $p\\_{\\intercal}^{(t + 1), \\star}(x) = \\frac{Z\\_{t + 1}}{Z\\_{t}} p\\_{\\intercal}^{t}(x) f(\\mathcal{D}\\_{ t + 1 } | x)$ is not related due to the fact that it does not refer to either $p_{F}^{t}(\\tau)$ or $p_{F}^{t+1}(\\tau)$ but to $p_{\\intercal}^{t}(x)$ and $p_{\\intercal}^{t+1}(x)$. Once again, we note that these minor corrections do not change the proof significantly and our conclusions still hold.\n\n> To what extent is the assumption that new data is independent of past data under the posterior predictive? This question is especially important in cases where you try to sample from the marginal likelihood over some object (for instance for structure learning), where you lose that independence under the posterior given only the structure (and not parameters for example). I suspect this might be the reason for performance degradation across streaming steps in the Bayesian phylogenetic inference case, given that Felsenstein's algorithm assigns a marginal likelihood.\n\nFirst, we would like to clarify the assumptions of our model. In the joint distribution, we only require that each data observation is conditionally independent given the parameters, i.e. the variable we want to compute the posterior on. In other words, given a joint distribution:\n$$\n\tp(\\theta, \\mathcal{D}\\_1, \\mathcal{D}\\_2, \\ldots, \\mathcal{D}\\_4) =\n\tp(\\theta) \\prod\\_{i=1}^4 p(\\mathcal{D}\\_i | \\theta),\n$$\nthis only assumes that, given $\\theta$, there is independence between the data $\\mathcal{D}_i$, however, if we marginalize $\\theta$ out of this distribution, the data distribution would be correlated. The fact that the marginal distribution is correlated is not an issue for our method. \n\nIn the Bayesian phylogenetics case, the $D_i$ comprises the i-th nucleobase for all biological species. We understand that our description in lines 302-304 might be misleading, since $D_1,\\ldots, D_M$ are independent given the tree structure $T$ but  $S_1, \\ldots, S_N$ are not independent. We will clarify this in the revised manuscript.\n\nPlease let us know if further clarifications are required.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "n6n0jgDkP6", "reviewer": "Reviewer_c2ea", "summary": "The authors provide a framework that allows training of GFlowNet models with streaming data by checkpointing a previously trained GFlowNet model and proposing the streaming balance condition. The problem as well as the approach is well motivated and theoretically sound and this work would be a valuable contribution to the research community. The authors conduct experiments on synthetic tasks and phylogenetic inference, and show improved scalability. Additionally, there is some theory that describes and bounds the approximation error based on errors on the checkpointed GFlowNet model and current estimation problem. Overall, I think the work is quite interesting but it would be nice to consider another experiment since currently most of the experiments are based on synthetic data.", "questions": "- It is not clear what do the authors mean by $\\mathbf{A}$ in their notation at the start of Section 2.\n- The authors seem to have a typo in the equation under Line 142, essentially the right hand equation should have $p_F^{t+1}$.\n- Can the authors provide a visualization of the Set generation task? If I understand correctly, the authors first sample $d$ points randomly and the corresponding $f_i(j)$ randomly as well where $i = 1, ..., K+1$ and $j = 1, ..., d$. Then they define $R_i$ over each set, and a tempered version of $R_i$ with temperature $\\alpha$. Is that it?", "limitations": "The authors have adequately talked about the limitations and impacts of their work.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The authors provide useful theoretical contributions that bound the approximation errors into two terms: error from solving the current distribution matching problem and error coming from suboptimal solution to the previous distribution matching problem.\n- The problem of training sampling methods that can incorporate streaming data is quite important and relevant to the field, especially for Bayesian Inference methodologies.\n- The authors do provide a good set of experiments, and show the benefits of their proposed approach over the standard training of GFlowNets.", "weaknesses": "- I found Definition 2 to be quite confusing to read and it is still not clear to me how the equivalence holds. Could the authors clarify what the KL divergence is between, why is it valid and how does it circumvent the problem of parameterizing a partition function?\n- The original setup proposed by the authors follows from a trivial extension of the GFlowNets framework which begs to question the novelty of the work.\n- It is unclear how their formulation solves the problem of permutation invariance / iid treatment of observations. In particular, given two sets $D_1$ and $D_2$, we know that the posterior distribution $p(x | D_1, D_2)$ can be equivalently written as\n$$\np(x | D_1, D_2) = \\frac{p(D_1, D_2 | x) p(x)}{p(D_1, D_2)} = \\frac{p(D_1 | x) p(D_2 | x) p(x)}{p(D_1, D_2)} = \\frac{p(D_1 | x) p(x|D_2)}{p(D_1 | D_2)} = \\frac{p(D_2 | x) p(x|D_1)}{p(D_2 | D_1)}\n$$\nThen, in particular, how is it maintained that given any split of the data $D$ into $D_1$ and $D_2$, the final GFN trained leads to the same solution. If the GFN learns the right solution, then it will follow simply, but it would be nice if the model could satisfy this permutation invariance inherently. Could the authors provide some ablations and control experiments to show to what extent is this constraint satisfied over $t$?\n- The authors should add an experiment on discovering causal graphs conditioned on observational samples, which would strengthen their paper considerably.\n- The authors should consider the following related work, which are either connected to GFlowNets, sampling from an unnormalized density, or modeling Bayesian posterior inference:\n\nCranmer, Kyle, Johann Brehmer, and Gilles Louppe. \"The frontier of simulation-based inference.\" Proceedings of the National Academy of Sciences 117.48 (2020): 30055-30062.\n\nMittal, Sarthak, et al. \"Exploring Exchangeable Dataset Amortization for Bayesian Posterior Inference.\" ICML 2023 Workshop on Structured Probabilistic Inference {\\&} Generative Modeling. 2023.\n\nZhang, Qinsheng, and Yongxin Chen. \"Path integral sampler: a stochastic control approach for sampling.\" arXiv preprint arXiv:2111.15141 (2021).\n\nRichter, Lorenz, et al. \"VarGrad: a low-variance gradient estimator for variational inference.\" Advances in Neural Information Processing Systems 33 (2020): 13481-13492.\n\nSendera, Marcin, et al. \"On diffusion models for amortized inference: Benchmarking and improving stochastic control and sampling.\" arXiv preprint arXiv:2402.05098 (2024).\n\nBerner, Julius, Lorenz Richter, and Karen Ullrich. \"An optimal control perspective on diffusion-based generative modeling.\" arXiv preprint arXiv:2211.01364 (2022).\n\nAkhound-Sadegh, Tara, et al. \"Iterated denoising energy matching for sampling from Boltzmann densities.\" arXiv preprint arXiv:2402.06121 (2024).\n\nRichter, Lorenz, Julius Berner, and Guan-Horng Liu. \"Improved sampling via learned diffusions.\" arXiv preprint arXiv:2307.01198 (2023).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the suggestion of additional experiments and references, which we will include in the revised manuscript. We hope our additional experiments and clarifications enhance your perception of our work. Please let us know if we have left blank spots or if further clarifications are needed.  \n\n> I found Definition 2 to be quite confusing to read and it is still not clear to me how the equivalence holds. Could the authors clarify what the KL divergence is between, why is it valid and how does it circumvent the problem of parameterizing a partition function?\n\nThank you for the opportunity to clarify Definition 2. In fact, there was a typo. The denominator in Eq. 5 should be $p\\_{F}^{t}(\\tau) f(\\mathcal{D}\\_{t+1}|x)$ instead of $p\\_{F}^{t}(\\tau) \\pi\\_{t+1}(x)$. That being said, Eq. 5 is the KL divergence between the trajectory-level distribution we are learning $p\\_{F}^{t + 1}(\\tau)$ and that of the previous timestep $p\\_{F}^{t}(\\tau)$ weighted by the likelihood $f(\\mathcal{D}\\_{t+1}|x)$ for batch $t+1$. The corrected version of eq. 5 is:\n\n$$\n\\mathcal{L}\\_{KL}(G\\_{t+1} ; G\\_{t}) = \\mathbb{E}\\_{\\tau \\sim p\\_{F}^{(t + 1)}} \\left[ \\log \\frac{p\\_{F}^{(t + 1)}(\\tau)}{p\\_{F}^{(t)}(\\tau) f(\\mathcal{D}\\_{t+1}|x) }\\right] \\overset{C}{=} \\mathcal{D}\\_{KL} \\left[ p\\_{F}^{(t + 1)}(\\tau) || p(\\tau) \\right], \n$$\n\nwhere $p(\\tau) \\propto p\\_{F}^{t}(\\tau) f(\\mathcal{D}\\_{t+1}|x)$. We hope this correction clarifies definition 2. Otherwise, please let us know. \n\n\n> The original setup proposed by the authors follows from a trivial extension of the GFlowNets framework which begs to question the novelty of the work.\n\nWe believe our main contribution is proposing and analyzing the first general-purpose method for streaming approximate Bayesian inference for discrete distributions, including distributions with highly structured supports (e.g., graphs). Additionally, as far as we know, this is the first work explicitly addressing the problem of training GFlowNets in a dynamically evolving environment. \n\n> It is unclear how their formulation solves the problem of permutation invariance / iid treatment of observations\n\nThanks for the interesting question. As you mentioned, the permutation invariance is clearly satisfied if we perfectly minimize $\\mathcal{L}\\_{SB}$ (Eq. 4) or $\\mathcal{L}\\_{KL}$ (Eq. 5) at each time step. However, when this is not the case, the learned distribution is not necessarily invariant to the order in which the datasets $D_{1}, \\ldots, D\\_{t}$ are observed. To illustrate this, we again examine the problem of phylogenetic inference for two different scenarios highlighted in Fig. 1a of the rebuttal PDF. In both scenarios, we consider that two datasets, $D_{1}$ and $D_{2}$, are observed in sequence and compare the distribution of an SB-GFlowNet trained on the tuples $(D_{1}, D_{2})$ and $(D_{2}, D_{1})$. Fig. 1a (left panel) shows the resulting distributions are not necessarily identical when the GFlowNets at $t = 1$ are not sufficiently well-trained. In contrast, Fig. 1a (right panel) highlights that both distributions are approximately the same when the GFlowNets at $t = 1$ and $t = 2$ are adequately trained. A similar observation holds for the set generation task in Fig. 1b. To the best of our knowledge, this sensibility to the observations’ ordering due to imperfectly approximated posteriors is inherent to any posterior propagation scheme (e.g., streaming VI). We will add this discussion and these experiments to the revised manuscript. \n\n> authors should add an experiment on discovering causal graphs conditioned on observational samples, which would strengthen their paper considerably.\n\nThank you for the suggestion. We adopted Deleu et al.’s DAG-GFlowNet [1] to address the causal discovery problem with GFlowNets and followed the experimental setup for Fig. 3 of the same paper. Six novel batches of 200 samples were observed --- one at a time for each streaming update. Notably, Fig. 2 of the rebuttal PDF shows SB-GFlowNet accurately matches the target distribution on each streaming update. Similarly, Fig. 3 highlights that the probability of the true DAG characterizing the distribution of the observed data tends to increase along training. We will include these additional experiments in the revised manuscript. \n\n[1] Bayesian Structure Learning with GFlowNets. Deleu et al. UAI 2022\n\n> authors should consider the following related work, which are either connected to GFlowNets, sampling from an unnormalized density, or modeling Bayesian posterior inference\n\nThank you for the recommendations. We will properly include all related works in the revised manuscript. \n\n> It is not clear what do the authors mean by $𝐴$ in their notation at the start of Section 2.\n\nIndeed, we missed including a definition $A$, which represents the adjacency matrix of the DAG $\\mathcal{G}$. We clarify this in the revised manuscript. \n\n> The authors seem to have a typo in the equation under Line 142, essentially the right hand equation should have $p\\_{F}^{t + 1}$ \n\nThanks for catching this typo. It should in fact be $p\\_{F}^{t +  1}$ instead of $p\\_{F}^{t}$. We will update the manuscript accordingly. \n\n> Can the authors provide a visualization of the Set generation task? \n\nWe have included an illustration of the set generation task in Fig. 4 of the rebuttal PDF. This task consists of learning to sample from a distribution over sets by iteratively adding elements to an initially empty set. Regarding the definition of each reward $R\\_{i}$, it is exactly as you described. Each $f\\_{i}(j)$ is independently sampled and fixed before training and, for a set $S$, we let $\\log R_{i}(S) = \\sum\\_{j \\in S} f\\_{i}(S)$ be the unnormalized log-probability of $S$. The additional presence of $\\alpha$ is to evaluate the impact of $R_{i}$’s sparsity on the efficacy of the learning objectives — we repeated the streaming experiment applying different temperatures $\\alpha$ to the rewards $R_{i}$ (Table 1), i.e., using $R_{i}^{1/\\alpha}$.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ORrwgcjuiU", "reviewer": "Reviewer_bMoM", "summary": "Generative flow nets are extended to streaming inference by using the prediction of the previous network as the prior for the current one. Two training methods are proposed: streaming balance which uses squared difference of target and learned log posteriors, and VI which uses KL divergence. Analytic results for both methods bound the error in terms of updating error and error from the previous step. Three experiments show the method can accurately track the true posterior across updating steps.", "questions": "(2): why do we need to sample from $p_B$ and importance weight, rather than sampling directly from $p_F$?\n\n(5),170,178: $\\pi_{t+1}$ should be $f(\\mathcal{D}_{t+1}|x)$, yes? The derivations all look correct if so.\n\n181: what is $\\delta$?\n\n$Z_{(t+1)}$ at 205 and $\\hat{Z}_{t+1}$ are the same?\n\nMore of a suggestion: could the method be extended to cases where the target objects grow at each time step? For example, phylogenetic trees would grow as new species are observed. It seems like a natural application of flow nets since you can expand the graph by appending a new terminal layer.", "limitations": "The set generation task is degenerate because the objective is additive. A model only needs to track the posterior on individual $x$s.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The methods are elegant and sound and the experiment results are strong. This is an important alternative to streaming VI that I'm excited to try out myself.", "weaknesses": "Not a major weakness, but the extension to the streaming case (this paper's main contribution) is fairly straightforward. It was clear from the beginning that the predicted distribution from the previous step would provide the prior for the next, with the obvious adaptations of the balance and VI objectives.\n\nBoth $p_B$ terms in the streaming balance condition (3) appear to be unnecessary. Prop 1 will hold without them. We just need a joint distribution over $(x,\\tau)$ (equivalently, a distribution over $\\tau$) with the desired marginal on $x$, and $f(\\mathcal{D}_{t+1}|x) p^t_F(\\tau)$ already has that property. Compare to the definition of the target distribution $p(\\tau)$ in (5).\n\nSec 3.2 just gives the objective and doesn’t explain how $p_F$ and $Z$ are learned (likewise with the argmax lines in Algo 1). Is it just SGD on $\\mathcal{L}_{SB}$?\n\nThe main text says nothing about how to construct the graph ($\\mathcal{S}$ and $\\boldsymbol{A}$). Details are given in the appendix -- in all cases the allowable paths build the desired objects from simpler ones -- but a sentence in the main text would help.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for carefully reading and appreciating our work. Below, we address each of your comments. \n\n> Both $𝑝\\_{𝐵}$ terms in the streaming balance condition (3) appear to be unnecessary. \n\nIndeed, when $p_{B}^{t}(\\cdot | s)$ is fixed, it does not depend on $t$ and the terms corresponding to the backward policy in Equation (3) cancel out. However, albeit relying on a fixed $p_{B}$ is an often implemented strategy for learning GFlowNets — that we adopt in our work —, $p_{B}$ can in principle be jointly learned with $p_{F}$. In this case, $p_{B}^{t}$ would also depend on $t$ and there would be no cancellations in Equation (3). Nonetheless, due to the lack of clear empirical evidence in the literature supporting the learning of $p_{B}$, we choose to fix it in all our experiments. We will emphasize this point and include the simplified version of Equation (3) in the revised manuscript. \n\n> Sec 3.2 just gives the objective and doesn’t explain how $𝑝\\_{𝐹}$ and $𝑍$ are learned\n\nThanks for noticing. You are correct: both $p_{F}$ and $Z$ in Section 3.2 and in Algorithm 1 are learned by minimizing the corresponding objectives via SGD. We will point this out in the updated manuscript. \n\n> The main text says nothing about how to construct the graph ($𝑆$ and $𝐴$). Details are given in the appendix -- in all cases the allowable paths build the desired objects from simpler ones -- but a sentence in the main text would help.\n\nThank you for the suggestion. Indeed, we briefly discuss this in the lines 81-84 of section 2 preliminaries and in more detail in the appendix, but we will make this more explicit in the revised manuscript.\nThe design of the graph ($S,A$) is application-dependent and should be considered on a problem-by-problem basis. As a general guideline, when the elements of the target’s support $\\mathcal{X}$ can be described as being built from simpler partial objects, then, the initial state can correspond to an empty object and the non-terminal states would be partial compositions with an edge from $s$ to $s’$ meaning that $s’$ differs from $s$ by a single additional atomic component. For example, a phylogeny is a rooted tree where all the leaves correspond to the species observed and the internal nodes correspond to unobserved ancestors. The states in the support are single trees and the non-terminal states are forests, i.e. disconnected collections of trees, where the construction follows by joining two trees in the forest by an unobserved node. In other words, the state graph is implicitly defined by iteratively applying actions to the initial state.\n\n\n> (2): why do we need to sample from $p_{B}$\n\nIn fact, we could directly sample from $p_{F}$ and count the relative frequency of $x$ to estimate the marginal $p_{T}(x)$. However, our experience suggests that the estimator in Equation (2), which is also used by Malkin et al. [1] and Zhout et al. [2], generally yields better estimates of $p_{T}(x)$. \n\n> (5), 170, 178:  $\\pi_{𝑡+1}$ should be $𝑓(\\mathcal{𝐷}_{𝑡+1}|𝑥)$, yes?\n\nYes! Thank you for the observation. We will fix this in the revised manuscript.\n\n> 181: what is $\\delta$?\n\nThis is a typo; we meant $\\nabla\\_{\\theta} \\mathbb{E}\\_{\\tau \\sim p_{F}^{t + 1}}[\\gamma(\\tau)]$ instead of $\\mathbb{E}[\\delta(\\tau)]$. \n\n> $𝑍_{(𝑡+1)}$ at 205 and $\\hat{𝑍}_{𝑡+1}$ are the same?\n\nYes! This too is a typo. \n\n> More of a suggestion: could the method be extended to cases where the target objects grow at each time step?\n\nThank you for the question. We do not believe that SB-GFlowNets can be immediately extended to accommodate the training of GFlowNets when the size of the generated objects grows. There are two main reasons for this. Firstly, the neural network that encodes the policy often receives a fixed-size input corresponding to a representation of the current state. When the size of this representation changes, it is unclear how the neural network should be updated. However, for many applications, this could be circumvented by using GNN- or DeepSet-based policy networks, which work well with inputs of varying size. Secondly, it is not totally clear what would happen with the terminal flows when additional nodes/transitions are added to the flow network. Which parts of the network remain balanced upon this addition? Should we learn a novel policy on the expanded network, or can we exploit the knowledge of past models to alleviate the learning problem? These questions are both important and interesting, deserving a work of their own. \n\n[1] GFlowNets and Variational Inference. Malkin et al. ICLR 2023. \n\n[2] PhyloGFN: Phylogenetic inference with generative flow networks. Zhou et al. ICLR 2024", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Dear reviewers and AC,

We are glad to know reviewers found our work to be valuable contribution to the research community (c2ea), our method to be elegant (bMoM) and our experiment results to be  strong (bMoM), diverse (ZbQa), and convincing (xCCn) — opening a path for new applications of Bayesian methods to large-scale streaming data problem (w9BG).

We are grateful for the reviewers' constructive feedback and suggestions to improve our work. We will incorporate all clarifications requested by the reviewers in the revised manuscript. At the request of reviewer c2ea, we also have run additional experiments in causal discovery.

Best regards,

Authors

Meta-review: This paper considers the setting of streaming Bayesian inference with discrete state spaces. A streaming Bayesian GFlownet (SB-GFlowNet) framework is introduced that uses a GFlowNet and sequentially updates it using new data. Theoretical analysis is provided to understand error propagation in the updates. Empirical studies are performed using SB-GFlowNets for problems involving set generation, linear preference learning, and phylogentic inference. 

Overall, reviewers found that the contributions of the paper were novel and that the experimental results were convincing. Several reviewers commented on clarity-related issues, such as notation, definitions, conditions used for the proposition(s), and typos. During the rebuttal phase, these concerns were mostly clarified and addressed, but these revisions still need to be corrected/added in the final manuscript. In particular, after the discussion phase, reviewers unanimously recommended accepting this paper. 

Thus, I am happy to recommend acceptance, and strongly encourage the authors to revise the manuscript according to the feedback from reviewers.
Decision: Accept (poster)